updated some comments

Makefile

@@ -1,24 +1,31 @@
-TEXBASE=susceptibility1
+TEXBASE=highbeats
 BIBFILE=references.bib
-REBUTTALBASE=
+REBUTTALBASE=rebuttal3
 
 TEXFILE=$(TEXBASE).tex
 PDFFILE=$(TEXBASE).pdf
 TXTFILE=$(TEXBASE).txt
 
-REVISION=
+REVISION=cb626892bd10b2908691bd95767f87c3d6a57dfe
+
+ALLPDFFIGURES=$(shell sed -n -e '/^[^%].*includegraphics/{s/^.*includegraphics.*{\([^}]*\)}.*/\1.pdf/;p}' $(TEXFILE))
+PDFFIGURES=$(filter-out toblerone_animated.pdf, $(ALLPDFFIGURES))
+
+PT=$(wildcard *.py)
+PYTHONFILES=$(filter-out plotstyle.py myfunctions.py numerical_compar_both.py, $(PT))
+PYTHONPDFFILES=$(PYTHONFILES:.py=.pdf)
 
 ifdef REBUTTALBASE
 REBUTTALTEXFILE=$(REBUTTALBASE).tex
 REBUTTALPDFFILE=$(REBUTTALBASE).pdf
 endif
-REBUTTALREVISION=
+REBUTTALREVISION=68900a1
 
 # all ###########################################################
 ifdef REBUTTALBASE
-all: bib rebuttalbib
+all: bib supplement rebuttalbib
 else
-all: bib
+all: bib supplement
 endif
 
 # python #########################################################
@@ -30,9 +37,9 @@ watchplots :
 	while true; do ! make -q plots && make plots; sleep 0.5; done
 
 
-# rescue_local_eod manuscript #################################################
+# manuscript #####################################################
 bib: $(TEXBASE).bbl
-$(TEXBASE).bbl: $(TEXFILE) $(BIBFILE)
+$(TEXBASE).bbl: $(TEXFILE) $(BIBFILE) $(PDFFIGURES)
 	lualatex $(TEXFILE)
 	bibtex $(TEXBASE)
 	lualatex $(TEXFILE)
@@ -43,12 +50,17 @@ $(TEXBASE).bbl: $(TEXFILE) $(BIBFILE)
 	@sed -n -e '1,/You.ve used/p' $(TEXBASE).blg
 
 pdf: $(PDFFILE)
-$(PDFFILE) : $(TEXFILE)
+$(PDFFILE) : $(TEXFILE) $(PDFFIGURES)
 	lualatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && lualatex -interaction=scrollmode $< || true
 
 again :
 	lualatex $(TEXFILE)
 
+supplement: supplement.pdf
+supplement.pdf : supplement.tex
+	lualatex $<
+	lualatex $<
+
 # watch files #######################################################
 watchpdf :
 	while true; do ! make -s -q pdf && make pdf; sleep 0.5; done
@@ -109,7 +121,7 @@ $(REBUTTALPDFFILE) : $(REBUTTALTEXFILE)
 	lualatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && lualatex -interaction=scrollmode $< || true
 
 watchrebuttal :
-	while true; do ! make -q rebuttal && make rebuttal; sleep 0.5; done
+	while true; do ! make -s -q rebuttal && make rebuttal; sleep 0.5; done
 
 rebuttaldiff :
 	latexdiff-git -r $(REBUTTALREVISION) --append-textcmd="response,issue" --pdf $(REBUTTALTEXFILE)
@@ -167,7 +179,7 @@ help :
 	@echo -e \
 	"make pdf:           make the pdf file of the paper.\n"\
 	"make bib:           run bibtex and make the pdf file of the paper.\n"\
-	"make again:         run pdflatex and make the pdf file of the paper,\n"\
+	"make again:         run lualatex and make the pdf file of the paper,\n"\
         "                   no matter whether you changed the .tex file or not.\n"\
 	"make watchpdf:      make the pdf file of the paper\n"\
         "                   whenever the tex file is modified.\n"\

cells_suscept.py

@@ -1,4 +1,4 @@
-from utils_suseptibility import ampullary_punit
+from utils_suseptibility import ampullary_punit, fr_name_rm
 from utils_all import p_units_to_show
 #from utils_all import load_folder_name
 #from plt_RAM import plt_punit
@@ -9,4 +9,4 @@ if __name__ == '__main__':
 
     #stack_file = pd.read_csv('..\calc_RAM\calc_nix_RAM-eod_2022-01-28-ag-invivo-1_all__amp_20.0_filename_InputArr_400hz_30_P-unitApteronotusleptorhynchus.csv')
     cells_plot2 = p_units_to_show(type_here = 'contrasts')#permuted = True,
-    ampullary_punit(eod_metrice = False, color_same = False,base_extra = True, fr_name = '$f_{base}$',   cells_plot2=[cells_plot2[0]], isi_delta = 5, titles=[''],tags_individual = True, xlim_p = [0,1.15])#Low-CV P-unit,
+    ampullary_punit(eod_metrice = False, color_same = False,base_extra = True, fr_name = fr_name_rm(),   cells_plot2=[cells_plot2[0]], isi_delta = 5, titles=[''],tags_individual = True, xlim_p = [0,1.15])#Low-CV P-unit,

cells_suscept_high_CV.py

@@ -1,4 +1,4 @@
-from utils_suseptibility import ampullary_punit
+from utils_suseptibility import ampullary_punit, fr_name_rm
 from utils_all import p_units_to_show
 #from utils_all import load_folder_name
 #from plt_RAM import plt_punit
@@ -7,4 +7,4 @@ from IPython import embed
 if __name__ == '__main__':
 
     cells_plot2 = p_units_to_show(type_here='contrasts')#permuted = True,
-    ampullary_punit(base_extra = True,eod_metrice = False, color_same = False, fr_name = '$f_{base}$',  tags_individual = True, isi_delta = 5, cells_plot2=[cells_plot2[1]], titles=['', 'Ampullary cell,'],)#High-CV P-unit,
+    ampullary_punit(base_extra = True,eod_metrice = False, color_same = False, fr_name = fr_name_rm(),  tags_individual = True, isi_delta = 5, cells_plot2=[cells_plot2[1]], titles=['', 'Ampullary cell,'],)#High-CV P-unit,

data_overview_mod.py

@@ -8,7 +8,7 @@ from utils_all_down import default_settings
 from utils_suseptibility import colors_overview
 from utils_suseptibility import default_figsize, NLI_scorename2,pearson_label, exclude_nans_for_corr, kernel_scatter, \
     scatter_with_marginals_colorcoded, \
-    version_final
+    version_final, basename, stimname
 from utils_all import update_cell_names, load_overview_susept, make_log_ticks, p_units_to_show, save_visualization, setting_overview_score
 from scipy import stats
 try:
@@ -128,7 +128,7 @@ def data_overview3():
         var_item_names = [var_it,var_it,var_it2]#,var_it2]#['Response Modulation [Hz]',]
         var_types = ['response_modulation','response_modulation','']#,'']#'response_modulation'
         max_x = max_xs[c]
-        x_axis_names = ['CV$_{Base}$','CV$_{stim}$','Response Modulation [Hz]']#$_{Base}$,'Fr$_{Base}$',]
+        x_axis_names = ['CV$'+basename()+'$','CV$'+stimname()+'$','Response Modulation [Hz]']#$'+basename()+'$,'Fr$'+basename()+'$',]
         #score = scores[0]
         score_n = ['Perc99/Med', 'Perc99/Med', 'Perc99/Med']
         score = scores[c]

elsarticle/doc/makefile

@@ -37,6 +37,11 @@ copy:
 	cp $(file).sty ../
 
 
+# python #########################################################
+plots: $(PYTHONPDFFILES)
+$(PYTHONPDFFILES): %.pdf: %.py plotstyle.py
+	python3 $<
+
 
 
 

flowchart__spikes_c_sig_0_a_fe_0.02.csv

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model_and_data.py

@@ -106,7 +106,7 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
             # ax.text(1.66, 0.5, nonlin_title(), rotation=90, ha='center', va='center',
             #        transform=ax.transAxes)
 
-            ax_external.arrow_spines('lb')
+            #ax_external.arrow_spines('lb')
 
         #embed()
         #plt.show()
@@ -448,6 +448,7 @@ if __name__ == '__main__':
 
     ##########################
     #embed()
+    print('hi')
     model_and_data2(eod_metrice = False, width=0.005, show=show, D_extraction_method=D_extraction_method,
                    label=r'$\frac{1}{mV^2S}$')  #r'$\frac{1}{mV^2S}$'
 

model_full.py

@@ -1,7 +1,7 @@
 from utils_suseptibility import *
 
-def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='all', chose_score='mean_nrs',
-                         detections=['MeanTrialsIndexPhaseSort'], sorted_on='LocalReconst0.2NormAm',ylim = [-1.25, 1.25], dfs = ['m1', 'm2']):
+def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_score='mean_nrs',
+               detections=['MeanTrialsIndexPhaseSort'], sorted_on='LocalReconst0.2NormAm', dfs = ['m1', 'm2']):
 
     plot_style()
     default_figsize(column=2, length=2.3)
@@ -24,17 +24,12 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='al
     f1 = 33
     f2 = 139
     color01, color012, color01_2, color02, color0_burst, color0 = colors_suscept_paper_dots()
-    #embed()
-    #ax.plot(fr_noise * f1/fr_waves, fr_noise*f2/fr_waves, 'o', ms = 5, markeredgecolor = color012, markerfacecolor="None")
-    #ax.plot(-fr_noise * f1 / fr_waves, fr_noise * f2 / fr_waves, 'o', ms = 5, markeredgecolor=color01_2, markerfacecolor="None")
-
-    # if len(cbar) > 0:
     ###############################
     # data part
 
 
-    data_extra = False
-    if data_extra:
+    data_square_extra = False
+    if data_square_extra:
         ax = plt.subplot(grid[0])
         axes.append(ax)
         cell = '2012-07-03-ak-invivo-1'
@@ -98,32 +93,43 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='al
 
     markers = ['s', 'o']
     ms = 14
-    DF1_desired, DF2_desired, fr, eod_fr, arrays_len = plt_data_full_model(c1, chose_score, detections, devs, dfs, end, grid[3], mult_type, sorted_on, ms = ms, markers = ['s', 'o'],clip_on = True, DF2_desired = DF2_desired_orig, DF1_desired = DF1_desired_orig, alpha = [1,1], log = log, ylim_log  = ylim_log, nfft = nfft, xlim_psd = xlim_psd)
+    data_extra = False
+    if data_extra:
+        DF1_desired, DF2_desired, fr, eod_fr, arrays_len = plt_data_full_model(c1, chose_score, detections, devs, dfs, end, grid[3], mult_type, sorted_on, ms = ms, markers = ['s', 'o'],clip_on = True, DF2_desired = DF2_desired_orig, DF1_desired = DF1_desired_orig, alpha = [1,1], log = log, ylim_log  = ylim_log, nfft = nfft, xlim_psd = xlim_psd)
+        fr_mult = fr / eod_fr
 
+        multwise = False
+        if multwise:
+            DF1_frmult = np.abs((np.array(DF1_desired)-1)/fr_mult)
+            DF2_frmult = np.abs((np.array(DF2_desired) - 1) / fr_mult)
+        else:
+            DF1_frmult = np.array(DF1_desired_orig)/fr
+            DF2_frmult = np.array(DF2_desired_orig) / fr
+        #embed()
+        DF1_frmult[0] = 1
+        DF2_frmult[1] = 0.32339256#1.06949369
+        print(DF1_frmult)
+        print(DF2_frmult)
+
+        grid0 = gridspec.GridSpecFromSubplotSpec(len(DF1_desired), 1, wspace=0.15, hspace=0.35,
+                                                 subplot_spec=grid[2])
+
+    else:
+        markers = ['o', 'o', 'o', 'o', ]
+        DF1_frmult = [0.28, 1.28, 1, 1]
+        DF2_frmult = [1-0.28, 0.28, 0.28, 0.48]  # 1.06949369
+
+        grid0 = gridspec.GridSpecFromSubplotSpec(2, 2, wspace=0.15, hspace=0.35,
+                                                 subplot_spec=grid[2:3])
 
     #################
     #################
     # power spectra model
 
 
-    grid0 = gridspec.GridSpecFromSubplotSpec(len(DF1_desired), 1, wspace=0.15, hspace=0.35,
-                                             subplot_spec=grid[2])
 
-    fr_mult = fr / eod_fr
 
-    multwise = False
-    if multwise:
-        DF1_frmult = np.abs((np.array(DF1_desired)-1)/fr_mult)
-        DF2_frmult = np.abs((np.array(DF2_desired) - 1) / fr_mult)
-    else:
-        DF1_frmult = np.array(DF1_desired_orig)/fr
-        DF2_frmult = np.array(DF2_desired_orig) / fr
-    #embed()
-    DF1_frmult[0] = 1
-    DF2_frmult[1] = 0.32339256#1.06949369
 
