From 03753caab237cd25e1935ead572b3dc099ddf0d9 Mon Sep 17 00:00:00 2001
From: saschuta <56550328+saschuta@users.noreply.github.com>
Date: Sun, 28 Apr 2024 21:04:50 +0200
Subject: [PATCH] text update

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 Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985, 
 
-While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
+While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < \feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
@@ -521,7 +521,7 @@ While the sensory periphery can often be well described by linear models, this i
   }
 \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 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}. 
+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 \fbase{}. 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 the P-unit 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 beat 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. 
 
@@ -551,18 +551,27 @@ transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.
 High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
 
 
-\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
-Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green). 
-
 \begin{figure*}[!ht]
 \includegraphics[width=\columnwidth]{ampullary}
   \caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
   }
 \end{figure*}
 
+
+\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
+Irrespective of the CV, neither P-unit shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green). 
+
+
+
 \subsection*{Model-based estimation of the nonlinear structure}
 Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory,  cannot be observed. Here we investigate how these discrepancies can be understood.
 
+\begin{figure*}[!h]
+  \includegraphics[width=\columnwidth]{model_and_data}
+  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 2.5\% contrast. The plot shows that \suscept{} surface for $N=11$. Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). The signal noise component \signalnoise{} and the intrinsic noise are shown in the center and bottom. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but $10^6$ stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise are treated as signal (center trace,  \signalnoise) while the intrinsic noise is reduced (see methods for details). Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).
+}
+\end{figure*}
+
 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 
@@ -572,41 +581,39 @@ Note, that the increased number of repetitions goes along with a substantial red
 
 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 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 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}).
+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 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 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. % less prominent,
 \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 $N=10^6$ 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}). 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}.
+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 \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. The \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). 
 
+%\suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}). the fading of the
 
 
-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}). 
+Is it possible based on this second-order susceptibility (\subfigrefb{fig:model_full}{A}) to predict nonlinearities in a three-fish setting? We can test this 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 two 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 is present 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 is present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met no nonlinearity is observed in the model, neither at the sum nor at the difference of the two beat frequencies (\subfigrefb{fig:model_full}{E}). 
 
+Even though the second-order susceptibilities here were estimated from data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{data_overview_mod}
-  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C, E}) and ampullary cells (\panel{B, D, F}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in 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.
+  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C, E}) and ampullary cells (\panel{B, D, F}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in  \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
   }
 \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 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. 
+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 are highlighted with dark markers in \subfigrefb{fig:data_overview_mod}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles --  \figrefb{fig:cells_suscept_high_CV}). 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 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}).
+The population of ampullary cells is generally more homogeneous, with lower CVs than 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{} values 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 one (\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 $r=-0.59$, $p < 0.0001$)
 
@@ -617,10 +624,10 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
 
 %\,\panel[iii]{C}
 \subsection*{Theory applies to systems with and without carrier}
-Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{fig:ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
+Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons (\figsref{fig:cells_suscept} and\,\ref{fig:ampullary}). Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
 
 \subsection*{Intrinsic noise limits nonlinear responses}
-Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
+Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\subfigref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_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 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}.
 
@@ -630,7 +637,7 @@ Our analysis is based on the neuronal responses to white noise stimulus sequence
 
 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}, \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.
+In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) during noise stimulation 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.
 
 
 
@@ -730,7 +737,7 @@ The power spectrum of the stimulus $s(t)$ was calculated as
   S_{ss}(\omega) = \frac{\langle \tilde s(\omega) \tilde s^* (\omega)\rangle}{T}
 	\end{split}
 \end{equation}
-with $\tilde s^* $ being the complex conjugate and $\langle ... \rangle$ denoting averaging over the segments. The power spectrum of the spike trains $S_{xx}$ was calculated accordingly. The cross-spectrum $S_{xs}(\omega)$ between stimulus and evoked spike trains was calculated according to
+with $\tilde s^* $ being the complex conjugate and $\langle ... \rangle$ denoting averaging over the segments. The power spectrum of the spike trains $S_{xx}(\omega)$ was calculated accordingly. The cross-spectrum $S_{xs}(\omega)$ between stimulus and evoked spike trains was calculated according to
 \begin{equation}
   \label{cross}
 	\begin{split}
@@ -784,7 +791,7 @@ Leaky integrate-and-fire (LIF) models with a carrier were constructed to reprodu
   \end{equation}
 with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
 
