fixes and todos

references.bib

@@ -33,6 +33,24 @@
   publisher={Springer}
 }
 
+@Inbook{Baker2019,
+author="Baker, Clare V. H.",
+editor="Carlson, Bruce A.
+and Sisneros, Joseph A.
+and Popper, Arthur N.
+and Fay, Richard R.",
+title="The Development and Evolution of Lateral Line Electroreceptors: Insights from Comparative Molecular Approaches",
+bookTitle="Electroreception: Fundamental Insights from Comparative Approaches",
+year="2019",
+publisher="Springer International Publishing",
+address="Cham",
+pages="25--62",
+abstract="In the jawless lampreys, most nonteleost jawed fishes, and aquatic-stage amphibians, the lateral line system has a mechanosensory division responding to local water movement (``distant touch'') and an electrosensory division responding to low-frequency cathodal (exterior-negative) electric stimuli, such as the weak electric fields surrounding other animals. The electrosensory division was lost in the ancestors of teleost fishes and their closest relatives and in the ancestors of frogs and toads. However, anodally sensitive lateral line electroreception evolved independently at least twice within teleosts, most likely via modification of the mechanosensory division. This chapter briefly reviews this sensory system and describes our current understanding of the development of nonteleost lateral line electroreceptors, both in terms of their embryonic origin from lateral line placodes and at the molecular level. Gene expression analysis, using candidate genes and more recent unbiased transcriptomic (differential RNA sequencing) approaches, suggests a high degree of conservation between nonteleost electroreceptors and mechanosensory hair cells both in their development and in aspects of their physiology, including transmission mechanisms at the ribbon synapse. Taken together, these support the hypothesis that electroreceptors evolved in the vertebrate ancestor via the diversification of lateral line hair cells.",
+isbn="978-3-030-29105-1",
+doi="10.1007/978-3-030-29105-1_2",
+url="https://doi.org/10.1007/978-3-030-29105-1_2"
+}
+
 @article{Barnett1971,
   title={Bispectrum analysis of electroencephalogram signals during waking and sleeping},
   author={Barnett, TP and Johnson, LC and Naitoh, P and Hicks, N and Nute, C},
@@ -4112,6 +4130,26 @@ We collected weakly electric gymnotoid fish in the vicinity of Manaus, Amazonas,
   year={1990}
 }
 
+@article{Machens2001,
+  Title                    = {Representation of acoustic communication signals by insect auditory receptor neurons},
+  Abstract                 = {Despite their simple auditory systems, some insect species recognize certain temporal aspects of acoustic stimuli with an acuity equal to that of vertebrates; however, the underlying neural mechanisms and coding schemes are only partially understood. In this study, we analyze the response characteristics of the peripheral auditory system of grasshoppers with special emphasis on the representation of species- specific communication signals. We use both natural calling songs and artificial random stimuli designed to focus on two low-order statistical properties of the songs: their typical time scales and the distribution of their modulation amplitudes. Based on stimulus reconstruction techniques and quantified within an information- theoretic framework, our data show that artificial stimuli with typical time scales of >40 msec can be read from single spike trains with high accuracy. Faster stimulus variations can be reconstructed only for behaviorally relevant amplitude distributions. The highest rates of information transmission (180 bits/sec) and the highest coding efficiencies (40%) are obtained for stimuli that capture both the time scales and amplitude distributions of natural songs. Use of multiple spike trains significantly improves the reconstruction of stimuli that vary on time scales <40 msec or feature amplitude distributions as occur when several grasshopper songs overlap. Signal-to-noise ratios obtained from the reconstructions of natural songs do not exceed those obtained from artificial stimuli with the same low-order statistical properties. We conclude that auditory receptor neurons are optimized to extract both the time scales and the amplitude distribution of natural songs. They are not optimized, however, to extract higher-order statistical properties of the song-specific rhythmic patterns},
+  Address                  = {Innovationskolleg Theoretische Biologie, Institut fur Biologie, Humboldt-Universitat zu Berlin, 10099 Berlin, Germany},
+  Author                   = {Machens, C.K. and Stemmler, M.B. and Prinz, P. and Krahe, R. and Ronacher, B. and Herz, A.V.},
+  Comment                  = {Machens_etal_2001.pdf},
+  ISSN                     = {1529-2401},
+  Journal                  = {J Neurosci},
+  Keywords                 = {*Animal Communication, Acoustic Stimulation/*methods, Action Potentials/physiology, Animal, Auditory Pathways/*physiology, Female, Grasshoppers, Insect, Male, Models,Neurological, Neurons, Neurons,Afferent/*physiology, Periodicity, Reaction Time/physiology, Receptors,Sensory/*physiology, Sensory Thresholds/physiology, Signal Processing,Computer-Assisted, Species Specificity, Support,Non-U.S.Gov't},
+  Month                    = may,
+  Number                   = {9},
+  Owner                    = {grewe},
+  Pages                    = {3215--3227},
+  Refid                    = {28},
+  Timestamp                = {2008.09.26},
+  Url                      = {PM:11312306},
+  Volume                   = {21},
+  Year                     = {2001}
+}
+
 @ARTICLE{Machnik2008,
   AUTHOR = {Peter Machnik and Bernd Kramer},
   TITLE = {Female choice by electric pulse duration: attractiveness of the males' communication signal assessed by female bulldog fish, \textit{Marcusenius pongolensis} {(Mormyridae, Teleostei)}.},

