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53
references.bib
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@ -878,6 +878,19 @@ pages = {811--824}
Volume = {47}
}
@article{Borst1999,
author = {Borst, A. and Theunissen, F.E.},
journal = {Nature Neurosci.},
keywords = {Information, Information Theory, neural code, review, theory},
number = {11},
owner = {grewe},
pages = {947--957},
refid = {216},
timestamp = {2008.09.26},
title = {Information theory and neural coding},
volume = {2},
year = {1999}}
@Book{Bradbury2011,
Title = {Principles of animal communication},
Author = {Bradbury, JW and Vehrencamp, SL},
@ -2612,6 +2625,17 @@ article{Egerland2020,
Timestamp = {2020.01.27}
}
@article{Haggard2023,
title={Coding of object location by heterogeneous neural populations with spatially dependent correlations in weakly electric fish},
author={Haggard, Myriah and Chacron, Maurice J},
journal={PLOS Computational Biology},
volume={19},
number={3},
pages={e1010938},
year={2023},
publisher={Public Library of Science San Francisco, CA USA}
}
@ARTICLE{Hagiwara1963,
AUTHOR = {S. Hagiwara and H. Morita},
TITLE = {Coding mechanisms of electroreceptor fibers in some electric fish.},
@ -4802,6 +4826,24 @@ and Keller, Clifford H.},
VOLUME = {40},
PAGES = {e106010} }
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Journal = {Nat Neurosci},
Year = {2010},
Month = {Oct},
Number = {10},
Pages = {1276--1282},
Volume = {13},
Abstract = {Although examples of variation and diversity exist throughout the nervous system, their importance remains a source of debate. Even neurons of the same molecular type have notable intrinsic differences. Largely unknown, however, is the degree to which these differences impair or assist neural coding. We examined the outputs from a single type of neuron, the mitral cells of the mouse olfactory bulb, to identical stimuli and found that each cell's spiking response was dictated by its unique biophysical fingerprint. Using this intrinsic heterogeneity, diverse populations were able to code for twofold more information than their homogeneous counterparts. In addition, biophysical variability alone reduced pair-wise output spike correlations to low levels. Our results indicate that intrinsic neuronal diversity is important for neural coding and is not simply the result of biological imprecision.},
Institution = {Department of Biological Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA.},
Keywords = {Action Potentials; Animals; Animals, Newborn; Biophysical Phenomena; Biophysics; Electric Stimulation; Entropy; Kv1.2 Potassium Channel; Mice; Mice, Inbred C57BL; Models, Neurological; Nerve Net; Neurons; Olfactory Bulb; Patch-Clamp Techniques; Statistics as Topic},
Pii = {nn.2630},
Pmid = {20802489},
Refid = {792},
Timestamp = {2011.10.20},
}
@article{Palmer1982,
title={Encoding of rapid amplitude fluctuations by cochlear-nerve fibres in the guinea-pig},
author={Palmer, Alan Richard},
@ -6690,3 +6732,14 @@ groups and electrogenic mechanisms.},
Pages = {159--173},
Volume = {192}
}
@article{Zhang2022,
title={A robust receptive field code for optic flow detection and decomposition during self-motion},
author={Zhang, Yue and Huang, Ruoyu and N{\"o}renberg, Wiebke and Arrenberg, Aristides B},
journal={Current Biology},
volume={32},
number={11},
pages={2505--2516},
year={2022},
publisher={Elsevier}
}

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susceptibility1.tex
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@ -426,19 +426,19 @@ The P-unit responses can be partially explained by simple linear filters. The li
Theoretical work shows that leaky-integrate-and-fire (LIF) model neurons show a distinct pattern of nonlinear stimulus encoding when the model is driven by two cosine signals. In the context of the weakly electric fish, such a setting is part of the animal's everyday life as the sinusoidal electric-organ discharges (EODs) of neighboring animals interfere with the own field and each lead to sinusoidal amplitude modulations (AMs) that are called beats and envelopes \citealp{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units, respond to such AMs of the underlying EOD carrier and their time-dependent firing rate carries information about the stimulus' time-course. P-units are heterogeneous in their baseline firing properties \citealp{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy, more regular, firing pattern whereas high-CV P-units show a less regular firing pattern in their baseline activity.