-In the model, the input \carrierinput 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 \carrierinput \rfloor_0 = \left\{ \begin{array}{rcl} \carrierinput & ; & \carrierinput \ge 0 \\ 0 & ; & \carrierinput < 0 \end{array} \right.
@@ -857,7 +864,7 @@ From each simulation run, the first second was discarded and the analysis was ba
 % The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
 \subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}% 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. 
+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_split}), 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 (\Eqnref{eq:Noise_split_intrinsic}). 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$c_{\rm{noise}} = 1-c_{\rm{signal}}$
@@ -886,8 +893,9 @@ According to previous works \cite{Lindner2022} the total noise of a LIF model ($
 
 
 
-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.
+In the here used model a small portion of the original noise was assigned to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}) and a big portion used as the signal component ($c_{\rm{signal}} = 0.9$, \subfigrefb{flowchart}\,\panel[iii]{A}). For the noise split to be valid \cite{Lindner2022} it is critical that both components add up to the initial 100\,$\%$ of the total noise and the baseline properties as the firing rate and the CV of the model are maintained. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. After the original noise was split into a signal component with $c_{\rm{signal}}$, the frequencies above 300\,Hz were discarded and the signal strength was reduced after the dendritic low pass filtering. To compensate for these transformations the initial signal component was multiplying with the factor $\rho$, keeping the baseline CV (only carrier) and the CV during the noise split comparable, and resulting in $s_\xi(t)$. $\rho$ was found by bisecting the plane of possible factors and minimizing the error between the CV during baseline and stimulation.
 
+%that was found by minimizing the error between the
 %Furutsu-Novikov Theorem \cite{Novikov1965, Furutsu1963}\Eqnref{eq:ram_split},  (red in \subfigrefb{flowchart}\,\panel[iii]{A})  bisecting the space of possible $\rho$ scaling factors
 %$\rho$ 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
@@ -936,7 +944,7 @@ 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 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}). 
+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-units, 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]
diff --git a/susceptibility1.tex.bak b/susceptibility1.tex.bak
index e5c0893..1a5149e 100644
--- a/susceptibility1.tex.bak
+++ b/susceptibility1.tex.bak
@@ -513,7 +513,7 @@
 
 Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985, 
 
-While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
+While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < \feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}. 
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
@@ -521,7 +521,7 @@ While the sensory periphery can often be well described by linear models, this i
   }
 \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 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}. 
+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 \fbase{}. 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 the P-unit 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 beat 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. 
 
@@ -551,18 +551,27 @@ transmission \cite{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017}.
 High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
 
 
-\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
-Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green). 
-
 \begin{figure*}[!ht]
 \includegraphics[width=\columnwidth]{ampullary}
   \caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
   }
 \end{figure*}
 
+
+\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
+Irrespective of the CV, neither P-unit shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green). 
+
+
+
 \subsection*{Model-based estimation of the nonlinear structure}
 Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory,  cannot be observed. Here we investigate how these discrepancies can be understood.
 
+\begin{figure*}[!h]
+  \includegraphics[width=\columnwidth]{model_and_data}
+  \caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the  electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 2.5\% contrast. The plot shows that \suscept{} surface for $N=11$. Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). The signal noise component \signalnoise{} and the intrinsic noise are shown in the center and bottom. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but $10^6$ stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise are treated as signal (center trace,  \signalnoise) while the intrinsic noise is reduced (see methods for details). Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).
+}
+\end{figure*}
+
 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 
@@ -572,41 +581,39 @@ Note, that the increased number of repetitions goes along with a substantial red
 
 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 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 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}).
+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 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 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. % less prominent,
 \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 $N=10^6$ 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}). 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}.
+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 \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. The \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}). 
 