-    print(DF1_frmult)
-    print(DF2_frmult)
     #DF1_frmult[1] = 0.4
     #DF2_frmult[1] = 1.8
     #DF1_frmult[1] = 1.45
@@ -167,12 +173,11 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='al
 
     # ATTENTION: Diese Zelle ('2012-07-03-ak-invivo-1') braucht längere Abschnitte, mindsetesn 5 Sekunden damit das Powerspectrum nicht so niosy ist!
     #embed()dev_spikes='original',
-    fr = plt_model_full_model2(grid0,  reshuffled=reshuffled, datapoints=50, limit=10.2,
-                               reshuffle=reshuffled, dev=0.0005, a_f1s=[a_size], af_2 = a_size, array_len=[20,20],
-                               stimulus_length=length, way=way, plus_q=plus_q,
-                               diagonal=diagonal,  runs=1,nfft = nfft,xlim_psd =  xlim_psd, ylim_log = ylim_log,
-                               cells=[cell], dev_spikes = '05', markers = markers,DF1_frmult = DF1_frmult, DF2_frmult = DF2_frmult,
-                               show=True, log = log,ms = ms,  clip_on = True)  #arrays_len a_f1s=[0.02]"2012-12-13-an-invivo-1"'2013-01-08-aa-invivo-1'
+    fr, eod_fr_m = plt_model_full_model2(grid0, reshuffled=reshuffled, dev=0.0005, a_f1s=[a_size], af_2 = a_size,
+                               stimulus_length=length, plus_q=plus_q,
+                               diagonal=diagonal, runs=1, nfft = nfft, xlim_psd =  xlim_psd, ylim_log = ylim_log,
+                               cells=[cell], dev_spikes = '05', markers = markers, DF1_frmult = DF1_frmult, DF2_frmult = DF2_frmult,
+                               log = log, ms = ms, clip_on = True)  #arrays_len a_f1s=[0.02]"2012-12-13-an-invivo-1"'2013-01-08-aa-invivo-1'
 
 
 
@@ -185,16 +190,24 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='al
     #ax.plot(-fr_noise * f1 / fr_waves, fr_noise * f2 / fr_waves, 'o', ms=5, markeredgecolor=color01_2,
     #        markerfacecolor="None", alpha = 0.5)
 
-    DF2_hz = np.abs(np.array(DF1_desired) * eod_fr-eod_fr)
-    DF1_hz = np.abs(np.array(DF2_desired) * eod_fr - eod_fr)
 
 
-    for f in range(len(DF1_hz)):
-        ax.plot(fr_noise *  DF1_hz[f] / fr_waves, fr_noise * DF2_hz[f] / fr_waves, markers[f], ms=5, markeredgecolor=color012,
-                markerfacecolor="None")#, alpha = alphas[f]
-        ax.plot(-fr_noise *  DF1_hz[f] / fr_waves, fr_noise * DF2_hz[f] / fr_waves, markers[f], ms=5, markeredgecolor=color01_2,
-                markerfacecolor="None")#, alpha = alphas[f]
 
+    if data_extra:
+        DF2_hz = np.abs(np.array(DF1_desired) * eod_fr - eod_fr)
+        DF1_hz = np.abs(np.array(DF2_desired) * eod_fr - eod_fr)
+        for f in range(len(DF1_hz)):
+            ax.plot(fr_noise *  DF1_hz[f] / fr_waves, fr_noise * DF2_hz[f] / fr_waves, markers[f], ms=5, markeredgecolor=color012,
+                    markerfacecolor="None")#, alpha = alphas[f]
+            ax.plot(-fr_noise *  DF1_hz[f] / fr_waves, fr_noise * DF2_hz[f] / fr_waves, markers[f], ms=5, markeredgecolor=color01_2,
+                    markerfacecolor="None")#, alpha = alphas[f]
+    else:
+        #embed()
+        for f in range(len(DF1_frmult)):
+            ax.plot((fr_noise*DF1_frmult[f]), (fr_noise*DF2_frmult[f]-1), markers[f], ms=5, markeredgecolor=color012,
+                    markerfacecolor="None")#, alpha = alphas[f]
+            ax.plot(-(fr_noise*DF1_frmult[f]-1), (fr_noise*DF2_frmult[f]-1), markers[f], ms=5, markeredgecolor=color01_2,
+                    markerfacecolor="None")#, alpha = alphas[f]
 
     #embed()
     #tag2(fig=fig, xoffs=[-4.5, -4.5, -4.5, -5.5], yoffs=1.25)
@@ -205,10 +218,11 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='al
     save_visualization()
 
 
-def plt_model_full_model(axp, min=0.2, cells=[], a_f2 = 0.1, perc = 0.05,  alpha = 1, trials_nr = 15, add_pp=50,
-                                          single_waves=['_SingleWave_', '_SeveralWave_', ], cell_start=13,
-                                          zeros='zeros', several_peaks_nr = 2, a_f1s=[0, 0.005, 0.01, 0.05, 0.1, 0.2, ], a_frs=[1],
-                                          add_half=0, log = 'log', xlim = [0, 350], freqs_mult1 = None, freqs_mult2 = None, show=False, nfft=int(2 ** 15), beat='', gain=1, us_name=''):
+def plt_model_full_model(axp, min=0.2, cells=[], a_f2 = 0.1, perc = 0.05, alpha = 1, trials_nr = 15,
+                         single_waves=['_SingleWave_', '_SeveralWave_', ], cell_start=13,
+                         several_peaks_nr = 2, a_f1s=[0, 0.005, 0.01, 0.05, 0.1, 0.2, ], a_frs=[1],
+                         add_half=0, log = 'log', xlim = [0, 350], freqs_mult1 = None, freqs_mult2 = None,
+                         nfft=int(2 ** 15), gain=1, us_name=''):
     model_cells = pd.read_csv(load_folder_name('calc_model_core') + "/models_big_fit_d_right.csv")
     if len(cells) < 1:
         cells = model_cells.cell.loc[range(cell_start, len(model_cells))]
@@ -364,32 +378,17 @@ def plt_model_full_model(axp, min=0.2, cells=[], a_f2 = 0.1, perc = 0.05,  alpha
                                 freqs_beat = [fr, np.abs(beat1), np.abs(beat2), np.abs(np.abs(beat2) + np.abs(beat1)),np.abs(np.abs(beat2) - np.abs(beat1))
                                               ]  # np.abs(beat2 - beat1),np.abs(beat2 + beat1),
 
-                                #colors_w, colors_wo, color_base, color_01, color_02, color_012 = colors_cocktailparty_all()
                                 colors = [color0,color_01, color_02, color_012, color01_2]  # 'blue'
                                 labels = ['','', '', '', '']  # small , '|B1-B2|'
-                                #labels = ['$f_{1}=%d$' % beat1 + '\,Hz', '$f_{2}=%d$' % beat2 + '\,Hz',
-                                #          '$f_{1} + f_{2}=f_{Base}=%d$' % (
-                                #                 beat1 + beat2 - 1) + '\,Hz']  # small , '|B1-B2|'
                                 add_texts = [nr, nr + 0.35, nr + 0.2, nr + 0.2, nr + 0.2]  # [1.1,1.1,1.1]
                                 texts_left = [-7, -7, -7, -7,-7]
-                                # ax[1].set_title(
-                                #    '$f_{1}=%d$' % beat1 + '\,Hz' + ', ' + '$f_{2}=%d$' % beat2 + '\,Hz' + ', ' + '$ f_{Base}=%d$' % (
-                                #                beat1 + beat2 - 1) + '\,Hz')
-
                             else:
                                 freqs_beat = [np.abs(beat1), np.abs(beat1) * 2, np.abs(beat1 * 3),
                                               np.abs(beat1 * 4)]  # np.abs(beat1) / 2,
                                 colors = ['black', 'orange', 'blue', 'purple', 'black']  # 'grey',
-                                #colors = colors_didactic()
                                 add_texts = [nr + 0.1, nr + 0.1, nr + 0.1, nr + 0.1]  # [1.1,1.1,1.1,1.1]
                                 texts_left = [3, 0, 0, 0]
                                 labels = labels_didactic2()  # colors_didactic, labels_didactic
-                                # labels = ['S1', 'S2 / B1', 'S3', 'S4 / B2', 'f0']#'',
-
-                            if 'Several' in single_wave:
-                                color_beat = 'black'
-                            else:
-                                color_beat = colors[0]
                             if (np.mean(stimulus) != 0) & (np.mean(stimulus) != 1):
                                 # try:
                                 eod_interp, eod_norm = extract_am(stimulus, time_array, sampling=sampling_rate,
@@ -397,18 +396,11 @@ def plt_model_full_model(axp, min=0.2, cells=[], a_f2 = 0.1, perc = 0.05,  alpha
                                                                   emb=False,
                                                                   extract='', norm=False)
                             for l in range(len(spikes)):
-                                # ax[2, aa].scatter(spikes[l]*1000, np.ones(len(spikes[l]))*(l+1), color = 'grey', s = 5)
                                 spikes[l] = (spikes[l] - min) * 1000
-                                # ax[5, ff].set_xlim(0.1,0.2)
-
-
                             pp, f = ml.psd(mat05 - np.mean(mat05), Fs=1 / deltat, NFFT=nfft,
                                            noverlap=nfft // 2)
-                            ref = (np.max(pp))
-
                             if log:
                                 pp_mean = 10 * np.log10(pp_mean / np.max(pp_mean))
-                                #    pp_mean = np.log
                             print(freqs_beat)
                             print(labels)
                             plt_peaks_several(freqs_beat, pp_mean, axp, pp_mean, f, labels, 0, colors, ha='center',
@@ -432,16 +424,15 @@ def plt_model_full_model(axp, min=0.2, cells=[], a_f2 = 0.1, perc = 0.05,  alpha
                         axp.set_xlabel('Frequency [Hz]')
     return fr
 
-def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=1000, dev=0.0005, limit=10.2, a_f1s=[0.03],
-                          pdf=True, printing=False, plus_q='minus', freq1_ratio=1 / 2, diagonal='diagonal',
-                          freq2_ratio=2 / 3, way='absolut', stimulus_length=0.5, runs=1, trials_nr=500, cells=[],
-                          show=False, nfft=int(4096), beat='', nfft_for_morph=4096 * 4, gain=1,
-                          sampling_factors=[''],DF2_frmult = [],DF1_frmult = [],
-                          fish_receiver='Alepto', end_f1=4645,array_len = [],
-                          fish_emitter='Alepto', marker = 'o',xlim_psd =  [], ylim_log = [],
-                          fish_jammer='Alepto', reshuffle='reshuffled', markers = [], clip_on = True,
-                          redo_level='celllevel',ms = 14,  step=10, zeros='zeros', corr='ratecorrrisidual',
-                          us_name='', dev_spikes='original', start_f1=20, log = '',plot=False):
+def plt_model_full_model2(grid0, reshuffled='reshuffled', af_2 = 0.1, dev=0.0005, a_f1s=[0.03],
+                          printing=False, plus_q='minus', diagonal='diagonal',
+                          stimulus_length=0.5, runs=1, cells=[],
+                          nfft=int(4096), beat='', nfft_for_morph=4096 * 4, gain=1,
+                          DF2_frmult = [], DF1_frmult = [],
+                          array_len = [20,20,20,20,20],
+                          xlim_psd =  [], ylim_log = [],
+                          fish_jammer='Alepto', markers = [], clip_on = True,
+                          ms = 14, us_name='', dev_spikes='original', log =''):
     plot_style()
     model_cells = pd.read_csv(load_folder_name('calc_model_core') + "/models_big_fit_d_right.csv")
 
@@ -458,24 +449,15 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
         #embed()
         ax = plt.subplot(grid0[g])
         axes.append(ax)
-        trials_nr = array_len[g]
-        marker = markers[g]
-        #embed()
-        #model_done = True
-        #if model_done:
-        #old = False
-        #if old:
-        #    fr = plt_model_full_model(ax, a_f1s=[a_size], af_2 = 0.1, cells=[cell],trials_nr = arrays_len[g], add_pp=250, single_waves=['_SeveralSumWave_', ], cell_start=11,
-        #                                      zeros='ones', perc = 0.25, several_peaks_nr = 2, alpha = alphas[g], log = log, nfft = nfft, freqs_mult1 = DF1_frmult[g], freqs_mult2 = DF2_frmult[g], xlim = [0, 170], a_frs=[1], add_half=0, show=True)#01
-
-
-        #else:
+        try:
+            trials_nr = array_len[g]
+        except:
+            print('array nr something')
+            embed()
 
         for cell_here in cells:
-            # sachen die ich variieren will
             ###########################################
             single_waves = ['_SeveralWave_']  # , '_SingleWave_']
-            ####### VARY HERE
             for single_wave in single_waves:
                 if single_wave == '_SingleWave_':
                     a_f2s = [0]  # , 0,0.2
@@ -484,7 +466,6 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
                 for a_f2 in a_f2s:
                     # ,0.05,0.01, 0.005, 0.1, 0.2]  # 0.001,
                     for a_f1 in a_f1s:
-                        #embed()
                         a_frs = [1]
                         titles_amp = ['base eodf']  # ,'baseline to Zero',]
                         for a, a_fr in enumerate(a_frs):
@@ -513,42 +494,10 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
                                                                                            fish_morph_harmonics_var=fish_morph_harmonics_var,
                                                                                            beat=beat)
                             sampling = 1 / deltat
-                            multiple = 0
-                            slope = 0
-                            add = 0
-                            plus = 0
-                            sig_val = (7, 1)
-                            variant = 'sinz'
-                            if exponential == '':
-                                v_exp = 1
-                                exp_tau = 0.001
                             # prepare for adapting offset due to baseline modification
 
