reread

data_overview_mod.py

@@ -6,9 +6,9 @@ from matplotlib import gridspec as gridspec, pyplot as plt
 import numpy as np
 from utils_all_down import default_settings
 from utils_suseptibility import colors_overview
-from utils_suseptibility import default_figsize, NLI_scorename2,pearson_label, exclude_nans_for_corr, kernel_scatter, \
+from utils_suseptibility import default_figsize, NLI_scorename2_small,pearson_label, exclude_nans_for_corr, kernel_scatter, \
     scatter_with_marginals_colorcoded, \
-    version_final, basename, stimname
+    version_final, basename_small, stimname_small
 from utils_all import update_cell_names, load_overview_susept, make_log_ticks, p_units_to_show, save_visualization, setting_overview_score
 from scipy import stats
 try:
@@ -128,14 +128,14 @@ def data_overview3():
         var_item_names = [var_it,var_it,var_it2]#,var_it2]#['Response Modulation [Hz]',]
         var_types = ['response_modulation','response_modulation','']#,'']#'response_modulation'
         max_x = max_xs[c]
-        x_axis_names = ['CV$'+basename()+'$','CV$'+stimname()+'$','Response Modulation [Hz]']#$'+basename()+'$,'Fr$'+basename()+'$',]
+        x_axis_names = ['CV$'+basename_small()+'$','CV$'+stimname_small()+'$','Response Modulation [Hz]']#$'+basename()+'$,'Fr$'+basename()+'$',]
         #score = scores[0]
         score_n = ['Perc99/Med', 'Perc99/Med', 'Perc99/Med']
         score = scores[c]
         scores_here = [score,score,score]#,score]
         score_name = ['max(diag5Hz)/med_diagonal_proj_fr','max(diag5Hz)/med_diagonal_proj_fr']#,'max(diag5Hz)/med_diagonal_proj_fr']#'Perc99/Med'
         score_name = ['Fr/Med', 'Fr/Med']#'Fr/Med']  # 'Perc99/Med'
-        score_name = [NLI_scorename2(), NLI_scorename2(), NLI_scorename2()]#NLI_scorename()]  # 'Fr/Med''Perc99/Med'
+        score_name = [NLI_scorename2_small(), NLI_scorename2_small(), NLI_scorename2_small()]#NLI_scorename()]  # 'Fr/Med''Perc99/Med'
         ax_j = []
         axls = []
         axss = []

model_and_data.py

@@ -175,11 +175,11 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
             tr_name = 1#'$c=1\,\%$','$c=0\,\%$'
         c = 2.5
         cs = ['$c=%.1f$' %(c)+'$\,\%$','$c=0\,\%$']
-        titles = ['Model\n$N=11$', 'Model\n$N=%s $' % (tr_name) +'\,million',
+        titles = ['Model\n$N=11$', 'Model\n'+'$N=10^6$',
                   'Model\,('+noise_name().lower()+')' + '\n' + '$N=11$',
-                  'Model\,('+noise_name().lower()+')' + '\n' + '$N=%s$' % (tr_name) + '\,million',
-                  ]#%
-        #'Model\,('+noise_name().lower()+')' + '\n' + '$N=11$\n $c=1\,\%$',
+                  'Model\,('+noise_name().lower()+')' + '\n' + '$N=10^6$'
+                  ]#%#%s$' % (tr_name) + '\,million'
+        #'Model\,('+noise_name().lower()+')' + '\n' + '$N=11$\n $c=1\,\%$',$N=%s $' % (tr_name) +'\,million'
         #          'Model\,('+noise_name().lower()+')' + '\n' + '$N=%s$' % (tr_name) + '\,million\n $c=1\,\%$ '
         ax_model = []
 
@@ -206,7 +206,7 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
         
             stack = load_model_susept(path, cells_save, save_name.split(r'/')[-1] + cell_add)
 
-            embed()
+            #embed()
             if len(stack)> 0:
                 add_nonlin_title, cbar, fig, stack_plot, im = plt_single_square_modl(ax_external, cell, stack, perc, titles[s],
                                                                                      width, eod_metrice = eod_metrice, titles_plot=True,
@@ -404,11 +404,11 @@ def start_pos_modeldata():
 
 
 def signal_component_name():
-    return r'$\xi_{signal}$'#signal noise'
+    return r'$s_\xi(t)$'#r'$\xi_{signal}$'#signal noise'
 
 
 def noise_component_name():#$\xi_{noise}$noise_name =
-    return r'$\xi_{noise}$'#'Noise component'#'intrinsic noise'
+    return 'Intrinsic noise'##r'$\xi_{noise}$'#'Noise component'#'intrinsic noise'
 
 
 def ypos_x_modelanddata():