susceptibility1.tex

@@ -391,9 +391,9 @@
 %\paragraph{Author Contributions:} All authors designed the study and discussed the results. AB performed the data analyses and modeling, AB and JG drafted the paper, all authors discussed and revised the manuscript.
   
 \begin{keywords}
-\item heterogeneity
+\item second-order susceptibility
 \item electric fish
-\item mutual information
+\item nonlinear coding
 \end{keywords}
 
 % Please keep the abstract below 300 words
@@ -440,8 +440,8 @@ Here we search for such weakly nonlinear responses in electroreceptors of the tw
 
 Theoretical work on leaky integrate-and-fire and conductance-based models suggests a distinct structure of the second-order response function for neurons with low levels of intrinsic noise driven in the supra-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we explored a large set of recordings of P-units and ampullary cells of the active and passive electrosensory systems of the brown ghost knifefish \textit{Apteronotus leptorhynchus} to search for such weakly nonlinear responses in real neurons. Our work is supported by simulations of LIF-based models of P-unit spiking. We start with demonstrating the basic concepts using example P-units and models.
 
-\subsection{Nonlinear responses in P-units stimulated with two beat frequencies}\notejg{stimulated with two beats?}
+\subsection{Nonlinear responses in P-units stimulated with two frequencies}
-Without external stimulation, a P-unit is driven by the fish's own EOD alone (with EOD frequency \feod{}) and spontaneously fires action potentials at the baseline rate \fbase{}. Accordingly, the power spectrum of the baseline activity has a peak at \fbase{} (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish \notejg{(only valid for $f_1 < f_{EOD}/2)$}. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - \feod > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note that $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate.
+Without external stimulation, a P-unit is driven by the fish's own EOD alone (with EOD frequency \feod{}) and spontaneously fires action potentials at the baseline rate \fbase{}. Accordingly, the power spectrum of the baseline activity has a peak at \fbase{} (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - \feod > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note that $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate.
 
 When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}). Thus, the cellular response is not equal to the sum of the responses to the two beats presented separately. These peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that by definition are absent in linear systems.
 
@@ -449,7 +449,7 @@ When stimulating with both foreign signals simultaneously, additional peaks appe
 \subsection{Linear and weakly nonlinear regimes}
 \begin{figure*}[tp]
   \includegraphics[width=\columnwidth]{nonlin_regime.pdf}
-  \caption{\label{fig:nonlin_regime} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone=30$\,Hz and $\btwo=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum of the stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively.}
+  \caption{\label{fig:nonlin_regime} \notejg{legend: switch sum and difference, fbase to the front} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone=30$\,Hz and $\btwo=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum of the stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively.}
 \end{figure*}
 
 The stimuli used in \figref{fig:motivation} had the same not-small amplitude. Whether this stimulus conditions falls into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}.
@@ -460,7 +460,7 @@ This linear regime is followed by the weakly nonlinear regime. In addition to pe
 
 At higher stimulus amplitudes (\subfigref{fig:nonlin_regime}{C \& D}) additional peaks appear in the response spectrum. The linear response and the weakly-nonlinear response start to deviate from their linear and quadratic dependence on amplitude (\subfigref{fig:nonlin_regime}{E}). The responses may even decrease for intermediate stimulus contrasts (\subfigref{fig:nonlin_regime}{D}). At high stimulus contrasts, additional nonlinearities in the system, in particular clipping of the firing rate, shape the responses.
 