-                            baseline_with_wave_damping, baseline_without_wave = prepare_baseline_array(time_array, eod_fr,
-                                                                                                       nfft_for_morph,
-                                                                                                       phaseshift_fr,
-                                                                                                       mimick, zeros,
-                                                                                                       cell_recording,
-                                                                                                       sampling,
-                                                                                                       stimulus_length,
-                                                                                                       fish_receiver,
-                                                                                                       deltat, nfft,
-                                                                                                       damping_type,
-                                                                                                       damping, us_name,
-                                                                                                       gain, beat=beat,
-                                                                                                       fish_morph_harmonics_var=fish_morph_harmonics_var)
-
                             spikes_base = [[]] * trials_nr
-                            color0 = 'green'  # 'orange'
-                            color01 = 'blue'
-                            color02 = 'red'
-                            color012 = 'orange'
-                            color01_2 = 'purple'
-
                             color01, color012, color01_2, color02, color0_burst, color0 = colors_suscept_paper_dots()
-
-                            #fig = plt.figure(figsize=(11.5, 5.4))
-                        
                             for run in range(runs):
                                 print(run)
                                 t1 = time.time()
@@ -569,22 +518,9 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
                                                                                                adaptation_yes_j=f2,
                                                                                                adaptation_yes_e=f1,
                                                                                                adaptation_yes_t=t,
-                                                                                               adaptation_upper_tol=upper_tol,
-                                                                                               adaptation_lower_tol=lower_tol,
                                                                                                power_variant=power_here,
                                                                                                power_alpha=alpha,
                                                                                                power_nr=n,
-                                                                                               reshuffle=reshuffled,
-                                                                                               tau_change_choice=constant_reduction,
-                                                                                               tau_change_val=dent_tau_change,
-                                                                                               sigmoidal_mult=1,
-                                                                                               sigmoidal_plus=plus,
-                                                                                               sigmoidal_slope=slope,
-                                                                                               sigmoidal_add=add,
-                                                                                               sigmoidal_sigmoidal_val=sig_val,
-                                                                                               LIF_exponential=exponential,
-                                                                                               LIF_exponential_tau=exp_tau,
-                                                                                               LIF_expontential__v=v_exp,
                                                                                                **model_params)
 
                                     if t == 0:
@@ -659,19 +595,10 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
 
                                     eod_stimulus = eod_fish1 + eod_fish2
 
-                                    v_mems, offset_new, mat01, mat02, mat012, smoothed01, smoothed02, smoothed012, stimulus_01, stimulus_02, stimulus_012, mat05_01, spikes_01,  mat05_02, spikes_02, mat05_012, spikes_012 = get_arrays_for_three(
-                                        cell, a_f2, a_f1, SAM, eod_stimulus, eod_fish_r, freq2, eod_fish1, eod_fish2,
-                                        stimulus_length, offset, model_params, n, 'sinz', adapt_offset, add, deltat,
-                                        f2, trials_nr, time_array, f1, freq1, eod_fr, reshuffle=reshuffled, dev=dev)
-                                        #cell, a_f2, a_f1,
-                                        #SAM, eod_stimulus, eod_fish_r, freq2, eod_fish1, eod_fish2, stimulus_length,
-                                        #baseline_with_wave_damping, offset, model_params, n, variant, adapt_offset,
-                                        #upper_tol, lower_tol, dent_tau_change, constant_reduction,
-                                        #exponential, plus, slope, add,
-                                        #deltat, alpha, sig_val, v_exp, exp_tau, f2,
-                                        #trials_nr, time_array,
-                                        #f1, freq1, damping_type,
-                                        #gain, eod_fr, damping, us_name, dev=dev, reshuffle=reshuffled)
+                                    v_mems, offset_new, mat01, mat02, mat012, smoothed01, smoothed02, smoothed012, stimulus_01, stimulus_02, stimulus_012, mat05_01, spikes_01,  mat05_02, spikes_02, mat05_012, spikes_012 = get_arrays_for_three(cell, a_f2, a_f1, SAM, eod_stimulus, eod_fish_r, freq2,
+                                                         eod_fish1, eod_fish2, stimulus_length, offset, model_params, n,
+                                                         'sinz', adapt_offset, deltat, f2, trials_nr, time_array, f1,
+                                                         freq1, eod_fr, reshuffle=reshuffled, dev=dev)
                                     if printing:
                                         print('Generation process' + str(time.time() - t1))
 
@@ -762,10 +689,6 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
         plt_peaks_several(freqs, p0_mean, ax, p0_mean, fs, labels, 0, colors, emb=False, marker=marker, add_log=2.5,
                           exact=False, perc_peaksize=0.08, ms=ms, clip_on=clip_on, log=log)
         ax.set_xlim(0, 300)
-        #ax.set_ylim(0 - 20,
-            #            np.max(np.max(ps)) + 70)  # np.min(np.min(p0_means))
-        #if j == 0:
-        #    ax.legend(ncol=2)
         if j == 0:
             ax.text(1, 1, 'Model', ha='right', va='top', transform=ax.transAxes)
         if j == 0:
@@ -779,21 +702,16 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled',af_2 = 0.1, datapoints=
             ax.set_ylim(ylim_log)
             ax.set_ylabel('dB')
     join_y(axes)#[1::]
-    #embed()
-    test = False
-    if test:
-        plt.plot(p0_means[0])
-        plt.plot(p0_means[1])
-
 
     axes.append(ax)
 
 
-    return fr
+    return fr, eod_fr
 
 
 
-def plt_data_full_model(c1, chose_score, detections, devs, dfs, end, grid, mult_type, sorted_on, ms = 14, log = 'log',markers = ['s', 'o'], marker = 'o', clip_on = False, alpha = [],DF2_desired = [-33, -100], DF1_desired = [133, 66], ylim_log = (-15, 3), nfft = 2 ** 13, xlim_psd = [0, 235]):
+def plt_data_full_model(c1, chose_score, detections, devs, dfs, end, grid, mult_type, sorted_on, ms = 14, log = 'log', markers = ['s', 'o'],
+                        clip_on = False, alpha = [], DF2_desired = [-33, -100], DF1_desired = [133, 66], ylim_log = (-15, 3), nfft =2 ** 13, xlim_psd = [0, 235]):
     # mean_type = '_MeanTrialsIndexPhaseSort_Min0.25sExcluded_'
     extract = ''
     datasets, data_dir = find_all_dir_cells()

model_full_spikes_012.csv

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+28,157.16500000012584,156.72000000066026
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+30,174.7900000005333,173.42000000078832
+31,176.53999999998032,174.9700000007403
+32,192.8400000002088,176.52000000069228
+33,194.34000000051438,192.7950000006428
+34,195.81500000054203,194.2950000000389
+35,197.81500000004,195.87000000026882
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+37,215.31499999996723,212.44499999991666
+38,217.01500000067736,215.22000000075485
+39,232.09000000001961,216.94499999992394
+40,234.84000000057986,233.32000000007673
+41,236.4900000007341,234.84500000066026
+42,238.29000000073702,236.39500000061224
+43,252.99000000045763,251.5950000004347
+44,254.5400000004096,254.34500000008546
+45,256.26500000048816,256.07000000016404
+46,271.290000000184,258.0200000000156
+47,273.91500000026406,272.44500000031684
+48,275.6650000006206,274.0449999999152
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+50,293.5650000004474,278.42000000035176
+51,295.0150000001971,293.4200000006792
+52,296.6900000006293,294.97000000063116
+53,312.96500000057983,296.52000000058314
+54,314.4900000002539,311.8449999999763
+55,316.09000000076173,314.39500000013203
+56,317.76500000028443,316.0700000005642
+57,332.6650000004096,320.77000000006655
+58,334.16500000071517,332.5950000005657
+59,335.7900000005915,334.37000000029064
+60,340.0400000005478,338.5200000000447
+61,353.615000000494,353.4950000000942
+62,355.11500000079957,355.0200000006777
+63,356.9150000008025,356.59499999999815
+64,373.09000000055073,359.77000000073593
+65,374.61500000022477,373.14500000027755
+66,376.1900000004547,374.6950000002295
+67,391.61500000005014,376.5200000005104
+68,394.16500000020585,392.59500000005636
+69,395.8650000000065,394.1950000005642
+70,397.41500000086796,395.76999999988465
+71,412.3650000006395,400.39500000037214
+72,415.1400000005682,413.6200000000651
+73,416.79000000072244,415.195000000295
+74,433.16499999996574,416.8200000001713
+75,434.7150000008272,431.8700000001451
+76,436.2650000007792,433.44500000037505
+77,438.1150000004285,436.1950000000258
+78,452.865000000705,437.9450000003823
+79,454.34000000073263,452.7949999999516
+80,456.0150000002553,454.2950000002572
+81,472.3900000004081,455.79500000056277
+82,473.86500000043577,458.84499999991084
+83,475.3900000001098,473.8700000005162
+84,478.31500000079666,475.4200000004682
+85,493.2650000005682,476.9950000006981
+86,494.8400000007981,492.12000000059624
+87,496.4400000003965,494.72000000039833
+88,499.44000000009817,496.4450000004769

motivation.py

@@ -18,10 +18,9 @@ from utils_suseptibility import *
 
 
 
-def motivation_all_small(dev_desired = '1',ylim=[-1.25, 1.25], c1=10, dfs=['m1', 'm2'], mult_type='_multsorted2_', top=0.94, devs=['2'],
-                   figsize=None, redo=False, save=True, end='0', cut_matrix='malefemale', chose_score='mean_nrs',
-                   a_fr=1, restrict='modulation', adapt='adaptoffsetallall2', step=str(30),
-                   detections=['AllTrialsIndex'], variant='no', sorted_on='LocalReconst0.2Norm'):
+def motivation_all_small(dev_desired = '1', ylim=[-1.25, 1.25], c1=10, dfs=['m1', 'm2'], mult_type='_multsorted2_', top=0.94, devs=['2'],
+                         figsize=None, save=True, end='0', chose_score='mean_nrs',
+                         detections=['AllTrialsIndex'], sorted_on='LocalReconst0.2Norm'):
     autodefines = [
         'triangle_diagonal_fr']  # ['triangle_fr', 'triangle_diagonal_fr', 'triangle_df2_fr','triangle_df2_eodf''triangle_df1_eodf', ]  # ,'triangle_df2_fr''triangle_df1_fr','_triangle_diagonal__fr',]
     cells = ['2021-08-03-ac-invivo-1']  ##'2021-08-03-ad-invivo-1',,[10, ][5 ]
@@ -29,7 +28,7 @@ def motivation_all_small(dev_desired = '1',ylim=[-1.25, 1.25], c1=10, dfs=['m1',
     c2s = [10]
 
     plot_style()
-    default_figsize(column=2, length=3.3)  #6.7 ts=12, ls=12, fs=12
+    default_figsize(column=2, length=4.3)  #6.7 ts=12, ls=12, fs=12
     show = True
     DF2_desired = [0.8]
     DF1_desired = [0.87]
@@ -188,9 +187,9 @@ def motivation_all_small(dev_desired = '1',ylim=[-1.25, 1.25], c1=10, dfs=['m1',
                                     fig = plt.figure(figsize=figsize)
                                 else:
                                     fig = plt.figure()
-                                grid = gridspec.GridSpec(1, 1, wspace=0.7, hspace=0.35, left=0.055, top=top,
+                                grid = gridspec.GridSpec(2, 1, wspace=0.7, hspace=0.35, left=0.055, top=top,
                                                          bottom=0.15,
-                                                         right=0.935)  # height_ratios=[1, 2], height_ratios = [1,6]bottom=0.25, top=0.8,
+                                                         right=0.935, height_ratios=[1, 5.3])  # height_ratios=[1, 2], height_ratios = [1,6]bottom=0.25, top=0.8,
                                 hr = [1, 0.35, 1.2, 0, 3, ]  # 1
                                 ##########################################################################
                                 # several coherence plot
@@ -204,10 +203,17 @@ def motivation_all_small(dev_desired = '1',ylim=[-1.25, 1.25], c1=10, dfs=['m1',
 