model_full.py

@@ -116,8 +116,8 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s
 
     else:
         markers = ['o', 'o', 'o', 'o', ]
-        DF1_frmult = [0.28, 0.28, 1, 1]
-        DF2_frmult = [1-0.28, 1.28, 0.28, 1.28]  # 1.06949369
+        DF1_frmult = [0.28, 0.28, 1, 0.81]
+        DF2_frmult = [1-0.28, 1.28, 0.28, 1.18]  # 1.06949369
 
         grid0 = gridspec.GridSpecFromSubplotSpec(2, 2, wspace=0.15, hspace=0.4,
                                                  subplot_spec=grid[2::])
@@ -210,7 +210,7 @@ def model_full(c1=10, mult_type='_multsorted2_', devs=['05'], end='all', chose_s
             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]
+                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)):
@@ -693,7 +693,7 @@ def plt_model_full_model2(grid0, reshuffled='reshuffled', af_2 = 0.1, dev=0.0005
         colors = [color01, color02,
                  color01_2, color012, color0]# color02,np.abs(DF2 * 2), color02,'DF2_H1',
         labels = [deltaf1_label(), deltaf2_label(),
-                  diff_label(), sum_label(), fbasename()]
+                  diff_label(), sum_label(), fbasename_small()]
         marker = markers[j]#'DF1_H1','DF1_H4',v
 
         if len(stack_final) > 0:

susceptibility1.tex

@@ -262,6 +262,9 @@
 \newcommand{\ffstimintro}{\ensuremath{f_{2}}}
 \newcommand{\ffeodintro}{\ensuremath{f_{1}}}
 
+\newcommand{\carrierinput}{\ensuremath{y(t)}}
+
+
 \newcommand{\baseval}{134}
 \newcommand{\bone}{$\Delta f_{1}$}
 \newcommand{\btwo}{$\Delta f_{2}$}
@@ -311,6 +314,9 @@
 \newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum 
 \newcommand{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies
 
+
+\newcommand{\signalnoise}{$s_\xi(t)$}%su\right m 
+
 \newcommand{\bsumb}{$\bsum{}=\fbase{}$}%su\right m 
 \newcommand{\btwob}{$\Delta f_{2}=\fbase{}$}%sum 
 \newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}%sum 
@@ -511,11 +517,11 @@ While the sensory periphery can often be well described by linear models, this i
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
-  \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting. Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both encountered fish lead to a beat contrast of 10\,\%. Top: Interference of the receiver EOD with the EODs of other fish. Second row: Spike trains of the P-unit. Third row: Firing rate, retrieved as the convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: Only the receiver is present. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate. \figitem{B} The receiver and the fish with EOD frequency $f_{1}=631$\,Hz are present. \figitem{C} The receiver and the fish with EOD frequency $f_{2}=797$\,Hz are present. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate.
+  \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting. Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both encountered fish lead to a beat contrast of 10\,\%. Top: Sheme of a nonlinear system. Second row: Interference of the receiver EOD with the EODs of other fish. Third row: Spike trains of the P-unit. Forth row: Firing rate, retrieved as the convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: Only the receiver is present. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate. \figitem{B} The receiver and the fish with EOD frequency $f_{1}=631$\,Hz are present. \figitem{C} The receiver and the fish with EOD frequency $f_{2}=797$\,Hz are present. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate.
   }
 \end{figure*}
 
-Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom).\notejg{more baseline trials, working on it}. Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, top). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom). The latter is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}. 
+Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials as a baseline response $r_{0}$ at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom). Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, second row). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom), thus the response of the system is not equal to the input (\subfigrefb{fig:motivation}{D}, top). The appearing difference peak is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}. 
 
 The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding. 
 
@@ -539,7 +545,7 @@ Second-order susceptibility is expected to be especially pronounced for low-CV c
 
 Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \Eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
 
-Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that with the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize the signal
+Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize signal
 transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.
 
 High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
@@ -560,46 +566,47 @@ Traces of the expected structure of second-order susceptibility are found in bot
 In the electrophysiological experiments, we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, with parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
 
 In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal 
-transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
+transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and \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]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of a statistical estimate that is based on 11 trials only. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response.
+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.
 
 With high levels of intrinsic noise, we would not expect the nonlinear response features to survive. Indeed, we do not find these features in a high-CV P-unit and its corresponding model (not shown).
 
 \begin{figure*}[!hb]
   \includegraphics[width=\columnwidth]{model_and_data}
-  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{\rm{noise}}$ and $\xi_{\rm{noise}}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{\rm{noise}} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{\rm{noise}}$) while the remaining part is treated as $\xi_{\rm{noise}}$ (see methods for details). Note that the signal component ($\xi_{\rm{noise}}$) is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the noise component ($\xi_{\rm{noise}}$). Adding the discarded high frequency components to $\xi_{\rm{noise}}$ does not affect the results shown here.
+  \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).
 }
 \end{figure*}
 
