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+216,1.8101 +217,1.81665 +218,1.824 +219,1.83385 +220,1.84135 +221,1.85 +222,1.85955 +223,1.8672000000000002 +224,1.8737000000000001 +225,1.8843500000000002 +226,1.89405 +227,1.90165 +228,1.9103 +229,1.91785 +230,1.92645 +231,1.93395 +232,1.94045 +233,1.949 +234,1.9587 +235,1.9673500000000002 +236,1.9781000000000002 +237,1.98465 +238,1.9921000000000002 diff --git a/data_overview_mod.pdf b/data_overview_mod.pdf index e9baa5c..a09d92e 100644 Binary files a/data_overview_mod.pdf and b/data_overview_mod.pdf differ diff --git a/data_overview_mod.py b/data_overview_mod.py index 2dcc426..75de526 100644 --- a/data_overview_mod.py +++ b/data_overview_mod.py @@ -12,7 +12,7 @@ from threefish.core_load import save_visualization from threefish.core_bursts import setting_overview_score from threefish.core_plot_labels import basename_small, label_NLI_scorename2_small, label_pearson, label_stimname_small, \ make_log_ticks -from threefish.utils1_suscept import plt_specific_cells, start_name +from threefish.utils1_RAM import plt_specific_cells, start_name from scipy import stats try: from plotstyle import plot_style, spines_params diff --git a/flowchart.pdf b/flowchart.pdf index 8c6b4fc..3c15b42 100644 Binary files a/flowchart.pdf and b/flowchart.pdf differ diff --git a/flowchart.png b/flowchart.png index 93b9b53..beee8b0 100644 Binary files a/flowchart.png and b/flowchart.png differ diff --git a/flowchart.py b/flowchart.py index 68e419d..63f93fc 100644 --- a/flowchart.py +++ b/flowchart.py @@ -5,7 +5,7 @@ # from threefish.utils0 import default_settings, resave_small_files # from plt_RAM import model_and_data, model_and_data_sheme, model_and_data_vertical2 -from threefish.plot_suscept import model_sheme_split2 +from threefish.plot_suscept import flowchart_core # from threefish.utils0 import default_settings, resave_small_files # from plt_RAM import model_and_data, model_and_data_sheme, model_and_data_vertical2 # from threefish.utils0 import default_settings, resave_small_files @@ -14,21 +14,19 @@ import matplotlib.gridspec as gridspec from plotstyle import plot_style from threefish.core_load import save_visualization -from threefish.plot_suscept import model_sheme_split2 +from threefish.plot_suscept import flowchart_core from threefish.defaults import default_figsize def flowchart(): - #fig = plt.figure(figsize=(6.8, 5.5)) plot_style() default_figsize(column=2, length=4.7) #5.3 5.8 grid_orig = gridspec.GridSpec(1, 1, wspace=0.15, bottom=0.07, hspace=0.1, left=0.05, right=0.96, top=0.9) # , height_ratios = [0.4,3] - model_sheme_split2(grid_orig[0]) # grid_sheme grid_lower[3] - # model_sheme_only(grid[2]) + flowchart_core(grid_orig[0]) # grid_sheme grid_lower[3] + save_visualization() - #fig.savefig() if __name__ == '__main__': diff --git a/model_and_data.pdf b/model_and_data.pdf index fabd1e8..4093251 100644 Binary files a/model_and_data.pdf and b/model_and_data.pdf differ diff --git a/model_and_data.py b/model_and_data.py index 6bc62bd..484ff1a 100644 --- a/model_and_data.py +++ b/model_and_data.py @@ -10,9 +10,8 @@ from plotstyle import plot_style from threefish.core_plot_subplots import plot_lowpass2, plt_single_square_modl, plt_time_arrays from threefish.core_plot_suscept import perc_model_full, plt_data_susept from threefish.core_filenames import overlap_cells -from threefish.core_load import save_visualization -from threefish.core_reformat import get_flowchart_params, load_model_susept -from threefish.core_calc_model import resave_small_files +from threefish.core_load import resave_small_files, save_visualization +from threefish.core_reformat_RAM import get_flowchart_params, load_model_susept from threefish.core import find_folder_name from threefish.core_plot_labels import label_noise_name, nonlin_title, remove_xticks, remove_yticks, set_xlabel_arrow, \ set_ylabel_arrow, title_find_cell_add, xlabel_xpos_y_modelanddata diff --git a/model_full.pdf b/model_full.pdf index 4416b23..0bddc3d 100644 Binary files a/model_full.pdf and b/model_full.pdf differ diff --git a/model_full.png b/model_full.png index e4c793c..277bd81 100644 Binary files a/model_full.png and b/model_full.png differ diff --git a/model_full.py b/model_full.py index 7c1e9e3..26157c6 100644 --- a/model_full.py +++ b/model_full.py @@ -3,9 +3,11 @@ import numpy as np from matplotlib import gridspec, pyplot as plt from threefish.core_calc_fft import create_full_matrix2 -from threefish.core_reformat import chose_mat_max_value, convert_csv_str_to_float, get_axis_on_full_matrix, get_psds_ROC, get_transfer_from_model, \ +from threefish.core_reformat import chose_mat_max_value, get_axis_on_full_matrix, get_psds_ROC, get_transfer_from_model, \ load_b_public, load_stack_data_susept -from threefish.core_plot_subplots import colors_suscept_paper_dots, plt_model_full_model2, plt_psds_ROC, plt_RAM_perc +from threefish.core_reformat_RAM import convert_csv_str_to_float +from threefish.core_plot_subplots import colors_suscept_paper_dots, plt_model_full_model2, plt_model_letters, \ + plt_psds_ROC, plt_RAM_perc from threefish.core_load import save_visualization from threefish.core_values import vals_model_full from threefish.core_cell_choice import find_all_dir_cells @@ -16,7 +18,7 @@ from plotstyle import plot_style import time from threefish.defaults import default_diagonal_points, default_figsize from threefish.core_plot_colorbar import colorbar_outside, rescale_colorbar_and_values -from threefish.utils1_suscept import check_var_substract_method, chose_certain_group, \ +from threefish.utils1_RAM import check_var_substract_method, chose_certain_group, \ extract_waves, load_cells_three, predefine_grouping_frame, restrict_cell_type, \ save_arrays_susept from threefish.core_plot_labels import label_deltaf1, label_deltaf2, label_diff, label_sum, \ @@ -34,7 +36,7 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s plot_style() default_figsize(column=2, length=3.5)#2.7 - grid = gridspec.GridSpec(2, 4, wspace=0.15, bottom = 0.2, width_ratios = [3,1, 1.5,1.5], height_ratios = [1, 5], hspace=0.55, top=0.85, left=0.095, right=0.98)#hspace=0.25, + grid = gridspec.GridSpec(2, 2, wspace=0.55, bottom = 0.15, height_ratios = [2, 5], width_ratios = [1.2, 1], hspace=1, top=0.92, left=0.06, right=0.98)#hspace=0.25, @@ -48,7 +50,7 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s # model part ls = '--' lw = 0.5 - axm = plt.subplot(grid[1,0]) + axm = plt.subplot(grid[:,0]) axes.append(axm) @@ -62,7 +64,7 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s ############################################# # plot coherence cross = get_transfer_from_model(stack_saved) - axc = plt.subplot(grid[0, 0]) + axc = plt.subplot(grid[0, 1]) axc.plot(stack_saved.index, cross, color = 'black') axc.set_xlabel('Frequency [Hz]') axc.set_ylabel(xlabel_transfer_hz()) @@ -78,7 +80,7 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s ################# # power spectra data - log = 'log'#'log'#'log'#'log'#'log' + log = ''#'log'#'log'#'log'#'log'#'log' ylim_log = (-14.2, 3)#(-14.2, 3) nfft = 20000#2 ** 13 xlim_psd = [0, 300] @@ -105,7 +107,7 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s DF1_frmult, DF2_frmult = vals_model_full(val = 0.30833333333333335) #0.16666666666666666 grid0 = gridspec.GridSpecFromSubplotSpec(2, 2, wspace=0.15, hspace=0.4, - subplot_spec=grid[:,2::]) + subplot_spec=grid[1,1]) @@ -120,18 +122,19 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s diagonal = 'diagonal1' diagonal = '' plus_q = 'plus' # 'minus'#'plus'##'minus' - length = 2*40 # 5 + length = 1#2*40 # 5 reshuffled = '' # , alphas = [1,0.5] #a_size = 0.0065#0.0125#25#0.04#0.015 - a_size = 0.0005#und 0.0005 für das lineare das ist vielleicht für das nichtlienare 0.0065#0.0085#0.01#0.0085#125 # 0.0025# 0.01 0.005 und davor 0.025, funktioniert gut 0.0085 + a_size = 0.0065#und 0.0005 für das lineare das ist vielleicht für das nichtlienare 0.0065#0.0085#0.01#0.0085#125 # 0.0025# 0.01 0.005 und davor 0.025, funktioniert gut 0.0085 + # ATTENTION: Diese Zelle ('2012-07-03-ak-invivo-1') braucht längere Abschnitte, mindsetesn 5 Sekunden damit das Powerspectrum nicht so niosy ist! fr_noise, eod_fr_mm, axes2 = plt_model_full_model2(grid0, stack_final = stack_final, reshuffled=reshuffled, dev=0.0005, a_f1s=[a_size], af_2 = a_size, stimulus_length=length, plus_q=plus_q, stack_saved = stack_saved, - diagonal=diagonal, runs=1, nfft = nfft, xlim_psd = xlim_psd, ylim_log = ylim_log, - cells=[cell], dev_spikes = 'original', markers = markers, DF1_frmult = DF1_frmult, DF2_frmult = DF2_frmult, - log = log, ms = ms, clip_on = False, array_len = [1,1,1,1,1]) #arrays_len a_f1s=[0.02]"2012-12-13-an-invivo-1"'2013-01-08-aa-invivo-1' + diagonal=diagonal, runs=1, nfft = nfft, xlim_psd = xlim_psd, + cells=[cell], dev_spikes ='original', markers = markers, DF1_frmult = DF1_frmult, DF2_frmult = DF2_frmult, + log = log, ms = ms, clip_on = False, trials_nr_all= [1, 1, 1, 1, 1]) #arrays_len a_f1s=[0.02]"2012-12-13-an-invivo-1"'2013-01-08-aa-invivo-1' @@ -145,32 +148,11 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s plt_model_letters(DF1_frmult, DF2_frmult, axm, color012, color01_2, fr_noise, markers) fig = plt.gcf()#[axes[0], axes[3], axes[4]] - fig.tag([axes[0], axes2[0], axes2[1], axes2[2], axes2[3]], xoffs=-4.5, yoffs=1.2) # ax_ams[3], + fig.tag([axes[0], axc, axes2[0], axes2[1], axes2[2], axes2[3]], xoffs=-4.5, yoffs=1.2) # ax_ams[3], #plt.show() save_visualization() -def plt_model_letters(DF1_frmult, DF2_frmult, ax, color012, color01_2, fr_noise, markers): - letters_plot = True - if letters_plot: - letters = ['B', 'C', 'D', 'E'] - for f in range(len(DF1_frmult)): - ax.text((fr_noise * DF1_frmult[f]), (fr_noise * DF2_frmult[f] - 1), letters[f], color=color012, ha='center', - va='center') # , alpha = alphas[f] - ax.text((fr_noise * DF1_frmult[f] - 1), -(fr_noise * DF2_frmult[f] - 1), letters[f], color=color01_2, - ha='center', va='center') # , alpha = alphas[f] - - else: - 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() - - def plt_data_matrix(ax, axes, cell, grid, ls, lw, perc, stack_final): ax = plt.subplot(grid[0]) axes.append(ax) @@ -551,7 +533,7 @@ def plt_data_full_model(c1, chose_score, detections, devs, dfs, end, grid, mult_ #embed() ax00, fr_isi = plt_psds_ROC(arrays, ax00, ax_ps, cell, colors_p, f, grid0, group_mean, nfft, p_means, p_means_all, ps, 4, - spikes_pure, time_array, range_plot=[3], names=names, + spikes_pure, range_plot=[3], names=names, ax01=ax00, ms = ms, clip_on = clip_on, xlim_psd=xlim_psd, alphas = alphas, marker = markers[gg], choice = choice, labels = labels, ylim_log=ylim_log, log=log, text_extra=False) # [arrays[-1]]arrays, ax00, ax_ps, cell, colors_p, f, [-1]grid0, group_mean, nfft, p_means, p_means_all, ps, row,spikes_pure, time_array, ax00.show_spines('b') diff --git a/motivation.pdf b/motivation.pdf index c362fd9..e53fbe6 100644 Binary files a/motivation.pdf and b/motivation.pdf differ diff --git a/motivation.py b/motivation.py index 64f0689..aa31467 100644 --- a/motivation.py +++ b/motivation.py @@ -8,7 +8,7 @@ from threefish.defaults import default_figsize from threefish.core_time import cr_spikes_mat from threefish.core import find_code_vs_not import time -from threefish.utils1_suscept import check_var_substract_method, chose_certain_group, circle_plot, extract_waves, load_cells_three, predefine_grouping_frame, restrict_cell_type, save_arrays_susept, ws_nonlin_systems +from threefish.utils1_RAM import check_var_substract_method, chose_certain_group, circle_plot, extract_waves, load_cells_three, predefine_grouping_frame, restrict_cell_type, save_arrays_susept, ws_nonlin_systems from threefish.core_plot_labels import title_motivation from threefish.core_plot_subplots import colors_suscept_paper_dots, plot_arrays_ROC_psd_single3, plot_shemes4 from threefish.core_load import save_visualization diff --git a/susceptibility1.tex b/susceptibility1.tex index a5b94b7..0c10dc3 100644 --- a/susceptibility1.tex +++ b/susceptibility1.tex @@ -543,7 +543,7 @@ Nonlinear encoding as quantified by the second-order susceptibility is expected \begin{figure*}[t] \includegraphics[width=\columnwidth]{cells_suscept} -\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, \notejb{cutoff frequency?}) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) reflects the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, of the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities. \notejb{``Calculated based on the first frozen noise repeat.'' Really? Only one trial has been used for computing the susceptibilities?