                                 # ax_cohs = plt.subplot(grid[0,1])
 
+                                ##########################################################################
+                                # shemes
+                                ##########################################################################
+                                # part with the power spectra
+                                grid00 = gridspec.GridSpecFromSubplotSpec(1, 4, wspace=0.15, hspace=0.35,
+                                                                         subplot_spec=grid[0, :])
+
                                 ##########################################################################
                                 # part with the power spectra
                                 grid0 = gridspec.GridSpecFromSubplotSpec(5, 4, wspace=0.15, hspace=0.35,
-                                                                         subplot_spec=grid[:, :],
+                                                                         subplot_spec=grid[1, :],
                                                                          height_ratios=hr)
 
                                 xlim = [0, 100]
@@ -366,6 +372,5 @@ def motivation_all_small(dev_desired = '1',ylim=[-1.25, 1.25], c1=10, dfs=['m1',
 
 
 if __name__ == '__main__':#2.5
-    motivation_all_small(dev_desired = '1', c1=10, mult_type='_multsorted2_', devs=['05'], redo=True, save=True, end='all',
-                   cut_matrix='malefemale', chose_score='mean_nrs', restrict='modulation_no_classes', step='50',
-                   detections=['MeanTrialsIndexPhaseSort'], sorted_on='LocalReconst0.2NormAm')#
+    motivation_all_small(dev_desired = '1', c1=10, mult_type='_multsorted2_', devs=['05'], save=True, end='all',
+                         chose_score='mean_nrs', detections=['MeanTrialsIndexPhaseSort'], sorted_on='LocalReconst0.2NormAm')#

motivation_stim.py

@@ -365,7 +365,7 @@ def motivation_all_small_stim(dev_desired = '1',ylim=[-1.25, 1.25], c1=10, dfs=[
                             # plt.show()
 
 
-if __name__ == '__main__':#2.5
-    motivation_all_small_stim(dev_desired = '1', c1=10, mult_type='_multsorted2_', devs=['05'], redo=True, save=True, end='all',
-                   cut_matrix='malefemale', chose_score='mean_nrs', restrict='modulation_no_classes', step='50',
-                   detections=['MeanTrialsIndexPhaseSort'], sorted_on='LocalReconst0.2NormAm')#
+#if __name__ == '__main__':#2.5
+#    motivation_all_small_stim(dev_desired = '1', c1=10, mult_type='_multsorted2_', devs=['05'], redo=True, save=True, end='all',
+#                   cut_matrix='malefemale', chose_score='mean_nrs', restrict='modulation_no_classes', step='50',
+#                   detections=['MeanTrialsIndexPhaseSort'], sorted_on='LocalReconst0.2NormAm')#

references.bib.bak

@@ -4022,6 +4022,16 @@ We collected weakly electric gymnotoid fish in the vicinity of Manaus, Amazonas,
   Volume                   = {21}
 }
 
+@article{Longtin1993,
+  title={Stochastic resonance in neuron models},
+  author={Longtin, Andr{\'e}},
+  journal={Journal of statistical physics},
+  volume={70},
+  pages={309--327},
+  year={1993},
+  publisher={Springer}
+}
+
 @Article{Longtin1996,
   title={Encoding with bursting, subthreshold oscillations, and noise in mammalian cold receptors},
   author={Longtin, Andre and Hinzer, Karin},

removed/makefile

@@ -36,7 +36,10 @@ diff:
 copy:
 	cp $(file).sty ../
 
-
+# python #########################################################
+plots: $(PYTHONPDFFILES)
+$(PYTHONPDFFILES): %.pdf: %.py plotstyle.py
+	python3 $<
 
 
 

removed2/Makefile

@@ -0,0 +1,197 @@
+TEXBASE=susceptibility1
+BIBFILE=references.bib
+REBUTTALBASE=
+
+TEXFILE=$(TEXBASE).tex
+PDFFILE=$(TEXBASE).pdf
+TXTFILE=$(TEXBASE).txt
+
+REVISION=
+
+PYTHONFILES=$(filter-out plotstyle.py myfunctions.py numerical_compar_both.py, $(PT))
+PYTHONPDFFILES=$(PYTHONFILES:.py=.pdf)
+
+ifdef REBUTTALBASE
+REBUTTALTEXFILE=$(REBUTTALBASE).tex
+REBUTTALPDFFILE=$(REBUTTALBASE).pdf
+endif
+REBUTTALREVISION=
+
+# all ###########################################################
+ifdef REBUTTALBASE
+all: bib rebuttalbib
+else
+all: bib
+endif
+
+# python #########################################################
+plots: $(PYTHONPDFFILES)
+$(PYTHONPDFFILES): %.pdf: %.py plotstyle.py
+	python3 $<
+
+watchplots :
+	while true; do ! make -q plots && make plots; sleep 0.5; done
+
+
+# rescue_local_eod manuscript #################################################
+bib: $(TEXBASE).bbl
+$(TEXBASE).bbl: $(TEXFILE) $(BIBFILE)
+	lualatex $(TEXFILE)
+	bibtex $(TEXBASE)
+	lualatex $(TEXFILE)
+	lualatex $(TEXFILE)
+	lualatex $(TEXFILE)
+	@echo
+	@echo "BibTeX log:"
+	@sed -n -e '1,/You.ve used/p' $(TEXBASE).blg
+
+pdf: $(PDFFILE)
+$(PDFFILE) : $(TEXFILE)
+	lualatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && lualatex -interaction=scrollmode $< || true
+
+again :
+	lualatex $(TEXFILE)
+
+# watch files #######################################################
+watchpdf :
+	while true; do ! make -s -q pdf && make pdf; sleep 0.5; done
+
+
+# make diffs ########################################################
+diff :
+	#latexdiff-git -r $(REVISION) --pdf $(TEXFILE)
+	latexdiff-git -r $(REVISION) $(TEXFILE)
+	-lualatex $(TEXBASE)-diff$(REVISION)
+	-bibtex $(TEXBASE)-diff$(REVISION)
+	-lualatex $(TEXBASE)-diff$(REVISION)
+	-lualatex $(TEXBASE)-diff$(REVISION)
+	-lualatex $(TEXBASE)-diff$(REVISION)
+	mv $(TEXBASE)-diff$(REVISION).pdf $(TEXBASE)-diff.pdf
+	mv $(TEXBASE)-diff$(REVISION).tex $(TEXBASE)-diff.tex
+	mv $(TEXBASE)-diff$(REVISION).bbl $(TEXBASE)-diff.bbl
+	rm $(TEXBASE)-diff$(REVISION).*
+
+
+# convert to txt file ################################################
+txt: $(PDFFILE)
+	#dvi2tty -w 132 -v 500000 -e-60 -q $(DVIFILE) | sed -n -e '/\cL/,+2!p' > $(TXTFILE)
+	pdftotext -nopgbrk $(PDFFILE) - | fold -s > $(TXTFILE)
+
+# convert to rtf file ################################################
+rtf :
+	latex2rtf $(TEXFILE)
+
+# remove all fancy commands from the tex file:
+simplify :
+	sed -e '/overall style/,/page style/d; /setdoublespacing/,+1d; /usepackage.*caption/s/{\(.*\)}/\1/; /figure placement/,/^%/d; /ifthenelse.*nofigs/,/#1/d; /begin{multicols}/d; /end{multicols}/d; /begin{keywords}/,/end{keywords}/d; /begin{contributions}/,/end{contributions}/d; /figurecaptions/d; /linenomath/d; s/captionc/caption/' $(TEXFILE) | perl -00 -lpe 's/\\showfigure{((\s|.)*?)}/$$1/' > $(TEXBASE)-simplified.tex
+
+# statistics #########################################################
+stats: $(PDFFILE)
+# use \pagestyle{empty} and don't include any pictures!
+	pdftotext -nopgbrk $(PDFFILE) - | fold -s > tmp.txt
+	@echo
+	@echo "     words: " `wc -w tmp.txt 2> /dev/null | cut -d ' ' -f 1` 
+	@echo "characters: " `wc -c tmp.txt 2> /dev/null | cut -d ' '  -f 1`
+	rm tmp.txt
+
+# rebuttal ##########################################################
+ifdef REBUTTALBASE
+rebuttalbib: $(REBUTTALBASE).bbl
+$(REBUTTALBASE).bbl: $(REBUTTALTEXFILE) $(BIBFILE)
+	lualatex $(REBUTTALTEXFILE)
+	bibtex $(REBUTTALBASE)
+	lualatex $(REBUTTALTEXFILE)
+	lualatex $(REBUTTALTEXFILE)
+	lualatex $(REBUTTALTEXFILE)
+	@echo
+	@echo "BibTeX log:"
+	@sed -n -e '1,/You.ve used/p' $(REBUTTALBASE).blg
+
+rebuttal: $(REBUTTALPDFFILE)
+$(REBUTTALPDFFILE) : $(REBUTTALTEXFILE)
+	lualatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && lualatex -interaction=scrollmode $< || true
+
+watchrebuttal :
+	while true; do ! make -q rebuttal && make rebuttal; sleep 0.5; done
+
+rebuttaldiff :
+	latexdiff-git -r $(REBUTTALREVISION) --append-textcmd="response,issue" --pdf $(REBUTTALTEXFILE)
+	mv $(REBUTTALBASE)-diff$(REBUTTALREVISION).pdf $(REBUTTALBASE)-diff.pdf
+	rm $(REBUTTALBASE)-diff$(REBUTTALREVISION).*
+endif
+
+# git ##############################################################
+pull :
+	git pull origin master
+
+ifdef REBUTTALBASE
+
+edit : pull
+	emacs $(TEXFILE) $(BIBFILE) $(REBUTTALTEXFILE) Makefile &
+	sleep 1
+	okular $(REBUTTALPDFFILE) $(PDFFILE) &
+
+prepare : pull bib diffrev rebuttalbib rebuttaldiff
+
+else
+
+edit : pull
+	emacs $(TEXFILE) $(BIBFILE) Makefile &
+	sleep 1
+	okular $(PDFFILE) &
+
+prepare : pull bib diffrev
+
+endif
+
+push : prepare
+	git commit -a
+	git push origin master 
+
+# convert figures to png files #######################################
+figures:
+	./latexfigures2png $(TEXFILE)
+
+# convert pdf figures to eps #########################################
+epsfigures:
+	#for i in $(PDFFIGURES); do echo $$i; rm -f $${i%.pdf}.eps; pdftops -level3 -eps $$i $${i%.pdf}.eps; done
+	for i in $(PDFFIGURES); do echo $$i; rm -f $${i%.pdf}.eps; gs -q -dNOCACHE -dNOPAUSE -dBATCH -dSAFER -sDEVICE=epswrite -sOutputFile=$${i%.pdf}.eps $$i; done
+
+# clean up ############################################################
+
+clean:
+	rm -rf auto *~ *.aux *.blg *.bbl *.dvi *.log *.out *.fff *.ttt $(PDFFIGURES) __pycache__
+
+cleanall: clean
+	rm -f $(PDFFILE) figure-??.png
+
+# help ################################################################
+help : 
+	@echo -e \
+	"make pdf:           make the pdf file of the paper.\n"\
+	"make bib:           run bibtex and make the pdf file of the paper.\n"\
+	"make again:         run pdflatex and make the pdf file of the paper,\n"\
+        "                   no matter whether you changed the .tex file or not.\n"\
+	"make watchpdf:      make the pdf file of the paper\n"\
+        "                   whenever the tex file is modified.\n"\
+	"make diff:          make a diff file against the specified revision (REVISION variable)\n"\
+	"make txt:           make a plain text version of the paper ($(TXTFILE)).\n"\
+	"make rtf:           convert the paper ($(TXTFILE)) to rtf format.\n"\
+	"make simplify:      strip all fancy commands from the paper ($(TXTFILE))\n"\
+	"make stats:         print number of words and characters.\n"\
+	"make rebuttalbib:   run bibtex and make the pdf file of the rebuttal.\n"\
+	"make rebuttal:      make the pdf file of the rebuttal.\n"\
+	"make watchrebuttal: make the pdf file of the rebuttal\n"\
+        "                   whenever the tex file is modified.\n"\
+	"make rebuttaldiff:  make a diff file of the rebuttal against the specified revision\n"\
+        "                   (REBUTTALREVISION variable).\n"\
+	"make pull:          pull from the git repository.\n"\
+	"make edit:          pull and open emacs and okular with te relevant files.\n"\
+	"make prepare:       pull and make the pdfs and diffs of the manuscript and the rebuttal.\n"\
+	"make push:          prepare, commit, and push to the git repository.\n"\
+	"make figures   :    convert all figures to png files.\n"\
+	"make epsfigures:    convert all included pdf figures to eps files.\n"\
+	"make clean:         remove all intermediate files,\n"\
+        "                   just leave the source files and the final .pdf files.\n"\
+	"make cleanup:       remove all intermediate files as well as\n"\
+        "                   the final .pdf files.\n"\

susceptibility1.tex

@@ -251,11 +251,11 @@
       {\protect\marginpar{\small\textbf{changed\ifthenelse{\equal{#1}{}}{}{ [#1]}}}{\bfseries #2}}}}% 3
 