 \subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
-We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
+We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{model_full}
-  \caption{\label{fig:model_full} \figitem[]{A} Full second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\Eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods). The colored markers highlight the nonlinear effects found in \subfigrefb{fig:motivation}{D}. \figitem{B} Power spectral density of model responses (left) and the respective recorded data (right) under pure sinewave stimulation. \figitem{C} Same as \panel[]{B} but for frequency combinations off the nonlinear structure. \note[TODO]{Better combination of off-axis frequency components. Still working on the units of the second-order susceptibility in this plot. In \panel[]{B} left the green and blue markers are at the same position.}}
+  \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{}.}
 \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 can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}, \subfigref{fig:model_full}{B} right) and in the model (\subfigref{fig:model_full}{B} left). 
+However, the second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}). The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the fading of the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
+
 
-The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the fading of the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). If we choose different frequency combinations, weak or no nonlinear spectral peaks are observed \subfigrefb{fig:model_full}{C}. Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
 
-If two frequencies not part of the triangular structure are chosen with pure sine wave stimulation no nonlinearity peaks appear (\subfigrefb{fig:model_full}{C}). 
+We can test the predictions if based on the second-order susceptibility \subfigrefb{fig:model_full}{A} we can predict nonlinearities in a three-fish setting, by providing two beats with weak amplitudes to the same model \subfigrefb{fig:model_full}{B--E}. If we chose a frequency combination where the sum of the beat frequencies is equal to \fbase{} a nonlinearity can be observed at the sum of the two beat frequencies in the power spectrum of the response (\subfigrefb{fig:model_full}{B}). If instead we choose two beat frequencies which difference is equal to \fbase{}, a nonlinear peak will peak be observed at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one beat frequency is equal to \fbase{}, a peak at the sum, as well as at the difference frequency will be present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of the conditions are met no nonlinearity will be observed in the model, neither at the sum nor at the difference frequencies of the two beat frequencies (\subfigrefb{fig:model_full}{E}). 
+
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{data_overview_mod}
-  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
+  \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 one cell, shown in the previous figures (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}, \figrefb{fig:cells_suscept_high_CV}). \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
   }
 \end{figure*}
 
 %\Eqnref{response_modulation}
 
 \subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
-The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
+The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark squares and circles, respectively (\subfigrefb{fig:data_overview_mod}{A, C, E}). The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds with pronounced nonlinearity thus depends on the baseline CV (i.e. the internal noise), the CV during stimulation and the response strength. 
 %(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
 %In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
 
-The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
+The population of ampullary cells is generally more homogeneous and with lower CVs compared to P-units. Accordingly, ampullary cells show much higher \nli{} values than P-units (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black squares. Again, we see that cells that are strongly driven by the stimulus are at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high nonlinearity values (\subfigrefb{fig:data_overview_mod}{F}).
 
 %(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
 
@@ -615,14 +622,15 @@ Theoretical work\cite{Voronenko2017} explained analytically the occurrence of no
 \subsection*{Intrinsic noise limits nonlinear responses}
 Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
 
-The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Furutsu-Novikov theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
+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}
-Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{French1973,Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In a our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \cite{Voronenko2017}.
+Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding\cite{French1973,Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In a our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \cite{Voronenko2017}.
 
-In the natural situation, the stimuli are periodic signals defined by the difference frequencies. Ho well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
+In the natural situation, the stimuli are periodic signals defined by the difference frequencies. How well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
 
-In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
+In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}, \subfigref{fig:model_full}{B--E}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
 
 
 
@@ -639,9 +647,7 @@ On the other hand, the nonlinearity was found only in low-CV P-units (with white
 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}
-The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation). 
-
-Note that in this work we operated in a regime of weak stimuli and that the envelope encoding addressed in \cite{Savard2011,Middleton2007} operates in a regime of strong stimuli, where the firing rate is saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli could be addressed in further P-unit studies. 
+The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation). Note that in this work we operated in a regime of weak stimuli and that the envelope encoding addressed in \cite{Savard2011,Middleton2007} operates in a regime of strong stimuli, where the firing rate is saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli could be addressed in further P-unit studies. 
 
 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.
 
@@ -715,7 +721,7 @@ where $*$ denotes the convolution. $r(t)$ is then calculated as the across-trial
 To quantify how strongly the neuron is driven by the stimulus we quantified the response modulation as the standard deviation $\sigma_{M} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t)^2\rangle_t}$, where $\langle \cdot \rangle_t$ indicates averaging over time.
 