}} +\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, \notejb{cutoff frequency?}) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) reflects the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, of the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities. \notejb{``Calculated based on the first frozen noise repeat.'' Really? Only one trial has been used for computing the susceptibilities?} \noteab{Yes. In the population statistics of the last 20 years I was trying to maximize the number cells and maximize the duration. In the end the susceptibility is calculated based on 10 seconds. The susceptibility has always to be calculated with newly initiated noise, with 10 seconds and 0.5 nfft durations this results in 20 trials. For the overview figure the mean of all NLI values for each frozen noise was calculated. It is not possible to take the mean over several frozen noise matrices since the baseline properties sometimes change and this disrupts the second order susceptibility.}} \end{figure*} Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus driven responses of sensory neurons by means of transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly to fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper?}. @@ -577,7 +577,7 @@ In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fi \begin{figure*}[t] \includegraphics[width=\columnwidth]{model_and_data} - \caption{\label{model_and_data} Estimating second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of the\notejb{same as in Fig 2?} low-CV P-unit \notejb{``(cell 2012-07-03-ak)'' werden cell Nummern irgendwo noch verwendet? Am besten sollten die cell Nummern in jeder Abbildung genannt werden.} driven with a weak RAM stimulus with 2.5\,\% contrast. Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ 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. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility. \notejb{to methods: ``Note that the signal component \signalnoise{} 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 intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).''} + \caption{\label{model_and_data} Estimating second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of the\notejb{same as in Fig 2?} \noteab{No it is a different cell. In this cell we do not have several contrasts recorded, but a model.} low-CV P-unit (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters) \noteab{auch für die Datenzellen die identifier raussuchen?} driven with a weak RAM stimulus. The stimulus strength was with 2.5\,\% contrast in the electrophysiologically recorded P-unit. The contrast for the P-unit model was with 0.009\,\%, thus reproducing the CV of the electrophysiolgoical recorded P-unit during the 2.5\,\% contrast stimulation. Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ 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. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility. \notejb{to methods: ``Note that the signal component \signalnoise{} 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 intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).''} } \end{figure*} @@ -597,7 +597,7 @@ We estimated the second-order susceptibility of P-unit responses using RAM stimu \begin{figure*}[t] \includegraphics[width=\columnwidth]{model_full} - \caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. Baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters correspond to the beat frequencies used for the stimulation with pure sine-waves in the subsequent panels and indicate the sum/difference of those beat frequencies. \figitem{B--E} Power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three fish scenario). \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{C} None of the two beat frequencies matches \fbase{}.} + \caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. Baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters correspond to the beat frequencies used for the stimulation with pure sine-waves in the subsequent panels and indicate the sum/difference of those beat frequencies. \figitem{B--E} Power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three fish scenario). \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{C} None of the two beat frequencies matches \fbase{}.} \end{figure*} However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\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 nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff) where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). diff --git a/susceptibility1.tex.bak b/susceptibility1.tex.bak index b7036df..1fb43f7 100644 --- a/susceptibility1.tex.bak +++ b/susceptibility1.tex.bak @@ -543,31 +543,31 @@ Nonlinear encoding as quantified by the second-order susceptibility is expected \begin{figure*}[t] \includegraphics[width=\columnwidth]{cells_suscept} -\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. 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} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) of the same P-unit. The stimulus contrast reflects the strength of the AM. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 10\% (light purple) and 20\% contrast (dark purple) RAM 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:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, \notejb{cutoff frequency?}) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) reflects the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, of the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities. \notejb{``Calculated based on the first frozen noise repeat.'' Really? Only one trial has been used for computing the susceptibilities?} \noteab{Yes. In the population statistics of the last 20 years I was trying to maximize the number cells and maximize the duration. In the end the susceptibility is calculated based on 10 seconds. The susceptibility has always to be calculated with newly initiated noise, with 10 seconds and 0.5 nfft durations this results in 20 trials. For the overview figure the mean of all NLI values for each frozen noise was calculated. It is not possible to take the mean over several frozen noise matrices since the baseline properties sometimes change and this disrupts the second order susceptibility.