 %%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newcommand{\fstim}{\ensuremath{f_{stim}}}
-\newcommand{\feod}{\ensuremath{f_{EOD}}}
+\newcommand{\fstim}{\ensuremath{f_{\rm{stim}}}}
+\newcommand{\feod}{\ensuremath{f_{\rm{EOD}}}}
 \newcommand{\feodsolid}{\ensuremath{f_{\rm{EOD}}}}
-\newcommand{\feodhalf}{\ensuremath{f_{EOD}/2}}
-\newcommand{\fbase}{\ensuremath{f_{base}}}
+\newcommand{\feodhalf}{\ensuremath{f_{\rm{EOD}}/2}}
+\newcommand{\fbase}{\ensuremath{f_{\rm{base}}}}
 \newcommand{\fbasesolid}{\ensuremath{f_{\rm{base}}}}
 \newcommand{\fstimintro}{\ensuremath{\rm{EOD}_{2}}}
 \newcommand{\feodintro}{\ensuremath{\rm{EOD}_{1}}}
@@ -284,12 +284,12 @@
 \newcommand{\cstable}{\ensuremath{c_{2}}}
 \newcommand{\fstable}{\ensuremath{f_{2}}}
 
-\newcommand{\aeod}{\ensuremath{A(f_{EOD})}}
+\newcommand{\aeod}{\ensuremath{A(f_{\rm{EOD}})}}
 \newcommand{\fbasecorrsolid}{\ensuremath{f_{\rm{BaseCorrected}}}}
-\newcommand{\fbasecorr}{\ensuremath{f_{BaseCorrected}}}
-\newcommand{\ffall}{$f_{EOD}$\&$f_{1}$\&$f_{2}$}
-\newcommand{\ffvary}{$f_{EOD}$\&$f_{1}$}%sum 
-\newcommand{\ffstable}{$f_{EOD}$\&$f_{2}$}%sum 
+\newcommand{\fbasecorr}{\ensuremath{f_{\rm{BaseCorrected}}}}
+\newcommand{\ffall}{$f_{\rm{EOD}}$\&$f_{1}$\&$f_{2}$}
+\newcommand{\ffvary}{$f_{\rm{EOD}}$\&$f_{1}$}%sum 
+\newcommand{\ffstable}{$f_{\rm{EOD}}$\&$f_{2}$}%sum 
 \newcommand{\colstableone}{blue}%sum 
 \newcommand{\colstabletwo}{cyan}%sum 
 \newcommand{\colvaryone}{brown}%sum 
@@ -316,12 +316,12 @@
 \newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}%sum 
 \newcommand{\bsumbtwo}{$\bsum{}=2 \fbase{}$}%sum 
 \newcommand{\bsumbc}{$\bsum{}=\fbasecorr{}$}%sum 
-\newcommand{\bsume}{$\bsum{}=f_{EOD}$}%sum 
-\newcommand{\bsumehalf}{$\bsum{}=f_{EOD}/2$}%sum 
+\newcommand{\bsume}{$\bsum{}=\feod{}$}%sum 
+\newcommand{\bsumehalf}{$\bsum{}=\feod{}/2$}%sum 
 \newcommand{\bdiffb}{$\bdiff{}=\fbase{}$}%diff of both beat frequencies
 \newcommand{\bdiffbc}{$\bdiff{}=\fbasecorr{}$}%diff of both beat frequencies
-\newcommand{\bdiffe}{$\bdiff{}=f_{EOD}$}%diff of both 
-\newcommand{\bdiffehalf}{$\bdiff{}=f_{EOD}/2$}%diff of both
+\newcommand{\bdiffe}{$\bdiff{}=\feod{}$}%diff of both 
+\newcommand{\bdiffehalf}{$\bdiff{}=\feod{}/2$}%diff of both
 %beat frequencies
 
 %%%%% tables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -353,12 +353,12 @@
 \newcommand{\fonebc}{$f_{1}=\fbasecorr{}$}%sum 
 \newcommand{\fsumbtwo}{$\fsum{}=2 \fbase{}$}%sum 
 \newcommand{\fsumbc}{$\fsum{}=\fbasecorr{}$}%sum 
-\newcommand{\fsume}{$\fsum{}=f_{EOD}$}%sum 
-\newcommand{\fsumehalf}{$\fsum{}=f_{EOD}/2$}%sum 
+\newcommand{\fsume}{$\fsum{}=\feod{}$}%sum 
+\newcommand{\fsumehalf}{$\fsum{}=\feod{}/2$}%sum 
 \newcommand{\fdiffb}{$\fdiff{}=\fbase{}$}%diff of both beat frequencies
 \newcommand{\fdiffbc}{$\fdiff{}=\fbasecorr{}$}%diff of both beat frequencies
-\newcommand{\fdiffe}{$\fdiff{}=f_{EOD}$}%diff of both 
-\newcommand{\fdiffehalf}{$\fdiff{}=f_{EOD}/2$}%diff of both
+\newcommand{\fdiffe}{$\fdiff{}=\feod{}$}%diff of both 
+\newcommand{\fdiffehalf}{$\fdiff{}=\feod{}/2$}%diff of both
 
 \newcommand{\fctwo}{\ensuremath{f_{\rm{Female}}}}%sum 
 \newcommand{\fcone}{\ensuremath{f_{\rm{Intruder}}}}%sum 
@@ -507,7 +507,7 @@
 
 Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985, 
 
-While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. P-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units ot the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
+While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
@@ -517,15 +517,16 @@ While the sensory periphery can often be well described by linear models, this i
 
 Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom).\notejg{more baseline trials, working on it}. Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, top). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom). The latter is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}. 
 
-The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Voronenko2017,Neiman2011fish,Nikias1993}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding. 
+The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding. 
 
 
 
 \section*{Results}
-Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs) that are called beats (two-fish interaction) and envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units encode such AMs of the underlying EOD carrier in  their time-dependent firing rates \cite{Bastian1981a,Walz2014}. We here explore nonlinear responses of different cells that exhibit distinctly different levels of output variability, quantified by the coefficient of variation (CV) of their interspike intervals (ISI). Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process.
+Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs) that are called beats (two-fish interaction) and envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units encode such AMs of the underlying EOD carrier in  their time-dependent firing rates \cite{Bastian1981a,Walz2014}. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process.
 %P-units are heterogeneous in their baseline firing properties \cite{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. 
 
 
+
 \subsection*{Nonlinear signal transmission in low-CV P-units} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
 
 Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
@@ -538,7 +539,8 @@ Second-order susceptibility is expected to be especially pronounced for low-CV c
 
 Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \Eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
 
-Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines).
+Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that with the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize the signal
+transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.
 
 High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
 
@@ -553,12 +555,12 @@ Irrespective of the CV, neither cell shows the complete proposed structure of no
 \end{figure*}
 
 \subsection*{Model-based estimation of the nonlinear structure}
-Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expec\-tations\cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory,  cannot be observed. Here we investigate how these discrepancies can be understood.
+Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory,  cannot be observed. Here we investigate how these discrepancies can be understood.
 
 In the electrophysiological experiments, we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, with parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
 
 In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal 
-transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{noise}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
+transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
 
 Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of a statistical estimate that is based on 11 trials only. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response.
 
@@ -566,19 +568,19 @@ With high levels of intrinsic noise, we would not expect the nonlinear response
 
 \begin{figure*}[!hb]
   \includegraphics[width=\columnwidth]{model_and_data}
-  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details). Note that the signal component ($\xi_{signal}$) is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the noise component ($\xi_{noise}$). Adding the discarded high frequency components to $\xi_{noise}$ does not affect the results shown here.
+  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{\rm{noise}}$ and $\xi_{\rm{noise}}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{\rm{noise}} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{\rm{noise}}$) while the remaining part is treated as $\xi_{\rm{noise}}$ (see methods for details). Note that the signal component ($\xi_{\rm{noise}}$) is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the noise component ($\xi_{\rm{noise}}$). Adding the discarded high frequency components to $\xi_{\rm{noise}}$ does not affect the results shown here.
 }
 \end{figure*}
 
 \subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
-We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
+We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{model_full}
   \caption{\label{fig:model_full} \figitem[]{A} Full second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\Eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods). The colored markers highlight the nonlinear effects found in \subfigrefb{fig:motivation}{D}. \figitem{B} Power spectral density of model responses (left) and the respective recorded data (right) under pure sinewave stimulation. \figitem{C} Same as \panel[]{B} but for frequency combinations off the nonlinear structure. \note[TODO]{Better combination of off-axis frequency components. Still working on the units of the second-order susceptibility in this plot. In \panel[]{B} left the green and blue markers are at the same position.}}
 \end{figure*}
 %section \ref{intrinsicsplit_methods}).\figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B}  The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.
 
-The second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}, \subfigref{fig:model_full}{B} right) and in the model (\subfigref{fig:model_full}{B} left). 
+However, the second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}, \subfigref{fig:model_full}{B} right) and in the model (\subfigref{fig:model_full}{B} left). 
 
 The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the fading of the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). If we choose different frequency combinations, weak or no nonlinear spectral peaks are observed \subfigrefb{fig:model_full}{C}. Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
 
@@ -586,18 +588,18 @@ If two frequencies not part of the triangular structure are chosen with pure sin
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{data_overview_mod}
-  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
+  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
   }
 \end{figure*}
 
 %\Eqnref{response_modulation}
 
 \subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
-The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
-%(Pearson's $r=-0.35$, $p<0.001$)222 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
-%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=222$*, $\n{}=222$******, $\n{}=222$
+The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
+%(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
+%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
 
-The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
+The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
 
 %(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
 
@@ -608,19 +610,21 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
 
 %\,\panel[iii]{C}
 \subsection*{Theory applies to systems with and without carrier}
-Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{fig:ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
+Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{fig:ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
 
 \subsection*{Intrinsic noise limits nonlinear responses}
-Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 222 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
+Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
 
-The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Novikov-Furutsu theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
+The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Furutsu-Novikov theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
 
 \subsection*{Noise stimulation approximates the real three-fish interaction}
 Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{French1973, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In a our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \cite{Voronenko2017}.
 
 In the natural situation, the stimuli are periodic signals defined by the difference frequencies. Ho well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
 
-In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
+In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
+
+
 
 % The nonlinearity of ampullary cells in paddlefish \cite{Neiman2011fish} has been previously accessed with bandpass limited white noise. 
 
@@ -628,7 +632,7 @@ In contrast to the situation with individual frequencies (direct sine-waves or s
 
 \subsection*{Selective readout versus integration of heterogeneous populations}% Nonlinearity might be influenced once integrating from a P-unit population with heterogeneous baseline properties}%Heterogeneity of P-units might influence 
 
-The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies. 
+The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. These nonlinear effects are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies. 
 P-units, however are very heterogeneous in their baseline firing properties\cite{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2)\cite{Barayeu2023}. It is thus likely that there are P-units which match approximately to the specificities of the different encounters. 
 
 On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\cite{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\cite{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \cite{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \cite{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\cite{Hladnik2023} and other systems\cite{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions. 
@@ -652,7 +656,7 @@ auditory nerve fibers and such nonlinear effects might also be expected in the a
 \subsection*{Experimental subjects and procedures}
 
 Within this project we re-evaluated datasets that were recorded between 2010 and 2023 at the Ludwig Maximilian University (LMU) M\"unchen and the Eberhard-Karls University T\"ubingen. All experimental protocols complied with national and European law and were approved by the respective Ethics Committees of the Ludwig-Maximilians Universität München (permit no. 55.2-1-54-2531-135-09) and the Eberhard-Karls Unversität Tübingen (permit no. ZP 1/13 and ZP 1/16).
-The final sample consisted of 222 P-units and 47 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
+The final sample consisted of 221 P-units and 47 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
 Germany). The fish were kept in tanks with a water temperature of $25\pm1\,^\circ$C and a conductivity of around $270\,\micro\siemens\per\centi\meter$ under a 12\,h:12\,h light-dark cycle. 
 