 \paragraph{Spectral analysis}\label{susceptibility_methods}
-The neuron is driven by the stimulus and thus the spiking response $x(t)$ (\eqref{eq:spikes}) depends on the stimulus $s(t)$. To investigate the relation between stimulus and response we calculated the first- and second-order susceptibility of the neuron to the stimulus in the frequency domain. The Fourier transforms of $s(t)$ and $x(t)$ are denoted as $\tilde s(\omega)$ and $\tilde x(\omega)$ and were calculated according to $\tilde x(\omega) = \int_{0}^{T} \, x(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. Stimuli had a duration of 10\,s and spectra of stimulus and response were calculated in separate segments of 0.5\,s with no overlap resulting in a spectral resolution of 2\,Hz.
+The neuron is driven by the stimulus and thus the spiking response $x(t)$, \Eqnref{eq:spikes}, depends on the stimulus $s(t)$. To investigate the relation between stimulus and response we calculated the first- and second-order susceptibility of the neuron to the stimulus in the frequency domain. The Fourier transforms of $s(t)$ and $x(t)$ are denoted as $\tilde s(\omega)$ and $\tilde x(\omega)$ and were calculated according to $\tilde x(\omega) = \int_{0}^{T} \, x(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. Stimuli had a duration of 10\,s and spectra of stimulus and response were calculated in separate segments of 0.5\,s with no overlap resulting in a spectral resolution of 2\,Hz.
 
 The power spectrum of the stimulus $s(t)$ was calculated as 
 \begin{equation}
@@ -757,7 +763,7 @@ The second-order susceptibility was calculated by dividing the higher-order cros
 %   \chi_{2}(\omega_{1}, \omega_{2}) = \frac{TN \sum_{n=1}^N \int_{0}^{T} dt\,r_{n}(t) e^{-i(\omega_{1}+\omega_{2})t} \int_{0}^{T}dt'\,s_{n}(t')e^{i \omega_{1}t'} \int_{0}^{T} dt''\,s_{n}(t'')e^{i \omega_{2}t''}}{2 \sum_{n=1}^N \int_{0}^{T} dt\, s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{T} dt\,s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{2}t'}}
 % 	\end{split}
 % \end{equation}
-The absolute value of a second-order susceptibility matrix is visualized in \figrefb{fig:model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $r(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $r(t)$ at the difference of the input frequencies.
+The absolute value of a second-order susceptibility matrix is visualized in \figrefb{fig:model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $x(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $x(t)$ at the difference of the input frequencies.
 
 \paragraph{Nonlinearity index}\label{projected_method}
 We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the peakedness of the nonlinearity (PNL) as
@@ -774,23 +780,23 @@ If the same frozen noise was recorded several times in a cell, each noise repeti
 Leaky integrate-and-fire (LIF) models with a carrier were constructed to reproduce the specific firing properties of P-units \cite{Chacron2001,Sinz2020}. The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD modeled as a cosine wave
 \begin{equation}
     \label{eq:eod}
-    x(t) = x_{EOD}(t) = \cos(2\pi f_{EOD} t)
+    \carrierinput = y_{EOD}(t) = \cos(2\pi f_{EOD} t)
   \end{equation}
 with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
 
-In the model, the input $x(t)$ was then first thresholded to model the synapse between the primary receptor cells and the afferent. 
+In the model, the input \carrierinput was then first thresholded to model the synapse between the primary receptor cells and the afferent. 
 \begin{equation}
   \label{eq:threshold2}
-  \lfloor x(t) \rfloor_0 = \left\{ \begin{array}{rcl} x(t) & ; & x(t) \ge 0 \\ 0 & ; & x(t) < 0 \end{array} \right.
+  \lfloor \carrierinput \rfloor_0 = \left\{ \begin{array}{rcl} \carrierinput & ; & \carrierinput \ge 0 \\ 0 & ; & \carrierinput < 0 \end{array} \right.
 \end{equation}
-$\lfloor x(t) \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (\subfigrefb{flowchart}{A}).
+$\lfloor \carrierinput \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (\subfigrefb{flowchart}{A}).
 
 The resulting receptor signal was then low-pass filtered to approximate passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B})
 \begin{equation}
   \label{eq:dendrite}
-  \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor x(t) \rfloor_{0}
+  \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor \carrierinput \rfloor_{0}
 \end{equation}
-with $\tau_{d}$ as the dendritic time constant. Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. Because the input was dimensionless, the dendritic voltage was dimensionless, too. The combination of threshold and low-pass filtering extracts the amplitude modulation of the input $x(t)$. 
+with $\tau_{d}$ as the dendritic time constant. Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. Because the input was dimensionless, the dendritic voltage was dimensionless, too. The combination of threshold and low-pass filtering extracts the amplitude modulation of the input \carrierinput. 
 
 The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
 \begin{equation}
@@ -823,7 +829,7 @@ 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]{D}) 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]{D}), there is only a peak at $f_{eod}$, while under the noise split condition (\panel[iii]{D}) 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}
@@ -844,15 +850,14 @@ The random amplitude modulation (RAM) input to the model was created by drawing
     \label{eq:ram_equation}
     y(t) = (1+ s(t)) \cdot \cos(2\pi f_{EOD} t)
 \end{equation}
-\note{fix stimulus x and y notation and RAM}
 From each simulation run, the first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
 % \subsection{Second-order susceptibility analysis of the model}
 % %\subsubsection{Model second-order nonlinearity}
 