}} \end{figure*} Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus driven responses of sensory neurons by means of transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly to fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper?}. -The second-order susceptibility, \Eqnref{eq:susceptibility}, quantifies for each combination of two stimulus frequencies \fone{} and \ftwo{} amplitude and phase of the stimulus-evoked response at the sum \fsum{} of these two frequencies. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that simulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF driven in the super-threshold regime with two sinewave stimuli, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} excatly match the neuron's baseline firing rate \fbase{} \cite{Voronenko2017}. Only then additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). +The second-order susceptibility, \Eqnref{eq:susceptibility}, quantifies amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that simulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF driven in the super-threshold regime with two sinewave stimuli, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} excatly match the neuron's baseline firing rate \fbase{} \cite{Voronenko2017}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). % DAS GEHOERT IN DIE DISKUSSION: % 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}). -For the low-CV P-unit we observe a band of slightly elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}). This structure vanishes for the stronger stimulus (\subfigref{fig:cells_suscept}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected susceptibility values onto the diagonal by taking the mean over all anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). For the low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:cells_suscept}{G}, dot on top line). For the higher RAM contrast, however, this peak vanishes and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude but also increases the total noise in the system. Increased noise is known to linearize signal transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease. +For the low-CV P-unit we observe a band of slightly elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}). This structure vanishes for the stronger stimulus (\subfigref{fig:cells_suscept}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected susceptibility values onto the diagonal by taking the mean over all anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). For the low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:cells_suscept}{G}, dot on top line). For the higher RAM contrast, however, this peak vanishes and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude, but also increases the total noise in the system. Increased noise is known to linearize signal transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease. In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounced nonlinearities even at low stimulus contrasts (\figrefb{fig:cells_suscept_high_CV}). \begin{figure*}[t] \includegraphics[width=\columnwidth]{ampullary} - \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. + \caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. \figitem{C} Bad-limited white noise stimulus (top, \notejb{cutoff frequency?}) added to the fish's self-generated electric field and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \Eqnref{linearencoding_methods}, of the responses to stimulation with 2\,\% (light green) and 20\,\% contrast (dark green). \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both stimulus contrasts as indicated. Pink triangles indicate baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. \notejb{`` Calculated based on the first frozen noise repeat.''} } \end{figure*} \subsection*{Ampullary afferents exhibit strong nonlinear interactions} -Irrespective of the CV, neither of the two example P-units shows the complete expected structure of nonlinear interactions. Electric fish 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 their baseline ISIs (CV$_{\text{base}}=0.08$--$0.22$)\notejb{in the figure the CV is 0.07, below this range!}\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \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 low-contrast noise stimulus (note: this is not an AM but a stimulus that is added to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projection onto the diagonal looses its distinct peak at \fsum{} and its overal level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). +Irrespective of the CV, neither of the two example P-units shows the complete expected structure of nonlinear interactions. Electric fish 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 their baseline ISIs (CV$_{\text{base}}=0.08$--$0.22$)\notejb{in the figure the CV is 0.07, below this range!}\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \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 low-contrast noise stimulus (note: this is not an AM but a stimulus that is added to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility, where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal looses its distinct peak at \fsum{} and its overal level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). @@ -577,31 +577,28 @@ In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fi \begin{figure*}[t] \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 2.5\% contrast. The plot shows that \suscept{} surface for $N=11$. 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). The signal noise component \signalnoise{} and the intrinsic noise are shown in the center and bottom. \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 $10^6$ 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 are treated as signal (center trace, \signalnoise) while the intrinsic noise is reduced (see methods for details). Note that the signal component \signalnoise{} 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 intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown). + \caption{\label{model_and_data} Estimating second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of the\notejb{same as in Fig 2?} \noteab{No it is a different cell. In this cell we do not have several contrasts recorded, but a model.