 Before surgery, the animals were deeply anesthetized via bath application with a solution of MS222 (120\,mg/l, PharmaQ, Fordingbridge, UK) buffered with Sodium Bicarbonate (120\,mg/l). The posterior anterior lateral line nerve (pALLN) was exposed by making a small cut into the skin covering the nerve. The cut was placed dorsal of the operculum just before the nerve descends towards the anterior lateral line ganglion (ALLNG). Those parts of the skin that were to be cut were locally anesthetized by cutaneous application of liquid lidocaine hydrochloride (20\,mg/ml, bela-pharm GmbH). During the surgery water supply was ensured by a mouthpiece to maintain anesthesia with a solution of MS222 (100\,mg/l) buffered with Sodium Bicarbonate (100\,mg/l). After surgery fish were immobilized by intramuscular injection of from 25\,$\micro$l to 50\,$\micro$l of tubocurarine (5\,mg/ml dissolved in fish saline; Sigma-Aldrich).
@@ -848,11 +852,11 @@ From each simulation run, the first second was discarded and the analysis was ba
 % The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
 \subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
-According to the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$ \note{hier umschreiben das xi muss ein RAM sein} and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{noise}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
-%\sqrt{\rho \, 2D \,c_{signal}} \cdot \xi(t) 
+According to the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$ \note{hier umschreiben das xi muss ein RAM sein} and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
+%\sqrt{\rho \, 2D \,c_{\rm{signal}}} \cdot \xi(t) 
 
-%(1-c_{signal})\cdot\xi$c_{noise} = 1-c_{signal}$
-%c_{signal} \cdot \xi
+%(1-c_{\rm{signal}})\cdot\xi$c_{\rm{noise}} = 1-c_{\rm{signal}}$
+%c_{\rm{signal}} \cdot \xi
 \begin{equation}
     \label{eq:ram_split}
     y(t) = (1+ s_\xi(t)) \cdot \cos(2\pi f_{EOD} t)
@@ -866,18 +870,18 @@ According to the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} the tot
 
 \begin{equation}
   \label{eq:Noise_split_intrinsic}
-  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{noise}} \cdot \xi(t)
+  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)
 \end{equation}
 % das stimmt so, das c kommt unter die Wurzel!
 
 %\begin{equation}
 %  \label{Noise_split_intrinsic}
-%  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{noise}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
+%  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{\rm{noise}}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
 %\end{equation}
 
 
 
-A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\Eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[iii]{A}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
+A big portion of the total noise was assigned to the signal component ($c_{\rm{signal}} = 0.9$) and the remaining part to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{\rm{noise}}(t)$ was scaled up by the factor $\rho$ (\Eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[iii]{A}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
 
 
 % See section \ref{lifmethods} for model and parameter description.
@@ -936,7 +940,7 @@ CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the sa
 
 \begin{figure*}[hp]%hp!
 \includegraphics[width=\columnwidth]{trialnr}  
-  \caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$. Grey -- 5th to 95th percentile of the second-order susceptibility.
+  \caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$. Black -- 99.9th percentile of the second-order susceptibility.
   }
 \end{figure*}
 

susceptibility1.tex.bak

@@ -251,11 +251,11 @@
       {\protect\marginpar{\small\textbf{changed\ifthenelse{\equal{#1}{}}{}{ [#1]}}}{\bfseries #2}}}}% 3
 
 %%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\newcommand{\fstim}{\ensuremath{f_{stim}}}
-\newcommand{\feod}{\ensuremath{f_{EOD}}}
+\newcommand{\fstim}{\ensuremath{f_{\rm{stim}}}}
+\newcommand{\feod}{\ensuremath{f_{\rm{EOD}}}}
 \newcommand{\feodsolid}{\ensuremath{f_{\rm{EOD}}}}
-\newcommand{\feodhalf}{\ensuremath{f_{EOD}/2}}
-\newcommand{\fbase}{\ensuremath{f_{base}}}
+\newcommand{\feodhalf}{\ensuremath{f_{\rm{EOD}}/2}}
+\newcommand{\fbase}{\ensuremath{f_{\rm{base}}}}
 \newcommand{\fbasesolid}{\ensuremath{f_{\rm{base}}}}
 \newcommand{\fstimintro}{\ensuremath{\rm{EOD}_{2}}}
 \newcommand{\feodintro}{\ensuremath{\rm{EOD}_{1}}}
@@ -284,12 +284,12 @@
 \newcommand{\cstable}{\ensuremath{c_{2}}}
 \newcommand{\fstable}{\ensuremath{f_{2}}}
 
-\newcommand{\aeod}{\ensuremath{A(f_{EOD})}}
+\newcommand{\aeod}{\ensuremath{A(f_{\rm{EOD}})}}
 \newcommand{\fbasecorrsolid}{\ensuremath{f_{\rm{BaseCorrected}}}}
-\newcommand{\fbasecorr}{\ensuremath{f_{BaseCorrected}}}
-\newcommand{\ffall}{$f_{EOD}$\&$f_{1}$\&$f_{2}$}
-\newcommand{\ffvary}{$f_{EOD}$\&$f_{1}$}%sum 
-\newcommand{\ffstable}{$f_{EOD}$\&$f_{2}$}%sum 
+\newcommand{\fbasecorr}{\ensuremath{f_{\rm{BaseCorrected}}}}
+\newcommand{\ffall}{$f_{\rm{EOD}}$\&$f_{1}$\&$f_{2}$}
+\newcommand{\ffvary}{$f_{\rm{EOD}}$\&$f_{1}$}%sum 
+\newcommand{\ffstable}{$f_{\rm{EOD}}$\&$f_{2}$}%sum 
 \newcommand{\colstableone}{blue}%sum 
 \newcommand{\colstabletwo}{cyan}%sum 
 \newcommand{\colvaryone}{brown}%sum 
@@ -316,12 +316,12 @@
 \newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}%sum 
 \newcommand{\bsumbtwo}{$\bsum{}=2 \fbase{}$}%sum 
 \newcommand{\bsumbc}{$\bsum{}=\fbasecorr{}$}%sum 
-\newcommand{\bsume}{$\bsum{}=f_{EOD}$}%sum 
-\newcommand{\bsumehalf}{$\bsum{}=f_{EOD}/2$}%sum 
+\newcommand{\bsume}{$\bsum{}=\feod{}$}%sum 
+\newcommand{\bsumehalf}{$\bsum{}=\feod{}/2$}%sum 
 \newcommand{\bdiffb}{$\bdiff{}=\fbase{}$}%diff of both beat frequencies
 \newcommand{\bdiffbc}{$\bdiff{}=\fbasecorr{}$}%diff of both beat frequencies
-\newcommand{\bdiffe}{$\bdiff{}=f_{EOD}$}%diff of both 
-\newcommand{\bdiffehalf}{$\bdiff{}=f_{EOD}/2$}%diff of both
+\newcommand{\bdiffe}{$\bdiff{}=\feod{}$}%diff of both 
+\newcommand{\bdiffehalf}{$\bdiff{}=\feod{}/2$}%diff of both
 %beat frequencies
 
 %%%%% tables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -353,12 +353,12 @@
 \newcommand{\fonebc}{$f_{1}=\fbasecorr{}$}%sum 
 \newcommand{\fsumbtwo}{$\fsum{}=2 \fbase{}$}%sum 
 \newcommand{\fsumbc}{$\fsum{}=\fbasecorr{}$}%sum 
-\newcommand{\fsume}{$\fsum{}=f_{EOD}$}%sum 
-\newcommand{\fsumehalf}{$\fsum{}=f_{EOD}/2$}%sum 
+\newcommand{\fsume}{$\fsum{}=\feod{}$}%sum 
+\newcommand{\fsumehalf}{$\fsum{}=\feod{}/2$}%sum 
 \newcommand{\fdiffb}{$\fdiff{}=\fbase{}$}%diff of both beat frequencies
 \newcommand{\fdiffbc}{$\fdiff{}=\fbasecorr{}$}%diff of both beat frequencies
-\newcommand{\fdiffe}{$\fdiff{}=f_{EOD}$}%diff of both 
-\newcommand{\fdiffehalf}{$\fdiff{}=f_{EOD}/2$}%diff of both
+\newcommand{\fdiffe}{$\fdiff{}=\feod{}$}%diff of both 
+\newcommand{\fdiffehalf}{$\fdiff{}=\feod{}/2$}%diff of both
 
 \newcommand{\fctwo}{\ensuremath{f_{\rm{Female}}}}%sum 
 \newcommand{\fcone}{\ensuremath{f_{\rm{Intruder}}}}%sum 
@@ -507,7 +507,7 @@
 
 Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985, 
 
-While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. P-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units ot the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
+While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
@@ -517,15 +517,16 @@ While the sensory periphery can often be well described by linear models, this i
 
 Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom).\notejg{more baseline trials, working on it}. Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, top). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom). The latter is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}. 
 
-The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Voronenko2017,Neiman2011fish,Nikias1993}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding. 
+The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding. 
 
 
 
 \section*{Results}
-Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs) that are called beats (two-fish interaction) and envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units encode such AMs of the underlying EOD carrier in  their time-dependent firing rates \cite{Bastian1981a,Walz2014}. We here explore nonlinear responses of different cells that exhibit distinctly different levels of output variability, quantified by the coefficient of variation (CV) of their interspike intervals (ISI). Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process.
+Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs) that are called beats (two-fish interaction) and envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units encode such AMs of the underlying EOD carrier in  their time-dependent firing rates \cite{Bastian1981a,Walz2014}. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process.
 %P-units are heterogeneous in their baseline firing properties \cite{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. 
 
 
+
 \subsection*{Nonlinear signal transmission in low-CV P-units} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
 
 Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
@@ -538,43 +539,48 @@ Second-order susceptibility is expected to be especially pronounced for low-CV c
 
 Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \Eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
 
-Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines).
+Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that with the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize the signal
+transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.
 
 High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
 
 
 \subsection*{Ampullary afferents exhibit strong nonlinear interactions}
-Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{ampullary}{G}, dark green). 
+Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green). 
 
 \begin{figure*}[!ht]
 \includegraphics[width=\columnwidth]{ampullary}
-  \caption{\label{ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
+  \caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
   }
 \end{figure*}
 
-\subsection*{High level of total noise hides parts of the nonlinearity structure}
-Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations \cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory,  cannot be observed. Here we investigate how these discrepancies can be understood.
+\subsection*{Model-based estimation of the nonlinear structure}
+Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory,  cannot be observed. Here we investigate how these discrepancies can be understood.
+
+In the electrophysiological experiments, we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, with parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
+
+In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal 
+transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
 
-In the electrophysiological experiments we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, with parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal 
-transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $\varepsilon s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods \eqref{eq:crosshigh}, \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise ($\sqrt{2D \, c_{noise}} \cdot \xi(t)$, with $c_\text{noise} = 0.1$, see methods \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude / standard deviation of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}). Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{trialnr}). This demonstrates the limited reliability of the statistics that is based on 11 trials. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response.
+Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of a statistical estimate that is based on 11 trials only. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response.
 
-In the high-CV P-unit we could not find such nonlinear structures, neither in the electrophysiologically recorded data nor in the respective model (not shown), corresponding to the theoretical predictions \cite{Voronenko2017}.
+With high levels of intrinsic noise, we would not expect the nonlinear response features to survive. Indeed, we do not find these features in a high-CV P-unit and its corresponding model (not shown).
 
 \begin{figure*}[!hb]
   \includegraphics[width=\columnwidth]{model_and_data}
-  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details). Note that the signal component ($\xi_{signal}$) is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the noise component ($\xi_{noise}$). Adding the discarded high frequency components to $\xi_{noise}$ does not affect the results shown here.
+  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{\rm{noise}}$ and $\xi_{\rm{noise}}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{\rm{noise}} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{\rm{noise}}$) while the remaining part is treated as $\xi_{\rm{noise}}$ (see methods for details). Note that the signal component ($\xi_{\rm{noise}}$) is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the noise component ($\xi_{\rm{noise}}$). Adding the discarded high frequency components to $\xi_{\rm{noise}}$ does not affect the results shown here.
 }
 \end{figure*}
 
 \subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
-We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
+We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{model_full}
   \caption{\label{fig:model_full} \figitem[]{A} Full second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\Eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods). The colored markers highlight the nonlinear effects found in \subfigrefb{fig:motivation}{D}. \figitem{B} Power spectral density of model responses (left) and the respective recorded data (right) under pure sinewave stimulation. \figitem{C} Same as \panel[]{B} but for frequency combinations off the nonlinear structure. \note[TODO]{Better combination of off-axis frequency components. Still working on the units of the second-order susceptibility in this plot. In \panel[]{B} left the green and blue markers are at the same position.}}
 \end{figure*}
 %section \ref{intrinsicsplit_methods}).\figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B}  The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.
 
-The second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}, \subfigref{fig:model_full}{B} right) and in the model (\subfigref{fig:model_full}{B} left). 
+However, the second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}, \subfigref{fig:model_full}{B} right) and in the model (\subfigref{fig:model_full}{B} left). 
 