+%\suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}). the fading of the
 
 
-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}). 
+Is it possible based on this second-order susceptibility (\subfigrefb{fig:model_full}{A}) to predict nonlinearities in a three-fish setting? We can test this 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 two 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 is present 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 is present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met no nonlinearity is observed in the model, neither at the sum nor at the difference of the two beat frequencies (\subfigrefb{fig:model_full}{E}). 
 
+Even though the second-order susceptibilities here were estimated from data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{data_overview_mod}
-  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C, E}) and ampullary cells (\panel{B, D, F}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in 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.
+  \caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C, E}) and ampullary cells (\panel{B, D, F}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in  \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli.  \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
   }
 \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 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. 
+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 are highlighted with dark markers in \subfigrefb{fig:data_overview_mod}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles --  \figrefb{fig:cells_suscept_high_CV}). 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 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}).
+The population of ampullary cells is generally more homogeneous, with lower CVs than 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{} values 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 one (\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 $r=-0.59$, $p < 0.0001$)
 
@@ -617,10 +624,10 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
 
 %\,\panel[iii]{C}
 \subsection*{Theory applies to systems with and without carrier}
-Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{fig:ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
+Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons (\figsref{fig:cells_suscept} and\,\ref{fig:ampullary}). Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
 
 \subsection*{Intrinsic noise limits nonlinear responses}
-Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
+Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\subfigref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_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 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}.
 
@@ -630,7 +637,7 @@ Our analysis is based on the neuronal responses to white noise stimulus sequence
 
 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}, \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.
+In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) during noise stimulation 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.
 
 
 
@@ -730,7 +737,7 @@ The power spectrum of the stimulus $s(t)$ was calculated as
   S_{ss}(\omega) = \frac{\langle \tilde s(\omega) \tilde s^* (\omega)\rangle}{T}
 	\end{split}
 \end{equation}
-with $\tilde s^* $ being the complex conjugate and $\langle ... \rangle$ denoting averaging over the segments. The power spectrum of the spike trains $S_{xx}$ was calculated accordingly. The cross-spectrum $S_{xs}(\omega)$ between stimulus and evoked spike trains was calculated according to
+with $\tilde s^* $ being the complex conjugate and $\langle ... \rangle$ denoting averaging over the segments. The power spectrum of the spike trains $S_{xx}(\omega)$ was calculated accordingly. The cross-spectrum $S_{xs}(\omega)$ between stimulus and evoked spike trains was calculated according to
 \begin{equation}
   \label{cross}
 	\begin{split}
@@ -784,7 +791,7 @@ Leaky integrate-and-fire (LIF) models with a carrier were constructed to reprodu
   \end{equation}
 with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
 
-In the model, the input \carrierinput 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 \carrierinput \rfloor_0 = \left\{ \begin{array}{rcl} \carrierinput & ; & \carrierinput \ge 0 \\ 0 & ; & \carrierinput < 0 \end{array} \right.
@@ -857,7 +864,7 @@ From each simulation run, the first second was discarded and the analysis was ba
 % The second-order susceptibility in the model was calculated with \Eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{fig:model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 
 
 \subsection*{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}% 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. 
+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_split}), 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 (\Eqnref{eq:Noise_split_intrinsic}). 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$c_{\rm{noise}} = 1-c_{\rm{signal}}$
@@ -886,7 +893,7 @@ According to previous works \cite{Lindner2022} the total noise of a LIF model ($
 
 
 
-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.
+In the here used model a small portion of the original noise was assigned to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}) and a big portion used as the signal component ($c_{\rm{signal}} = 0.9$, \subfigrefb{flowchart}\,\panel[iii]{A}). For the noise split to be valid \cite{Lindner2022} it is critical that both components add up to the initial 100\,$\%$ of the total noise and the baseline properties as the firing rate and the CV of the model cell are maintained. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. After the original noise was split into a signal component with $c_{\rm{signal}}$ and the frequencies above 300\,Hz were discarded and the signal strength was transformed during the dendritic low pass filtering. To compensate for these transformations the signal 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.