} low-CV P-unit (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters) \notesr{auch für die Datenzellen die identifier raussuchen?} driven with a weak RAM stimulus. The stimulus strength was with 2.5\,\% contrast in the electrophysiologically recorded P-unit. The contrast for the P-unit model was with 0.009\,\%, thus reproducing the CV of the electrophysiolgoical recorded P-unit during the 2.5\,\% contrast stimulation. Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ 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. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility. \notejb{to methods: ``Note that the signal component \signalnoise{} 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 intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).''} } \end{figure*} -One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore responses to only a limited number of trials, repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental 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, and 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 as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). +One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore responses to only a limited number of trials, i.e. repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental 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, and 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 trials as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). -In simulations of the model we can increase the number of repetitions beyond what would be experimentally possible, here to one million (\subfigrefb{model_and_data}\,\panel[iii]{B}). The estimate of the second-order susceptibility indeed improves. It gets less noisy and the diagonal at \fsum{} is emphasized. However, the expected vertical and horizontal lines at \foneb{} and \ftwob{} are still missing. +In simulations of the model we can increase the number of trials beyond what would be experimentally possible, here to one million (\subfigrefb{model_and_data}\,\panel[iii]{B}). The estimate of the second-order susceptibility indeed improves. It gets less noisy and the diagonal at \fsum{} is emphasized. However, the expected vertical and horizontal lines at \foneb{} and \ftwob{} are still missing. Using a broadband stimulus increases the effective input-noise level and this may linearize signal transmission and suppress potential nonlinear responses \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full expected structure of the second-order susceptibility 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 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 \signalnoise, 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]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). 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]{C}), 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. +Note, that the increased number of trials goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{C}), that saturate in its peak values for $N>10^5$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of an estimate of the second-order susceptibility 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. 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). - - \subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation} We estimated the second-order susceptibility of P-unit responses using RAM stimuli. In particular, we found pronounced nonlinear responses in the limit of weak stimulus amplitudes. How do these findings relate to the situation of two pure sinewave stimuli with finite amplitudes that approximates the interference of EODs of real animals? For the P-units the relevant signals are the beat frequencies \bone{} and \btwo{} that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line of the second-order susceptibility estimated for a vanishing external RAM stimulus (\subfigrefb{model_and_data}\,\panel[iii]{C}). In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red dot, \subfigrefb{fig:motivation}{D}), that is not covered by the so-far shown part of the second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iii]{C}), in which only the response at the sum of the two stimulus frequencies is addressed. % less prominent, \begin{figure*}[t] \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 $N=10^6$ stimulus realizations and the intrinsic noise split (see methods). The position of the orange letters corresponds to the beat frequencies used for the pure sinewave stimulation in the subsequent panels. The position of the red letters correspond to the difference of those beat frequencies. \figitem{B--E} Power spectral density of model responses under pure sinewave stimulation. \figitem{B} The two beat frequencies are in sum equal to \fbase{}. \figitem{C} The difference of the two beat frequencies is equal to \fbase{}. \figitem{D} Only one beat frequency is equal to \fbase{}. \figitem{C} No beat frequencies is equal to \fbase{}.} + \caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. Baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters correspond to the beat frequencies used for the stimulation with pure sine-waves in the subsequent panels and indicate the sum/difference of those beat frequencies. \figitem{B--E} Power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three fish scenario). \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{C} None of the two beat frequencies matches \fbase{}.} \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}. However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\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 nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff) where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). Is it possible based on the second-order susceptibility estimated by means of RAM stimuli (\subfigrefb{fig:model_full}{A}) to predict nonlinear responses in a three-fish setting? We can test this by stimulating the same model with two beats with weak amplitudes \notejb{specify the contrast at least in the figure legend!