 The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the fading of the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). If we choose different frequency combinations, weak or no nonlinear spectral peaks are observed \subfigrefb{fig:model_full}{C}. Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
 
@@ -582,18 +588,18 @@ If two frequencies not part of the triangular structure are chosen with pure sin
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{data_overview_mod}
-  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
+  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
   }
 \end{figure*}
 
 %\Eqnref{response_modulation}
 
 \subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
-The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
-%(Pearson's $r=-0.35$, $p<0.001$)222 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
-%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=222$*, $\n{}=222$******, $\n{}=222$
+The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
+%(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
+%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
 
-The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
+The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
 
 %(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
 
@@ -604,15 +610,21 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
 
 %\,\panel[iii]{C}
 \subsection*{Theory applies to systems with and without carrier}
-Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
+Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{fig:ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
 
 \subsection*{Intrinsic noise limits nonlinear responses}
-Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 222 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
+Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
 
-The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Novikov-Furutsu theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
+The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Furutsu-Novikov theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
 
 \subsection*{Noise stimulation approximates the real three-fish interaction}
-Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. In true white noise, each frequency component contributes with equal amplitude. This has the methodological advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{Borst1999,Chacron2005, Grewe2017}. In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
+Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{French1973, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In a our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \cite{Voronenko2017}.
+
+In the natural situation, the stimuli are periodic signals defined by the difference frequencies. Ho well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
+
+In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
+
+
 
 % The nonlinearity of ampullary cells in paddlefish \cite{Neiman2011fish} has been previously accessed with bandpass limited white noise. 
 
@@ -620,7 +632,7 @@ Our analysis is based on the neuronal responses to white noise stimulus sequence
 
 \subsection*{Selective readout versus integration of heterogeneous populations}% Nonlinearity might be influenced once integrating from a P-unit population with heterogeneous baseline properties}%Heterogeneity of P-units might influence 
 
-The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies. 
+The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. These nonlinear effects are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies. 
 P-units, however are very heterogeneous in their baseline firing properties\cite{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2)\cite{Barayeu2023}. It is thus likely that there are P-units which match approximately to the specificities of the different encounters. 
 
 On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\cite{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\cite{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \cite{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \cite{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\cite{Hladnik2023} and other systems\cite{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions. 
@@ -644,7 +656,7 @@ auditory nerve fibers and such nonlinear effects might also be expected in the a
 \subsection*{Experimental subjects and procedures}
 
 Within this project we re-evaluated datasets that were recorded between 2010 and 2023 at the Ludwig Maximilian University (LMU) M\"unchen and the Eberhard-Karls University T\"ubingen. All experimental protocols complied with national and European law and were approved by the respective Ethics Committees of the Ludwig-Maximilians Universität München (permit no. 55.2-1-54-2531-135-09) and the Eberhard-Karls Unversität Tübingen (permit no. ZP 1/13 and ZP 1/16).
-The final sample consisted of 222 P-units and 47 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
+The final sample consisted of 221 P-units and 47 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
 Germany). The fish were kept in tanks with a water temperature of $25\pm1\,^\circ$C and a conductivity of around $270\,\micro\siemens\per\centi\meter$ under a 12\,h:12\,h light-dark cycle. 
 
 Before surgery, the animals were deeply anesthetized via bath application with a solution of MS222 (120\,mg/l, PharmaQ, Fordingbridge, UK) buffered with Sodium Bicarbonate (120\,mg/l). The posterior anterior lateral line nerve (pALLN) was exposed by making a small cut into the skin covering the nerve. The cut was placed dorsal of the operculum just before the nerve descends towards the anterior lateral line ganglion (ALLNG). Those parts of the skin that were to be cut were locally anesthetized by cutaneous application of liquid lidocaine hydrochloride (20\,mg/ml, bela-pharm GmbH). During the surgery water supply was ensured by a mouthpiece to maintain anesthesia with a solution of MS222 (100\,mg/l) buffered with Sodium Bicarbonate (100\,mg/l). After surgery fish were immobilized by intramuscular injection of from 25\,$\micro$l to 50\,$\micro$l of tubocurarine (5\,mg/ml dissolved in fish saline; Sigma-Aldrich).
@@ -766,6 +778,20 @@ Leaky integrate-and-fire (LIF) models with a carrier were constructed to reprodu
   \end{equation}
 with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
 
+%\subsection*{Stimuli for the model}
+The model neurons were driven with similar stimuli as the real neurons in the original experiments. To mimic the interaction with one or two foreign animals the receiving fish's EOD (\Eqnref{eq:eod}) was normalized to an amplitude of one and the respective EODs of a second or third fish were added.%\Eqnref{ eq.\,\ref{eq:eod}
+
+The random amplitude modulation (RAM) input to the model was created by drawing random amplitude and phases from Gaussian distributions for each frequency component in the range 0--300 Hz. An inverse Fourier transform was applied to get the final amplitude RAM time-course. The input to the model was then
+\begin{equation}
+    \label{eq:ram_equation}
+    y(t) = (1+ s(t)) \cdot \cos(2\pi f_{EOD} t)
+\end{equation}
+\note{fix stimulus x and y notation and RAM}
+From each simulation run, the first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
+% \subsection{Second-order susceptibility analysis of the model}
+% %\subsubsection{Model second-order nonlinearity}
+% The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
+
 In the model, the input $x(t)$ was then first thresholded to model the synapse between the primary receptor cells and the afferent. 
 \begin{equation}
   \label{eq:threshold2}
@@ -824,27 +850,13 @@ The model's ODEs were integrated by the Euler forward method with a time-step of
 \subsection*{Model parameters}\label{paramtext}
 The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step-like increases and decreases in EOD amplitude (onset-state and steady-state responses, effective adaptation time constant). For each simulation, the start parameters $A$, $V_{d}$ and $V_{m}$ were drawn from a random starting value distribution, estimated from a 100\,s baseline simulation after an initial 100\,s of simulation that was discarded as a transient.
 
-\subsection*{Stimuli for the model}
-The model neurons were driven with similar stimuli as the real neurons in the original experiments. To mimic the interaction with one or two foreign animals the receiving fish's EOD (\Eqnref{eq:eod}) was normalized to an amplitude of one and the respective EODs of a second or third fish were added.%\Eqnref{ eq.\,\ref{eq:eod}
-
-The random amplitude modulation (RAM) input to the model was created by drawing random amplitude and phases from Gaussian distributions for each frequency component in the range 0--300 Hz. An inverse Fourier transform was applied to get the final amplitude RAM time-course. The input to the model was then
-\begin{equation}
-    \label{eq:ram_equation}
-    y(t) = (1+ s(t)) \cdot \cos(2\pi f_{EOD} t)
-\end{equation}
-\note{fix stimulus x and y notation and RAM}
-From each simulation run, the first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
-% \subsection{Second-order susceptibility analysis of the model}
-% %\subsubsection{Model second-order nonlinearity}
-
-% The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
 \subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
-According to the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$ \note{hier umschreiben das xi muss ein RAM sein} and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{noise}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
-%\sqrt{\rho \, 2D \,c_{signal}} \cdot \xi(t) 
+According to the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$ \note{hier umschreiben das xi muss ein RAM sein} and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
+%\sqrt{\rho \, 2D \,c_{\rm{signal}}} \cdot \xi(t) 
 
-%(1-c_{signal})\cdot\xi$c_{noise} = 1-c_{signal}$
-%c_{signal} \cdot \xi
+%(1-c_{\rm{signal}})\cdot\xi$c_{\rm{noise}} = 1-c_{\rm{signal}}$
+%c_{\rm{signal}} \cdot \xi
 \begin{equation}
     \label{eq:ram_split}
     y(t) = (1+ s_\xi(t)) \cdot \cos(2\pi f_{EOD} t)
@@ -858,18 +870,18 @@ According to the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} the tot
 
 \begin{equation}
   \label{eq:Noise_split_intrinsic}
-  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{noise}} \cdot \xi(t)
+  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)
 \end{equation}
 % das stimmt so, das c kommt unter die Wurzel!
 
 %\begin{equation}
 %  \label{Noise_split_intrinsic}
-%  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{noise}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
+%  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{\rm{noise}}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
 %\end{equation}
 
 
 
-A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\Eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[iii]{A}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
+A big portion of the total noise was assigned to the signal component ($c_{\rm{signal}} = 0.9$) and the remaining part to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{\rm{noise}}(t)$ was scaled up by the factor $\rho$ (\Eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[iii]{A}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
 
 
 % See section \ref{lifmethods} for model and parameter description.
@@ -928,7 +940,7 @@ CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the sa
 
 \begin{figure*}[hp]%hp!
 \includegraphics[width=\columnwidth]{trialnr}  
-  \caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$. Grey -- 5th to 95th percentile of the second-order susceptibility.
+  \caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$. Black -- 99.9th percentile of the second-order susceptibility.
   }
 \end{figure*}
 

trialnr.py

@@ -139,11 +139,11 @@ def trialnr(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1], cells
                 if len(stack)> 0:
                     model_show, stack_plot, stack_plot_wo_norm = get_stack(cell, stack)
                     stacks.append(stack_plot)
-                    perc95.append(np.percentile(stack_plot,99.9))
+                    perc95.append(np.percentile(stack_plot,99.99))
                     perc05.append(np.percentile(stack_plot, 0))
                     median.append(np.percentile(stack_plot, 50))
                     stacks_wo_norm.append(stack_plot_wo_norm)
-                    perc95_wo_norm.append(np.percentile(stack_plot_wo_norm,99.9))
+                    perc95_wo_norm.append(np.percentile(stack_plot_wo_norm,99.99))
                     perc05_wo_norm.append(np.percentile(stack_plot_wo_norm, 0))
                     median_wo_norm.append(np.percentile(stack_plot_wo_norm, 50))
 
@@ -158,53 +158,18 @@ def trialnr(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1], cells
                     perc05_wo_norm.append(float('nan'))
                     median_wo_norm.append(float('nan'))
 
-        fig, ax = plt.subplots(2,2)
+        fig, ax = plt.subplots(1,1)
         #ax.plot(trial_nrs_here, perc05, color = 'grey')
-        ax[0,0].plot(trial_nrs_here, perc95, color = 'grey')
+        ax.plot(trial_nrs_here, perc95, color = 'grey', clip_on = False)
         #ax.plot(trial_nrs_here, median, color = 'black', label = 'median')
         #ax.scatter(trial_nrs_here, perc05, color = 'grey')
-        ax[0,0].scatter(trial_nrs_here, perc95, color = 'black')
+        ax.scatter(trial_nrs_here, perc95, color = 'black', clip_on = False)
 
-        ax[0,1].plot(trial_nrs_here, perc95_wo_norm / trial_nrs_here, color = 'grey')
-        #ax.plot(trial_nrs_here, median, color = 'black', label = 'median')
-        #ax.scatter(trial_nrs_here, perc05, color = 'grey')
-        ax[0,1].scatter(trial_nrs_here, perc95_wo_norm / trial_nrs_here, color = 'black')
-
-        ##################################
-        ax[1,0].plot(trial_nrs_here, median, color = 'grey')
-        #ax.plot(trial_nrs_here, median, color = 'black', label = 'median')
-        #ax.scatter(trial_nrs_here, perc05, color = 'grey')
-        ax[1,0].scatter(trial_nrs_here, median, color = 'black')
 
-        ax[1,1].plot(trial_nrs_here, median_wo_norm / trial_nrs_here, color = 'grey')
-        #ax.plot(trial_nrs_here, median, color = 'black', label = 'median')
-        #ax.scatter(trial_nrs_here, perc05, color = 'grey')
-        ax[1,1].scatter(trial_nrs_here, median_wo_norm / trial_nrs_here, color = 'black')
-        #ax[2].plot(trial_nrs_here, perc95, color = 'grey')
-        #ax.plot(trial_nrs_here, median, color = 'black', label = 'median')
-        #ax.scatter(trial_nrs_here, perc05, color = 'grey')
-        #ax[2].scatter(trial_nrs_here, perc95, color = 'black')
-        #ax.scatter(trial_nrs_here, median, color = 'black', label = 'median')
-        #ax.fill_between(trial_nrs_here, perc05, perc95, color='grey')
-
-        ax[0,0].set_xscale('log')
-        ax[0,0].set_yscale('log')
-        ax[0,1].set_xscale('log')
-        ax[0,1].set_yscale('log')
-        ax[1,0].set_xscale('log')
-        ax[1,0].set_yscale('log')
-        ax[1,1].set_xscale('log')
-        ax[1,1].set_yscale('log')
-
-        ax[0,0].set_xlabel('Trials [$N$]')
-        ax[0,0].set_ylabel('$\chi_{2}$\,[Hz]')
-        ax[1,0].set_xlabel('Trials [$N$]')
-        ax[1,0].set_ylabel('$\chi_{2}$\,[Hz]')
-
-        ax[0,1].set_xlabel('Trials [$N$]')
-        ax[0,1].set_ylabel('$cross$\,[Hz]')
-        ax[1,1].set_xlabel('Trials [$N$]')
-        ax[1,1].set_ylabel('$cross$\,[Hz]')
+        ax.set_xscale('log')
+        ax.set_yscale('log')
+        ax.set_xlabel('Trials [$N$]')
+        ax.set_ylabel('$\chi_{2}$\,[Hz]')
         '''        ax = plt.subplot(1,3,2)
         ax.plot(trial_nrs_here, perc05_wo_norm, color = 'grey')
         ax.plot(trial_nrs_here, perc95_wo_norm, color = 'grey')

utils_all_down.py

@@ -1817,10 +1817,6 @@ def model_sheme_split(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_len
 