-For this example, we have chosen two specific stimulus (beat) frequencies. One of these matching the P-unit's baseline firing rate. In the following, however, we are interested in how the nonlinear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime.
+For this example, we have chosen two specific stimulus (beat) frequencies. One of these matching the P-unit's baseline firing rate. In the following, however, we are interested in how the nonlinear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime. For the sake of simplicity we will drop the $\Delta$ notation event though P-unit stimuli are beats.
 
 
 \subsection{Nonlinear signal transmission in low-CV P-units}
@@ -488,7 +488,7 @@ In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounc
   \caption{\label{fig:ampullary} Linear and nonlinear stimulus encoding in an ampullary afferent (cell identifier ``2012-04-26-ae"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. \figitem{C} Bad-limited white noise stimulus (top, with a cutoff frequency of 150\,Hz) added to the fish's self-generated electric field and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \eqnref{linearencoding_methods}, of the responses to stimulation with 2\,\% (light green) and 20\,\% contrast (dark green) of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for both stimulus contrasts as indicated. Pink triangles indicate the baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. }
 \end{figure*}
 
-Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs (CV$_{\text{base}}=0.06$--$0.22$) \citep{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \fbase{} and its harmonics. Since the cells do not respond to the self-generated EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a low-contrast noise stimulus (note: this is no longer an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility, where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at \fsum{} and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). 
+Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs (CV$_{\text{base}}=0.06$ to $0.22$, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \fbase{} and its harmonics. Since the cells do not respond to the self-generated EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a low-contrast noise stimulus (note: this is no longer an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility, where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at \fsum{} and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). 
 
 
 \subsection{Model-based estimation of the nonlinear structure}
@@ -513,11 +513,11 @@ With high levels of intrinsic noise, we would not expect the nonlinear response
 
 
 \subsection{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
-Using the RAM stimulation we found pronounced nonlinear responses in the limit to weak stimulus amplitudes. How do these findings relate to the situation of pure sinewave stimuli with finite amplitudes that approximate the interference of EODs of real animals? For the P-units, the relevant signals are the beat frequencies \bone{} and \btwo{} that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line of the second-order susceptibility estimated for a vanishing external RAM stimulus (\subfigrefb{model_and_data}\,\panel[iii]{C}). In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red dot, \subfigrefb{fig:motivation}{D}), that is not covered by the so-far shown part of the second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iii]{C}), in which only the response at the sum of the two stimulus frequencies is addressed. % less prominent,
+Using the RAM stimulation we found pronounced nonlinear responses in the limit to weak stimulus amplitudes. How do these findings relate to the situation of pure sinewave stimuli with finite amplitudes that approximate the interference of EODs of real animals? For the P-units, the relevant signals are the beat frequencies \fone{} and \ftwo{} that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, \ftwo{} was similar to \fbase{}, corresponding to the horizontal line of the second-order susceptibility estimated for a vanishing external RAM stimulus (\subfigrefb{model_and_data}\,\panel[iii]{C}). In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red dot, \subfigrefb{fig:motivation}{D}), that is not covered by the so-far shown part of the second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iii]{C}), in which only the response at the sum of the two stimulus frequencies is addressed. % less prominent,
 
 \begin{figure*}[t]
   \includegraphics[width=\columnwidth]{model_full.pdf}
-  \caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility,  \eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants.  Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies.  \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02.  Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
+  \caption{\label{fig:model_full} \notejg{Legend without Delta, order of legend, dots higher} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility,  \eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants.  Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies.  \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02.  Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
 \end{figure*}
 
 However, the second-order susceptibility \eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff), where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}). 
@@ -578,37 +578,22 @@ Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectru
 The population of ampullary cells is very homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties \citep{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar in the full population of ampullary cells (at the baseline firing frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract such nonlinear responses. How such nonlinear effects might influence prey detection should be addressed in future studies.
 