} (\subfigrefb{fig:model_full}{B--E}). If we choose a frequency combination where the sum of the two beat frequencies is equal to the model's baseline firing rate \fbase{}, a peak at the sum of the two beat frequencies appears in the power spectrum of the response (\subfigrefb{fig:model_full}{B}), as expected from \suscept. If instead we choose two beat frequencies that differ by \fbase{}, a peak is present at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one beat frequency is equal to \fbase{}, both a peak at the sum and at the difference frequency is present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met, neither a peak at the sum nor at the difference of the two beat frequencies appears (\subfigrefb{fig:model_full}{E}). @@ -611,24 +608,24 @@ Is it possible based on the second-order susceptibility estimated by means of RA % TO DISCUSSION: %Even though the second-order susceptibilities here were estimated from 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}. -\begin{figure*}[!ht] +\begin{figure*}[tp] \includegraphics[width=\columnwidth]{data_overview_mod} - \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C, E}) and ampullary cells (\panel{B, D, F}). 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. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \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} \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. - % The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:data_overview_mod}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles -- \figrefb{fig:cells_suscept_high_CV}). - % Several of the recorded neurons contribute with two samples to the population analysis as their responses have been recorded to two different contrast of the same RAM stimulus. 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). + \caption{\label{fig:data_overview} Nonlinear responses in P-units and ampullary cells. The second-order susceptibility is condensed into the peakedness of the nonlinearity, \nli{} \Eqnref{eq:nli_equation}, that relates the amplitude of the projected susceptibility at a cell's baseline firing rate to its median (see \subfigrefb{fig:cells_suscept}{G}). Each of the recorded neurons contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \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} \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. + % The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:data_overview}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles -- \figrefb{fig:cells_suscept_high_CV}). + % Several of the recorded neurons contribute with two samples to the population analysis as their responses have been recorded to two different contrast of the same RAM stimulus. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview}{A}, see methods). % The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{} values are highlighted with black squares. } \end{figure*} %\Eqnref{response_modulation} \subsection*{Low CVs and weak stimuli are associated with strong nonlinearity} -All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For a comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the nonlinearity \nli{} \Eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:cells_suscept}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:data_overview_mod}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:data_overview_mod}{C}). +All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For a comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the nonlinearity \nli{} \Eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:cells_suscept}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:data_overview}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:data_overview}{C}). -The effective stimulus strength also plays an important role. We quantify the effect a stimulus has on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. P-units are heterogeneous in their sensitivity, their response modulations to the same stimulus contrast vary a lot \cite{Grewe2017}. Cells with weak responses to a stimulus, be it an insensitive cell or a weak stimulus, have higher \nli{} values and thus a more pronounced ridge in the second-order susceptibility at \fsumb{} in comparison to strongly responding cells that basically have flat second-order susceptibilities (\subfigrefb{fig:data_overview_mod}{E}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and both the CV and response strength during stimulation (effective output noise). +The effective stimulus strength also plays an important role. We quantify the effect a stimulus has on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. P-units are heterogeneous in their sensitivity, their response modulations to the same stimulus contrast vary a lot \cite{Grewe2017}. Cells with weak responses to a stimulus, be it an insensitive cell or a weak stimulus, have higher \nli{} values and thus a more pronounced ridge in the second-order susceptibility at \fsumb{} in comparison to strongly responding cells that basically have flat second-order susceptibilities (\subfigrefb{fig:data_overview}{E}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and both the CV and response strength during stimulation (effective output noise). %(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, with lower baseline CVs than P-units. Accordingly, \nli{} values of ampullary cells are indeed much higher than in P-units by about a factor of ten. Ampullary cells also show a negative correlation with baseline CV. Again, sensitive cells with strong response modulations are at the bottom of the distribution and have \nli{} values close to one (\subfigrefb{fig:data_overview_mod}{B, D}). The weaker the response modulation because of less sensitive cells or weaker stimulus amplitudes the stronger the nonlinear component of a cell's response (\subfigrefb{fig:data_overview_mod}{F}). +The population of ampullary cells is generally more homogeneous, with lower baseline CVs than P-units. Accordingly, \nli{} values of ampullary cells are indeed much higher than in P-units by about a factor of ten. Ampullary cells also show a negative correlation with baseline CV. Again, sensitive cells with strong response modulations are at the bottom of the distribution and have \nli{} values close to one (\subfigrefb{fig:data_overview}{B, D}). The weaker the response modulation, because of less sensitive cells or weaker stimulus amplitudes, the stronger the nonlinear component of a cell's response (\subfigrefb{fig:data_overview}{F}). %(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$) \section*{Discussion} @@ -641,9 +638,9 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract 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 (\subfigref{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. +Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\subfigref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview}{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 apply a noise-split \cite{Lindner2022} based on the 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 nonlinear 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 apply a noise-split \cite{Lindner2022} based on the 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 nonlinear 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} @@ -664,7 +661,7 @@ In contrast to the situation with individual frequencies (direct sine-waves or s 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. +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}\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. A possible readout mechanism should be the topic of future studies that need to take into account that the nonlinearities are stronger in pure sine-wave stimulation and the fraction of cells that show it under naturalistic stimulation might be larger than expected from the distribution of CVs. \subsection*{Behavioral relevance of nonlinear interactions} @@ -672,7 +669,7 @@ The behavioral relevance of the weak signal detection in P-units is evident from Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, nonlinearities were demonstrated to be strong for weak stimuli but were shown to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \lepto{} usually prefers small groups of fish\cite{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons. -The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview_mod}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\cite{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\cite{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\cite{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\cite{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous activity also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\cite{Grewe2017} while it is within the linear coding range in paddlefish\cite{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:data_overview_mod}{F}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\cite{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies. +The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\cite{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\cite{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\cite{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\cite{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous activity also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\cite{Grewe2017} while it is within the linear coding range in paddlefish\cite{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:data_overview}{F}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\cite{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies. \subsection*{Conclusion} We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferences of the weakly electric fish \lepto{}, systems that are very often characterized using linear methods. The observed nonlinearities match the expectations from previous theoretical studies\cite{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-oder susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian @@ -707,9 +704,9 @@ The stimulus was isolated from the ground (ISO-02V, npi-electronics, Tamm, Germa \end{figure*} \subsection*{White noise stimulation}\label{rammethods} -The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The stimulus intensity is given as the contrast, i.e. the standard deviation of the white noise stimulus in relation to the fish's EOD amplitude. The contrast varied between 1 and 20\,$\%$. Only cell recordings with at least 10\,s of white noise stimulation were included for the analysis. When ampullary cells were recorded, the white noise was directly applied as the stimulus. To create random amplitude modulations (RAM) for P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation profile (MXS-01M; npi electronics). +The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The stimulus intensity is given as the contrast, i.e. the standard deviation of the white noise stimulus in relation to the fish's EOD amplitude. The contrast varied between 1 and 20\,\%. Only cell recordings with at least 10\,s of white noise stimulation were included for the analysis. When ampullary cells were recorded, the white noise was directly applied as the stimulus. To create random amplitude modulations (RAM) for P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation profile (MXS-01M; npi electronics). -% and between 2.5 and 40\,$\%$ for \eigen +% and between 2.5 and 40\,\% for \eigen \subsection*{Data analysis} Data analysis was performed with Python 3 using the packages matplotlib\cite{Hunter2007}, numpy\cite{Walt2011}, scipy\cite{scipy2020}, pandas\cite{Mckinney2010}, nixio\cite{Stoewer2014}, and thunderfish (\url{https://github.com/bendalab/thunderfish}). @@ -794,7 +791,7 @@ We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. \end{equation} For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $\fbase{}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $\fbase{} \pm 5$\,Hz (\subfigrefb{fig:cells_suscept}{G}) and dividing it by the median of $D(f)$. -If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} values was used for the population statistics in \figref{fig:data_overview_mod}. +If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} values was used for the population statistics in \figref{fig:data_overview}. \subsection*{Leaky integrate-and-fire models}\label{lifmethods} @@ -850,7 +847,7 @@ Whenever the membrane voltage $V_m(t)$ crossed the spiking threshold $\theta=1$ \begin{figure*}[!ht] \includegraphics[width=\columnwidth]{flowchart} \caption{\label{flowchart} - Flowchart of a LIF P-unit model with EOD carrier. The main steps of the model are depicted in the left model column (Model cell identifier 2012-07-03-ak, see table~\ref{modelparams} for model parameters). The three other columns show the relevant signals in three different settings. (i) the baseline situation, no external stimulus, only the animal's self-generated EOD (i.e. the carrier) is present (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\% contrast) stimulus, (iii) Noise split condition in which 90\% of the internal noise is used as a driving stimulus scaled with the correction factor $\rho$ (see text). Note: that the firing rate and the CV of the ISI distribution is the same in this and the baseline condition. \figitem{A} Thresholding: a simple linear threshold was applied to the EOD carrier, \Eqnref{eq:eod}. The red line on top depicts the amplitude modulation (AM). \figitem{B} Dendritic low-pass filtering attenuates the carrier. \figitem{C} A Gaussian noise is added to the signal in B. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model in response to the addition of B and C. \figitem{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (\panel[i]{E}) there are several peaks, from left to right, at the baseline firing rate $\fbase{}$, $f_{EOD} - \fbase{}$ $f_{EOD}$, and $f_{EOD} + \fbase{}$. In the stimulus driven regime (\panel[ii]{E}), there is only a peak at \feod, while under the noise split condition (\panel[iii]{E}) again all peaks are present.} + Flowchart of a LIF P-unit model with EOD carrier. The main steps of the model are depicted in the left model column (Model cell identifier 2012-07-03-ak, see table~\ref{modelparams} for model parameters). The three other columns show the relevant signals in three different settings. (i) the baseline situation, no external stimulus, only the animal's self-generated EOD (i.e. the carrier) is present (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\,\% contrast) stimulus, (iii) Noise split condition in which 90\,\% of the internal noise is used as a driving stimulus scaled with the correction factor $\rho$ (see text). Note: that the firing rate and the CV of the ISI distribution is the same in this and the baseline condition. \figitem{A} Thresholding: a simple linear threshold was applied to the EOD carrier, \Eqnref{eq:eod}. The red line on top depicts the amplitude modulation (AM). \figitem{B} Dendritic low-pass filtering attenuates the carrier. \figitem{C} A Gaussian noise is added to the signal in B. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model in response to the addition of B and C. \figitem{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (\panel[i]{E}) there are several peaks, from left to right, at the baseline firing rate $\fbase{}$, $f_{EOD} - \fbase{}$ $f_{EOD}$, and $f_{EOD} + \fbase{}$. In the stimulus driven regime (\panel[ii]{E}), there is only a peak at \feod, while under the noise split condition (\panel[iii]{E}) again all peaks are present.} \end{figure*} \subsection*{Numerical implementation} @@ -907,7 +904,7 @@ According to previous works \cite{Lindner2022} the total noise of a LIF model ($ -In the here used model a small portion of the original noise was assigned to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}) and a big portion used as the signal component ($c_{\rm{signal}} = 0.9$, \subfigrefb{flowchart}\,\panel[iii]{A}). For the noise split to be valid \cite{Lindner2022} it is critical that both components add up to the initial 100\,$\%$ of the total noise and the baseline properties as the firing rate and the CV of the model are maintained. 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. After the original noise was split into a signal component with $c_{\rm{signal}}$, the frequencies above 300\,Hz were discarded and the signal strength was reduced after the dendritic low pass filtering. To compensate for these transformations the initial signal component was multiplying with the factor $\rho$, keeping the baseline CV (only carrier) and the CV during the noise split comparable, and resulting in $s_\xi(t)$. $\rho$ was found by bisecting the plane of possible factors and minimizing the error between the CV during baseline and stimulation. +In the here used model a small portion of the original noise was assigned to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}) and a big portion used as the signal component ($c_{\rm{signal}} = 0.9$, \subfigrefb{flowchart}\,\panel[iii]{A}). For the noise split to be valid \cite{Lindner2022} it is critical that both components add up to the initial 100\,\% of the total noise and the baseline properties as the firing rate and the CV of the model are maintained. 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. After the original noise was split into a signal component with $c_{\rm{signal}}$, the frequencies above 300\,Hz were discarded and the signal strength was reduced after the dendritic low pass filtering. To compensate for these transformations the initial signal component was multiplying with the factor $\rho$, keeping the baseline CV (only carrier) and the CV during the noise split comparable, and resulting in $s_\xi(t)$. $\rho$ was found by bisecting the plane of possible factors and minimizing the error between the CV during baseline and stimulation. %that was found by minimizing the error between the %Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963}\Eqnref{eq:ram_split}, (red in \subfigrefb{flowchart}\,\panel[iii]{A}) bisecting the space of possible $\rho$ scaling factors diff --git a/trialnr.pdf b/trialnr.pdf index 5872d6b..9d88919 100644 Binary files a/trialnr.pdf and b/trialnr.pdf differ diff --git a/trialnr.py b/trialnr.py index 67372d9..ad8b6f9 100644 --- a/trialnr.py +++ b/trialnr.py @@ -7,11 +7,10 @@ from plotstyle import plot_style from threefish.core_plot_labels import title_find_cell_add from threefish.defaults import default_figsize, default_settings from threefish.core_filenames import overlap_cells -from threefish.core_load import save_visualization -from threefish.core_reformat import get_stack, load_model_susept -from threefish.core_calc_model import resave_small_files +from threefish.core_load import resave_small_files, save_visualization +from threefish.core_reformat_RAM import get_stack, load_model_susept from threefish.core import find_folder_name -from threefish.utils1_suscept import trial_nrs_ram_model +from threefish.utils1_RAM import trial_nrs_ram_model import itertools as it ##from update_project import *