 % The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
-\subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
-According to the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$ \note{hier umschreiben das xi muss ein RAM sein} and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
+\subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}% the Furutsu-Novikov Theorem  with the same correlation function
+According to previous works \cite{Lindner2022} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$, a RAM stimulus where frequencies above 300\,Hz are discarded (\Eqnref{eq:ram_equation}), and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. 
 %\sqrt{\rho \, 2D \,c_{\rm{signal}}} \cdot \xi(t) 
 
 %(1-c_{\rm{signal}})\cdot\xi$c_{\rm{noise}} = 1-c_{\rm{signal}}$
@@ -881,12 +886,16 @@ According to the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} the tot
 
 
 
-A big portion of the total noise was assigned to the signal component ($c_{\rm{signal}} = 0.9$) and the remaining part to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{\rm{noise}}(t)$ was scaled up by the factor $\rho$ (\Eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[iii]{A}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
+A big portion of the total noise was assigned to the signal component ($c_{\rm{signal}} = 0.9$) and the remaining part to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the noise split to be valid \cite{Lindner2022} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, after the noise $\xi(t)$ was split into a signal component with $c_{\rm{signal}}$ and the frequencies above 300\,Hz were discarded, the signal component was multiplying with the factor $\rho$, then resulting in $s_\xi(t)$. $\rho$ was found by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
+
+%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
+%$\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. 
+%In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
 
 
 % See section \ref{lifmethods} for model and parameter description.
 \begin{table*}[hp!]
-  \caption{\label{modelparams} Model parameters of LIF models, fitted to 2 electrophysiologically recorded P-units (\cite{Ott2020}).}
+  \caption{\label{modelparams} Model parameters of LIF models, fitted to 2 electrophysiologically recorded P-units \cite{Ott2020}.}
   \begin{center}
   \begin{tabular}{lrrrrrrrr}
     \hline
@@ -927,7 +936,7 @@ A big portion of the total noise was assigned to the signal component ($c_{\rm{s
 
 
 \paragraph*{S1 Second-order susceptibility of high-CV P-unit}
-CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}). 
+CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis as in (\figrefb{fig:cells_suscept}) for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}). 
 
 \label{S1:highcvpunit}
 \begin{figure*}[!ht]
@@ -940,7 +949,7 @@ CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the sa
 
 \begin{figure*}[hp]%hp!
 \includegraphics[width=\columnwidth]{trialnr}  
-  \caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$. Black -- 99.9th percentile of the second-order susceptibility.
+  \caption{\label{fig:trialnr} Saturation of the second-order susceptibility depending on the stimulus repetition number $\n{}$. Gray line -- 99.9th percentile of the second-order susceptibility matrix.
   }
 \end{figure*}
 

susceptibility1.tex.bak

@@ -262,6 +262,9 @@
 \newcommand{\ffstimintro}{\ensuremath{f_{2}}}
 \newcommand{\ffeodintro}{\ensuremath{f_{1}}}
 
+\newcommand{\carrierinput}{\ensuremath{y(t)}}
+
+
 \newcommand{\baseval}{134}
 \newcommand{\bone}{$\Delta f_{1}$}
 \newcommand{\btwo}{$\Delta f_{2}$}
@@ -311,6 +314,9 @@
 \newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum 
 \newcommand{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies
 
+
+\newcommand{\signalnoise}{$s_\xi(t)$}%su\right m 
+
 \newcommand{\bsumb}{$\bsum{}=\fbase{}$}%su\right m 
 \newcommand{\btwob}{$\Delta f_{2}=\fbase{}$}%sum 
 \newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}%sum 
@@ -511,11 +517,11 @@ While the sensory periphery can often be well described by linear models, this i
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
-  \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting. Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both encountered fish lead to a beat contrast of 10\,\%. Top: Interference of the receiver EOD with the EODs of other fish. Second row: Spike trains of the P-unit. Third row: Firing rate, retrieved as the convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: Only the receiver is present. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate. \figitem{B} The receiver and the fish with EOD frequency $f_{1}=631$\,Hz are present. \figitem{C} The receiver and the fish with EOD frequency $f_{2}=797$\,Hz are present. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate.
+  \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting. Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both encountered fish lead to a beat contrast of 10\,\%. Top: Sheme of a nonlinear system. Second row: Interference of the receiver EOD with the EODs of other fish. Third row: Spike trains of the P-unit. Forth row: Firing rate, retrieved as the convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: Only the receiver is present. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate. \figitem{B} The receiver and the fish with EOD frequency $f_{1}=631$\,Hz are present. \figitem{C} The receiver and the fish with EOD frequency $f_{2}=797$\,Hz are present. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate.
   }
 \end{figure*}
 
-Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom).\notejg{more baseline trials, working on it}. Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, top). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom). The latter is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}. 
+Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials as a baseline response $r_{0}$ at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom). Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, second row). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom), thus the response of the system is not equal to the input (\subfigrefb{fig:motivation}{D}, top). The appearing difference peak is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}. 
 
 The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding. 
 