 \subsection{Nonlinear encoding in P-units}
-Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, however, the natural stimuli encoded by P-units are periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Natural interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities estimated using noise-stimuli for the encoding of distinct frequencies?
+Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, however, the natural stimuli encoded by P-units are periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Natural interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities observed under noise stimulation for the encoding of distinct frequencies?
 
-We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. The total signal power in noise stimuli is uniformly distributed over a wide frequency band while it is spectrally focused in pure sinewave stimuli.
+We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility.
-% If both stimuli have the same total power, i.e. are presented with the same amplitude (contrast), then at the frequency of the sinewave stimulus the power of the noise stimulus is much lower than the one of the sinewave stimulus.
+%The total signal power in noise stimuli is uniformly distributed over a wide frequency band while the power is spectrally focused in pure sinewave stimuli.
-The linearizing effect of such narrow-band signals is much weaker in comparison to a broad-band noise stimulus of the same total power \notejb{Benjamin: can we say it like that? No. Is there a reference for it? No.}. \notejb{Periodische SIgnale vor dem Hintergrund des intrinsischen Zellrauschens, oder: Breitbandrauschstimuli (mit Bestimmung von Spektren hoeherer Ordnung) tragen zusaetzliches Rauschen bei, die die Dynamik linearisieren.}. This explains why we can observe nonlinear interactions between sinewave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{model_and_data}). As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals. 
+Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, pure sinewave stimulation is spectrally focussed and drives the system on the background of the intrinsic noise. This explains why we can observe nonlinear interactions between sinewave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{model_and_data}). As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals. 
 %We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here, in accordance with the the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies. 
 
 Extracting the AM from the stimulus already requires a (threshold) nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. This nonlinearity, however, does not show up in our estimates of the susceptibilities, because in our analysis we directly relate the AM waveform to the recorded cellular responses. Encoding the time-course of the AM, however, has been shown to be linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. In contrast, we here have demonstrated nonlinear interactions originating from the spike generator for broad-band noise stimuli in the limit of vanishing amplitudes and for stimulation with two distinct frequencies. Both settings have not been studied yet.
 
-The encoding of secondary or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires an additional nonlinearity in the system that was initially attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information for strong stimuli that evoke firing rates that hit saturation nonlinearities at zero rate or the maximum rate at the EOD frequency \citep{Savard2011}. Based on our work here, we would predict that P-units with low CVs encode the social envelopes even under weak stimulation, whenever the resulting beat frequencies match or add up to the baseline firing rate. Then difference frequencies show up in the response spectrum that characterize the slow envelope. The exact transitions on nonlinear encoding from a regime of weak stimuli to a regime of strong stimuli should be addressed in further studies (\figrefb{fig:nonlin_regime}).
+The encoding of secondary or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires an additional nonlinearity in the system that was initially attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information whenever the firing rate saturates at zero or the maximum rate at the EOD frequency \citep{Savard2011}. Based on our work, we predict that P-units with low CVs encode the social envelopes even under weak stimulation, whenever the resulting beat frequencies match or add up to the baseline firing rate. Then difference frequencies show up in the response spectrum that characterize the slow envelope.
 
-The weakly nonlinear interactions in low-CV P-units could facilitate the detectability of faint signals during three-animal interactions as found in courtship contexts in the wild \citep{Henninger2018}, the electrosensory cocktail party. The detection of a faint distant intruder male could be improved by the presence of a nearby strong female stimulus, because of the nonlinear interaction terms \citep{Schlungbaum2023}. This boosting effect is, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The population of P-units is very heterogeneous in their baseline firing rates and CVs (50--450\,Hz and 0.1--1.4, respectively, \figref{fig:dataoverview}\panel{A}, \citealp{Grewe2017, Hladnik2023}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions \citep{Henninger2018, Henninger2020}, while the number of low-CV P-units is small. Whether and how this information is specifically maintained and read out by pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain onto which P-units converge \citep{Krahe2014, Maler2009a} is an open question.
+The weakly nonlinear interactions in low-CV P-units could facilitate the detectability of faint signals during three-animal interactions as found in courtship contexts in the wild \citep{Henninger2018}, the electrosensory cocktail party. The detection of a faint, distant intruder male could be improved by the presence of a nearby strong female stimulus, because of the nonlinear interaction terms \citep{Schlungbaum2023}. This boosting effect is, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The population of P-units is very heterogeneous in their baseline firing rates and CVs (50--450\,Hz and 0.1--1.4, respectively, \figref{fig:dataoverview}\panel{A}, \citealp{Grewe2017, Hladnik2023}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions \citep{Henninger2018, Henninger2020}, while the number of low-CV P-units is small. Whether and how this information is specifically maintained and read out by pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain onto which P-units converge \citep{Krahe2014, Maler2009a} is an open question.
 