         # FIRST Row: Rectified stimulus
 
-        plot_point = [[], [], [], 'yes']
-        # counter, ax_rec[counter_here], ff, pp, axp = plot_rec_stimulus(eod_fr,grid_lowpass, time_transform, stimulus_here, color, color,
-        #                            time, counter, eod_fr, titles, g, deltat,
-        #                            fft_type, nfft, log, counterp,shift = shift, delta_f  = delta_f,  plot_point =   plot_point[counter_here], lw = lw, xlim = xlim)
         power_extra = False
         wr2 = [1, 1.2]
         if power_extra:
@@ -1840,7 +1836,7 @@ def model_sheme_split(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_len
                                                                    shift, nfft, time_transform, color, fft_type, lw=lw,
                                                                    xlim=xlim)
 
-        # ax_rec[counter.]
+
         ax_rec[counter_here].show_spines('b')
         remove_xticks(ax_rec[counter_here])
         axt_stims.append(ax_rec[counter_here])
@@ -1853,33 +1849,13 @@ def model_sheme_split(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_len
             axps_stimulus.append(axp)
         colors_chosen.append(color)
 
-        # if counter_here == 0:
-        #    ax_rec[counter_here].text(-7, 0, '0', color='black', ha='center', va='center')
-
-        # if colorful_title:
-        #    rainbow_title(fig, ax_rec[counter_here], titles[g], add_pos[g], color_add_pos[g])
-        # else:#add_pos[g]
 
         ax_rec[counter_here].text(0, 0.935, titles[c] + '\n' + r'$\rm{CV}=%s$' % (
             np.round(np.std(np.diff(spike_times)) / np.mean(np.diff(spike_times)), 2)) + '\n$f_{Base}=%s$' % (
                                       int(np.round(1 / np.mean(np.diff(spike_times))))) + '\,Hz',
                                   transform=ax_rec[counter_here].transAxes,
                                   va='bottom')  # verticalalignment='right',
-        # And plot correspoding sheme
-        # if g == 0:
-
-        # REMAINING Rows: dendridic filter / LIF /EIF stimulus
-
-        # for ee, exponential in enumerate(exponentials):
-
-        # model
-        # v_offset,  model_params, load_name= implement_three_core(cell,amp_frame, titles, g, cell_nr = cell_nr)
-        # for hard coding the offset here i check the change of the baseline
-
-        # if (ee == 0):
-        # SECOND Row: Dendridic Low pass filter
-        plot_point = [[], [], [], 'yes']
-        # ax_low[counter_here]
+        #embed()
         grid_lowpass = gridspec.GridSpecFromSubplotSpec(1, col,
                                                         subplot_spec=grid_power_col[1], wspace=ws, hspace=1.3,
                                                         width_ratios=wr)
@@ -1991,15 +1967,7 @@ def model_sheme_split(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_len
 
         # plt.show()
         counter_here += 1
-
-    devide = np.max(np.max(pps))
-    # plot_points = np.array([[], [], [], 'yes',
-    #                       [], [], [], 'yes'
-    #                     ,'yes','yes',[],'yes'
-    #                ,'yes','yes',[],'yes'])
-    # von oben nach unten von links nach rechts
-
-    # plot psd with shared log lim
+        #embed()
 
     ####################################
     # cut first parts
@@ -2052,11 +2020,9 @@ def model_sheme_split(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_len
     ax = np.transpose([axt_stims[0::2], axt_stims[1::2], ax_noise, axt_IF1, axps_stimulus])
 
     fig = plt.gcf()
-    # fig.tag(ax, xoffs=-3.5, yoffs = 1.5)
     nr = 1.7  # 2#.24
     tag2(fig, axshemes, xoffs=-3.5, yoffs=5.5)
-    tag2(fig, np.transpose(ax), xoffs=-3.5)  # yoffs = [5.5,nr,nr,nr,nr-0.2]
-    # get_ylim_same
+    tag2(fig = fig, axes = np.transpose(ax), xoffs=-3.5)  # yoffs = [5.5,nr,nr,nr,nr-0.2]
 
     #########################################################
     # plot psds
@@ -2130,6 +2096,7 @@ def model_sheme_split2(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_le
     # a_fes =
     nrs = [1, 2, 3, 4]
     a_fes = [0, 0.02, 0]  # alpha\alpha0.1,
+    a_fes = [0, 0.02, 0]  # alpha\alpha0.1,
     titles = ['Baseline', r'Contrast$\,=2\,\%$', noise_name()]  # r'Contrast$\,=20\,\%$',
 
     #########################################################################################
@@ -2201,25 +2168,18 @@ def model_sheme_split2(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_le
 
         noise_final_c, spike_times, stimulus, stimulus_here, time, v_dent_output, v_mem_output, frame = get_flowchart_params(
             a_fes, a_fr, c, c_sig, cell, deltat, eod_fr, model_params, stimulus_length, v_offset, var_types, eod_fe,
-            color_p1, color_p3, mult_nr=mult_nr, load_name=load_name, exp_tau=exp_tau, v_exp=v_exp)
+            color_p1, color_p3, mult_nr=mult_nr, load_name=load_name, exp_tau=exp_tau, v_exp=v_exp, redo = False)
         print(len(stimulus_here))
+        extracted, _ = extract_am(v_dent_output, stimulus_here / 1000, sampling=1 / deltat, eodf=eod_fr, norm=False)
+        print('noise var'+str(np.var(noise_final_c))+' stimulus var '+str(np.var(extracted))+' CV '+str(np.round(np.std(np.diff(spike_times)) / np.mean(np.diff(spike_times)), 2)))
         ##############################################
-        # titles = [titles[1]]
-        # for g, stimulus_here in enumerate([stimuli[1]]):
-        add = 0
         color = colors[counter_here]
 
         # FIRST Row: Rectified stimulus
 
-        plot_point = [[], [], [], 'yes']
-        # counter, ax_rec[counter_here], ff, pp, axp = plot_rec_stimulus(eod_fr,grid_lowpass, time_transform, stimulus_here, color, color,
-        #                            time, counter, eod_fr, titles, g, deltat,
-        #                            fft_type, nfft, log, counterp,shift = shift, delta_f  = delta_f,  plot_point =   plot_point[counter_here], lw = lw, xlim = xlim)
         power_extra = False
-        wr2 = [1, 1.2]
         if power_extra:
             wr = [1, 1.2]
-
             col = 2
         else:
             col = 1
@@ -2479,7 +2439,7 @@ def model_sheme_split2(grid_sheme_orig, time_transform=1000, ws=0.1, stimulus_le
 
 def get_flowchart_params(a_fes, a_fr, c, c_sig, cell, deltat, eod_fr, model_params, stimulus_length, v_offset,
                          var_types, eod_fe=[750], color_p1='black', color_p3='black', colorful_title=False, mult_nr=1,
-                         load_name='models_big_fit_d_right.csv', exp_tau=0.1, v_exp=1):  # v_exp=1, exp_tau=0.1
+                         load_name='models_big_fit_d_right.csv', exp_tau=0.1, v_exp=1, redo = False):  # v_exp=1, exp_tau=0.1
     # ok das hier scheint umständlich, aber ich hab einmal die eod_fe schleife und einmal die stimulus schleife
     # einfach eine stimulus schleife zu machen würde mehrere änderungen bedeutetn
     try:
@@ -2513,18 +2473,19 @@ def get_flowchart_params(a_fes, a_fr, c, c_sig, cell, deltat, eod_fr, model_para
                                                                                                           stimulus_length,
                                                                                                           v_exp,
                                                                                                           v_offset,
-                                                                                                          version_comp)
+                                                                                                          version_comp, redo = redo)
     time = np.arange(0, len(stimulus_here) * deltat, deltat)
 
     return noise_final_c, spike_times, stimulus, stimulus_here, time, v_dent_output, v_mem_output, frame
 
 
 def save_flowchart_susept(var_type, cell, a_fes, c, c_sig, deltat, eod_fish_rs, eod_fr, exp_tau, g, load_function,
-                          load_name, model_params, stimulus_length, v_exp, v_offset, version_comp, exponential=''):
+                          load_name, model_params, stimulus_length, v_exp, v_offset, version_comp, exponential='', redo = False):
     save_arrays = load_function + '_' + var_type + '_' + '_c_sig_' + str(c_sig) + '_a_fe_' + str(a_fes[c]) + '.csv'
     save_spikes = load_function + '_' + var_type + '_' + 'spikes_c_sig_' + str(c_sig) + '_a_fe_' + str(
         a_fes[c]) + '.csv'
-    redo = False
+    
+    #TODO: hier gibt es noch ein Problem!
     if (version_comp == 'code') | (version_comp == 'develop'):
         if (not os.path.exists(save_spikes)) | (redo == True):
             print('redo flowchart')
@@ -2731,7 +2692,7 @@ def save_flowchart_susept(var_type, cell, a_fes, c, c_sig, deltat, eod_fish_rs,
 
                 carrier_RAM[carrier_RAM < 0] = 0
                 stimulus_here = carrier_RAM
-
+                #embed()
             frame = pd.DataFrame()
 
             frame['v_dent_output'] = v_dent_output
@@ -2925,6 +2886,8 @@ def find_load_function():
                 if name in st[-1][1]:
                     load_function = st[-1][1].split('.py')[0].split(name)[-1].split('thesis')[-1]  # [1::]
                     found = True
+        if not found:
+            load_function = st[-1][1].split('.py')[0].split('suseptibility')[-1].split('thesis')[-1]  # [1::]
 
     # if '\\' in load_function:
     try:
@@ -2932,6 +2895,7 @@ def find_load_function():
     except:
         print('load_function_missing')
         embed()
+    #embed()
     return load_function
 
 
@@ -5252,7 +5216,7 @@ def egerland_noise(stimulus_length, deltat, cut_off, noise_added='', noise_stren
     # THIS is the one to find the correct CV
     elif 'additiv_cv_adapt_factor' in var:
         # 'additiv_cv_adapt_factor_scaled'
-        # embed()
+        #embed()
         frame = pd.read_csv(load_path + '.csv', index_col=0)  # "2012-12-21-ai-invivo-1","2012-06-27-an-invivo-1",
         try:
             height = float(frame.loc['height', cell])
@@ -5424,7 +5388,7 @@ def egerland_noise(stimulus_length, deltat, cut_off, noise_added='', noise_stren
     stimulus_spikes = noise_bef * 1
     white_noise_eng_abs = np.abs(noise_bef_cut_off)
 
-    # embed()
+    #embed()
     test = False
     if test:
         test_egerland()
@@ -7654,6 +7618,7 @@ def simulate(cell, v_offset, stimulus, emb=False, cut=True, EODf='eod_fr', delta
     else:
         noise_strength_new = np.sqrt(noise_strength * 2)
         noise_final *= noise_strength_new / np.sqrt(deltat)  # 0.05370289258320868 0.0015532069917408744
+    #embed()
 
     v_mem, adapt, v_dend = reshuffled_start_vals(reshuffle, a_zero, stimulus, v_zero, reshuffle_start_vals, cell,
                                                 reshuffle_stimulus_length_orig, reshuffle_transient)
@@ -13839,7 +13804,10 @@ def tag2(fig=None, axes=None, xoffs=None, yoffs=None,
             xoffs = fig.tags_xoffs
         else:
             xoffs = xo
-        xoffs = [xoffs] * len(axes_list)
+            xoffs = [xoffs] * len(axes_list)
+        if len(axes_list) != len(xoffs):
+            xoffs = xo
+            xoffs = [xoffs] * len(axes_list)
     elif (type(xoffs) != float) & (type(xoffs) != int):
         xoffs = np.array(xoffs)
         try:
@@ -13855,11 +13823,17 @@ def tag2(fig=None, axes=None, xoffs=None, yoffs=None,
         xoffs *= 0.6 * fs / w
         xoffs = [xoffs] * len(axes_list)
     if yoffs == 'auto':
+
         if hasattr(fig, 'tags_yoffs'):
             yoffs = fig.tags_yoffs
         else:
             yoffs = yo - 1.0 / h  # minus one pixel
-        yoffs = [yoffs] * len(axes_list)
+            yoffs = [yoffs] * len(axes_list)
+
+        if len(axes_list) != len(yoffs):
+            yoffs = yo - 1.0 / h  # minus one pixel
+            yoffs = [yoffs] * len(axes_list)
+        #embed()
     elif (type(yoffs) != float) & (type(yoffs) != int):
         yoffs = np.array(yoffs)
         try:
@@ -13893,8 +13867,12 @@ def tag2(fig=None, axes=None, xoffs=None, yoffs=None,
         except:
             print('y something')
             embed()
+        #try:
         if y >= 1.0:
             y = 1.0
+        #except:
+        #    print('y something')
+        #    embed()
         ax.text(x, y, l, transform=fig.transFigure, ha='left', va='top', **fkwargs)
         count += 1