@@ -539,7 +545,7 @@ Second-order susceptibility is expected to be especially pronounced for low-CV c
 
 Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \Eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
 
-Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that with the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize the signal
+Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize signal
 transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.
 
 High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
@@ -560,46 +566,47 @@ Traces of the expected structure of second-order susceptibility are found in bot
 In the electrophysiological experiments, we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, with parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
 
 In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal 
-transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
+transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.  Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs.  Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and \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]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of a statistical estimate that is based on 11 trials only. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response.
+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.
 
 With high levels of intrinsic noise, we would not expect the nonlinear response features to survive. Indeed, we do not find these features in a high-CV P-unit and its corresponding model (not shown).
 
 \begin{figure*}[!hb]
   \includegraphics[width=\columnwidth]{model_and_data}
-  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{\rm{noise}}$ and $\xi_{\rm{noise}}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{\rm{noise}} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{\rm{noise}}$) while the remaining part is treated as $\xi_{\rm{noise}}$ (see methods for details). Note that the signal component ($\xi_{\rm{noise}}$) is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the noise component ($\xi_{\rm{noise}}$). Adding the discarded high frequency components to $\xi_{\rm{noise}}$ does not affect the results shown here.
+  \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).
 }
 \end{figure*}
 
 \subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
-We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
+We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\Eqnref{eq:crosshigh}).
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{model_full}
-  \caption{\label{fig:model_full} \figitem[]{A} Full second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\Eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods). The colored markers highlight the nonlinear effects found in \subfigrefb{fig:motivation}{D}. \figitem{B} Power spectral density of model responses (left) and the respective recorded data (right) under pure sinewave stimulation. \figitem{C} Same as \panel[]{B} but for frequency combinations off the nonlinear structure. \note[TODO]{Better combination of off-axis frequency components. Still working on the units of the second-order susceptibility in this plot. In \panel[]{B} left the green and blue markers are at the same position.}}
+  \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{}.}
 \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 can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}, \subfigref{fig:model_full}{B} right) and in the model (\subfigref{fig:model_full}{B} left). 
+However, the second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows increased \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}). The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the fading of the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
+
 
-The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the fading of the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). If we choose different frequency combinations, weak or no nonlinear spectral peaks are observed \subfigrefb{fig:model_full}{C}. Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
 
-If two frequencies not part of the triangular structure are chosen with pure sine wave stimulation no nonlinearity peaks appear (\subfigrefb{fig:model_full}{C}). 
+We can test the predictions if based on the second-order susceptibility \subfigrefb{fig:model_full}{A} we can predict nonlinearities in a three-fish setting, by providing two beats with weak amplitudes to the same model \subfigrefb{fig:model_full}{B--E}. If we chose a frequency combination where the sum of the beat frequencies is equal to \fbase{} a nonlinearity can be observed at the sum of the two beat frequencies in the power spectrum of the response (\subfigrefb{fig:model_full}{B}). If instead we choose two beat frequencies which difference is equal to \fbase{}, a nonlinear peak will peak be observed at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one beat frequency is equal to \fbase{}, a peak at the sum, as well as at the difference frequency will be present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of the conditions are met no nonlinearity will be observed in the model, neither at the sum nor at the difference frequencies of the two beat frequencies (\subfigrefb{fig:model_full}{E}). 
+
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{data_overview_mod}
-  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
+  \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 one cell, shown in the previous figures (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}, \figrefb{fig:cells_suscept_high_CV}). \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
   }
 \end{figure*}
 
 %\Eqnref{response_modulation}
 
 \subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
-The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength. 
+The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark squares and circles, respectively (\subfigrefb{fig:data_overview_mod}{A, C, E}). The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds with pronounced nonlinearity thus depends on the baseline CV (i.e. the internal noise), the CV during stimulation and the response strength. 
 %(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
 %In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
 
-The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
+The population of ampullary cells is generally more homogeneous and with lower CVs compared to P-units. Accordingly, ampullary cells show much higher \nli{} values than P-units (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black squares. Again, we see that cells that are strongly driven by the stimulus are at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high nonlinearity values (\subfigrefb{fig:data_overview_mod}{F}).
 
 %(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
 
@@ -615,14 +622,15 @@ Theoretical work\cite{Voronenko2017} explained analytically the occurrence of no
 \subsection*{Intrinsic noise limits nonlinear responses}
 Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
 
-The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Furutsu-Novikov theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
+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}
-Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{French1973,Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In a our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \cite{Voronenko2017}.
+Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding\cite{French1973,Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In a our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \cite{Voronenko2017}.
 