-%For the response components arising due to the nonlinearities described here to be behaviorally relevant requires this information to survive the convergence onto the pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain with different tuning properties and levels of convergence \citep{Krahe2014, Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \citep{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citep{Hladnik2023} the pyramidal cell input population will be heterogeneous which contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions. A possible readout mechanism should be the topic of future studies that need to take into account that the nonlinearities are stronger in pure sine-wave stimulation and the fraction of cells that show it under naturalistic stimulation might be larger than expected from the distribution of CVs.
+\subsection{Conclusions}
 
-%The observed nonlinear effects can possibly facilitate the detectability of faint signals during a three-fish setting, the electrosensory cocktail party \citep{Henninger2018}. These nonlinear effects are, however, very specific with respect to the relation of stimulus frequencies and a given P-unit's baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish \citep{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies. 
+We have demonstrated pronounced nonlinear responses in primary electrosensory afferents at weak stimulus amplitudes and sufficiently low intrinsic noise levels. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. The resulting nonlinear components introduce spectral components not present in the original stimulus, but may provide an edge in the context of signal detection problems at stimulus amplitudes close to threshold \citep{Schlungbaum2023}. Electrosenory afferents share an evolutionary history with hair cells \citep{Baker2019} and share many response properties with mammalian auditory nerve fibers \citep{Barayeu2023, Joris2004}. Thus, we expect weakly nonlinear responses for near-threshold stimulation in auditory nerve fibers.
-%P-units, however, are very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2) \citep{Barayeu2023}. It is thus likely that there are P-units that approximately match the specificities of the different encounters. 
-
-%On the other hand, this nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:dataoverview}\panel{A}) in our sample. Only a small fraction of the P-units have a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL \citep{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons \citep{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \citep{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citep{Hladnik2023} the pyramidal cell input population will be heterogeneous. Even though heterogeneity was shown to be generally advantageous for the encoding in this \citep{Hladnik2023} and other systems \citep{Padmanabhan2010, Beiran2018} it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
-
-%\subsection{Nonlinear encoding in P-units}
-%Outside courtship behavior, the encoding of secondary or social envelopes is a common need \citep{Stamper2012Envelope}. In a previous study, it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise through nonlinear processing downstream in the ELL \citep{Middleton2006, Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CV cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and, high CVs could indeed encode social envelopes \citep{Savard2011}. These findings are in contrast to the previously mentioned work \citep{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness \citep{Chacron2004}. Bursting was not explicitly addressed in the previous work, still, the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting \citep{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in the following works \citep{Barayeu2024, Schlungbaum2024}. Note that in this work we operated in a regime of weak stimuli and that the envelope-encoding addressed in \citep{Savard2011, Middleton2007} operates in a regime of strong stimuli, where firing rates are saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli should  be addressed in further P-unit studies. 
-
-% Envelope extraction requires a nonlinearity:
-% In the context of social signaling among three fish, we observe an AM of the AM, also referred to as second-order envelope or just social envelope \citep{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities \citep{Middleton2006} and it was shown that a subpopulation of P-units is sensitive to envelopes \citep{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli \citep{Nelson1997, Chacron2004}.
-
-
-
-\subsection{Conclusions and outlook}
-We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferents of the weakly electric fish \lepto{}. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-order susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
-auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citep{Joris2004}.  Nevertheless, the theory holds for systems that are directly driven by a sinusoidal input (ampullary cells) and systems that are driven by amplitude modulations of a carrier (P-units) and is thus widely applicable.
 
 
 \section{Methods}