-In the natural situation, the stimuli are periodic signals defined by the difference frequencies. Ho well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
+In the natural situation, the stimuli are periodic signals defined by the difference frequencies. How well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
 
-In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
+In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}, \subfigref{fig:model_full}{B--E}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
 
 
 
@@ -639,9 +647,7 @@ On the other hand, the nonlinearity was found only in low-CV P-units (with white
 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}
-The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation). 
-
-Note that in this work we operated in a regime of weak stimuli and that the envelope encoding addressed in \cite{Savard2011,Middleton2007} operates in a regime of strong stimuli, where the firing rate is saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli could be addressed in further P-unit studies. 
+The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation). Note that in this work we operated in a regime of weak stimuli and that the envelope encoding addressed in \cite{Savard2011,Middleton2007} operates in a regime of strong stimuli, where the firing rate is saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli could be addressed in further P-unit studies. 
 
 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.
 
@@ -715,7 +721,7 @@ where $*$ denotes the convolution. $r(t)$ is then calculated as the across-trial
 To quantify how strongly the neuron is driven by the stimulus we quantified the response modulation as the standard deviation $\sigma_{M} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t)^2\rangle_t}$, where $\langle \cdot \rangle_t$ indicates averaging over time.
 
 \paragraph{Spectral analysis}\label{susceptibility_methods}
-The neuron is driven by the stimulus and thus the spiking response $x(t)$ (\eqref{eq:spikes}) depends on the stimulus $s(t)$. To investigate the relation between stimulus and response we calculated the first- and second-order susceptibility of the neuron to the stimulus in the frequency domain. The Fourier transforms of $s(t)$ and $x(t)$ are denoted as $\tilde s(\omega)$ and $\tilde x(\omega)$ and were calculated according to $\tilde x(\omega) = \int_{0}^{T} \, x(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. Stimuli had a duration of 10\,s and spectra of stimulus and response were calculated in separate segments of 0.5\,s with no overlap resulting in a spectral resolution of 2\,Hz.
+The neuron is driven by the stimulus and thus the spiking response $x(t)$, \Eqnref{eq:spikes}, depends on the stimulus $s(t)$. To investigate the relation between stimulus and response we calculated the first- and second-order susceptibility of the neuron to the stimulus in the frequency domain. The Fourier transforms of $s(t)$ and $x(t)$ are denoted as $\tilde s(\omega)$ and $\tilde x(\omega)$ and were calculated according to $\tilde x(\omega) = \int_{0}^{T} \, x(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. Stimuli had a duration of 10\,s and spectra of stimulus and response were calculated in separate segments of 0.5\,s with no overlap resulting in a spectral resolution of 2\,Hz.
 
 The power spectrum of the stimulus $s(t)$ was calculated as 
 \begin{equation}
@@ -757,7 +763,7 @@ The second-order susceptibility was calculated by dividing the higher-order cros
 %   \chi_{2}(\omega_{1}, \omega_{2}) = \frac{TN \sum_{n=1}^N \int_{0}^{T} dt\,r_{n}(t) e^{-i(\omega_{1}+\omega_{2})t} \int_{0}^{T}dt'\,s_{n}(t')e^{i \omega_{1}t'} \int_{0}^{T} dt''\,s_{n}(t'')e^{i \omega_{2}t''}}{2 \sum_{n=1}^N \int_{0}^{T} dt\, s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{T} dt\,s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{2}t'}}
 % 	\end{split}
 % \end{equation}
-The absolute value of a second-order susceptibility matrix is visualized in \figrefb{fig:model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $r(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $r(t)$ at the difference of the input frequencies.
+The absolute value of a second-order susceptibility matrix is visualized in \figrefb{fig:model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $x(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $x(t)$ at the difference of the input frequencies.
 
 \paragraph{Nonlinearity index}\label{projected_method}
 We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the peakedness of the nonlinearity (PNL) as
@@ -774,23 +780,23 @@ If the same frozen noise was recorded several times in a cell, each noise repeti
 Leaky integrate-and-fire (LIF) models with a carrier were constructed to reproduce the specific firing properties of P-units \cite{Chacron2001,Sinz2020}. The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD modeled as a cosine wave
 \begin{equation}
     \label{eq:eod}
-    x(t) = x_{EOD}(t) = \cos(2\pi f_{EOD} t)
+    \carrierinput = y_{EOD}(t) = \cos(2\pi f_{EOD} t)
   \end{equation}
 with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
 
-In the model, the input $x(t)$ was then first thresholded to model the synapse between the primary receptor cells and the afferent. 
+In the model, the input \carrierinput was then first thresholded to model the synapse between the primary receptor cells and the afferent. 
 \begin{equation}
   \label{eq:threshold2}
-  \lfloor x(t) \rfloor_0 = \left\{ \begin{array}{rcl} x(t) & ; & x(t) \ge 0 \\ 0 & ; & x(t) < 0 \end{array} \right.
+  \lfloor \carrierinput \rfloor_0 = \left\{ \begin{array}{rcl} \carrierinput & ; & \carrierinput \ge 0 \\ 0 & ; & \carrierinput < 0 \end{array} \right.
 \end{equation}
-$\lfloor x(t) \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (\subfigrefb{flowchart}{A}).
+$\lfloor \carrierinput \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (\subfigrefb{flowchart}{A}).
 
 The resulting receptor signal was then low-pass filtered to approximate passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B})
 \begin{equation}
   \label{eq:dendrite}
-  \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor x(t) \rfloor_{0}
+  \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor \carrierinput \rfloor_{0}
 \end{equation}
-with $\tau_{d}$ as the dendritic time constant. Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. Because the input was dimensionless, the dendritic voltage was dimensionless, too. The combination of threshold and low-pass filtering extracts the amplitude modulation of the input $x(t)$. 
+with $\tau_{d}$ as the dendritic time constant. Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. Because the input was dimensionless, the dendritic voltage was dimensionless, too. The combination of threshold and low-pass filtering extracts the amplitude modulation of the input \carrierinput. 
 
 The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
 \begin{equation}
@@ -823,7 +829,7 @@ 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]{D}) 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]{D}), there is only a peak at $f_{eod}$, while under the noise split condition (\panel[iii]{D}) 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}
@@ -844,15 +850,14 @@ The random amplitude modulation (RAM) input to the model was created by drawing
     \label{eq:ram_equation}
     y(t) = (1+ s(t)) \cdot \cos(2\pi f_{EOD} t)
 \end{equation}
-\note{fix stimulus x and y notation and RAM}
 From each simulation run, the first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
 % \subsection{Second-order susceptibility analysis of the model}
 % %\subsubsection{Model second-order nonlinearity}
 
 % The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
-\subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
-According to the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$ \note{hier umschreiben das xi muss ein RAM sein} and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
+\subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}% the Furutsu-Novikov Theorem  with the same correlation function
+According to previous works \cite{Lindner2022} the total noise of a LIF model ($\xi$) can be split up into several independent noise processes. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$, a RAM stimulus where frequencies above 300\,Hz are discarded (\Eqnref{eq:ram_equation}), and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. 
 %\sqrt{\rho \, 2D \,c_{\rm{signal}}} \cdot \xi(t) 
 
 %(1-c_{\rm{signal}})\cdot\xi$c_{\rm{noise}} = 1-c_{\rm{signal}}$
@@ -881,12 +886,16 @@ According to the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} the tot
 
 
 
-A big portion of the total noise was assigned to the signal component ($c_{\rm{signal}} = 0.9$) and the remaining part to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{\rm{noise}}(t)$ was scaled up by the factor $\rho$ (\Eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[iii]{A}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
+A big portion of the total noise was assigned to the signal component ($c_{\rm{signal}} = 0.9$) and the remaining part to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the noise split to be valid \cite{Lindner2022} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, after the noise $\xi(t)$ was split into a signal component with $c_{\rm{signal}}$ and the frequencies above 300\,Hz were discarded, the signal component was multiplying with the factor $\rho$, then resulting in $s_\xi(t)$. $\rho$ was found by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
+
+%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
+%$\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass. 
+%In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
 
 
 % See section \ref{lifmethods} for model and parameter description.
 \begin{table*}[hp!]
-  \caption{\label{modelparams} Model parameters of LIF models, fitted to 2 electrophysiologically recorded P-units (\cite{Ott2020}).}
+  \caption{\label{modelparams} Model parameters of LIF models, fitted to 2 electrophysiologically recorded P-units \cite{Ott2020}.}
   \begin{center}
   \begin{tabular}{lrrrrrrrr}
     \hline
@@ -927,7 +936,7 @@ A big portion of the total noise was assigned to the signal component ($c_{\rm{s
 
 
 \paragraph*{S1 Second-order susceptibility of high-CV P-unit}
-CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}). 
+CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis as in (\figrefb{fig:cells_suscept}) for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}). 
 
 \label{S1:highcvpunit}
 \begin{figure*}[!ht]
@@ -940,7 +949,7 @@ CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the sa
 
 \begin{figure*}[hp]%hp!
 \includegraphics[width=\columnwidth]{trialnr}  
-  \caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$. Black -- 99.9th percentile of the second-order susceptibility.
+  \caption{\label{fig:trialnr} Saturation of the second-order susceptibility depending on the stimulus repetition number $\n{}$. Gray line -- 99.9th percentile of the second-order susceptibility matrix.
   }
 \end{figure*}
 

utils_paper.py

@@ -2269,7 +2269,7 @@ def plt_single_square_modl(ax, cell, model, perc, titles, width, bias_factor=1,
                            titles_plot=False, xpos=1.1, resize=False, ls=8):
     model_show, stack_plot, stack_plot_wo_norm = get_stack(cell, model, bias_factor=bias_factor)
     print(np.max(np.max(stack_plot)))
-    embed()
+    #embed()
     if resize:
         stack_plot, add_nonlin_title, resize_val = rescale_colorbar_and_values(stack_plot)
     else:
@@ -34881,6 +34881,8 @@ def fbasenamehz():
 def fbasename():
     return r'$f' + basename() + '$'
 
+def fbasename_small():
+    return r'$f' + basename_small() + '$'
 
 def stimname():
     rm_var = rem_variable()