\subsection{Low-CV P-units exhibit nonlinear interactions} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
Second-order susceptibility is expected to be especially pronounced for low-CV cells \citealp{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\citealp{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{cells_suscept}{B}).
Second-order susceptibility is expected to be especially pronounced for low-CV cells \citealp{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\citealp{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
\begin{figure*}[!h]
\includegraphics{cells_suscept}
\caption{\label{cells_suscept} \notejg{dashed lines still a little faint} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. Calculated based on the first frozen noise repeat. \textbf{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \textbf{B} Power-spectrum of the baseline response. \textbf{C} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) of the same P-unit. The stimulus contrast reflects the strength of the AM. \textbf{D} Transfer function (first-order susceptibility, \eqnref{linearencoding_methods}) of the responses to 10\% (light purple) and 20\% contrast (dark purple) RAM stimulation. \textbf{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{}. \textbf{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.}
\caption{\label{fig:cells_suscept} \notejg{dashed lines still a little faint} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. Calculated based on the first frozen noise repeat. \textbf{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \textbf{B} Power-spectrum of the baseline response. \textbf{C} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) of the same P-unit. The stimulus contrast reflects the strength of the AM. \textbf{D} Transfer function (first-order susceptibility, \eqnref{linearencoding_methods}) of the responses to 10\% (light purple) and 20\% contrast (dark purple) RAM stimulation. \textbf{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{}. \textbf{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*}
Noise stimuli, the random amplitude modulations (RAM, \subfigref{cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{cells_suscept}{C}). The linear encoding (see \eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{eq:susceptibility}, \subfigrefb{cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}{G}, compare light and dark purple lines).
Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines).
\subsection{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023}. \Figref{cells_suscept_high_CV} shows the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{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{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{cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{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{cells_suscept_high_CV}{F}).
CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023}. \Figref{fig:cells_suscept_high_CV} shows the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
\subsection{Ampullary afferents exhibit strong nonlinear interactions}
@ -486,114 +486,56 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
\begin{figure*}[ht!]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity index \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
\caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity index \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
}
\end{figure*}
%\eqnref{response_modulation}
\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{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 (Pearson's $r=-0.16$, $p<0.01$). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{A, C}. 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{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) 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 nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \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 (Pearson's $r=-0.16$, $p<0.01$). The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C}. 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\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{C}) 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 nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
%(Pearson's $r=-0.35$, $p<0.001$)
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{data_overview_mod}{B}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{data_overview_mod}{D}).
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{D}).
%(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
\section{Discussion}
Nonlinearities are ubiquitous in nervous systems, they are essential to extract certain features such as amplitude modulations of a carrier. Here, we analyzed the nonlinearity in primary electroreceptor afferents of the weakly electric fish \lepto{}. Under natural stimulus conditions, when the electric fields of two or more animals produce interference patterns that contain information of the context of the encounter, such nonlinear encoding is required to enable responding to AMs or also so-called envelopes. The work presented here is inspired by observations of electric fish interactions in the natural habitat. In these observed electrosensory cocktail-parties, a second male is intruding the conversations of a courting dyad. The courting male detects the intruder at a large distance where the foreign electric signal is strongly attenuated due to spatial distance and tiny compared to the signal emitted by the close-by female. Such a three-animal situation is exemplified in \figref{motivation} for a rather specific combination of interfering EODs. There, we observe that the electroreceptor response contains components that result from nonlinear interference (\subfigref{motivation}{D}) that might help to solve this detection task. Based on theoretical work we work out the circumstances under which electroreceptor afferents show such nonlinearities.
\subsection*{Theory applies to systems with and without carrier}
Theoretical work\citealp{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\citealp{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}{C}{iii}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory \citealp{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
\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{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.1 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\citealp{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\citealp{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\citealp{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\citealp{Barayeu2023}. We can use this model and the noise-split according to the Novikov-Furutsu theorem\citealp{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\citealp{Voronenko2017}.
%\fsum{} or \fdiff{} were equal to \fbase{} \bsum{} or \bdiff{},
\subsection{Noise stimulation approximates the real three-fish interaction}
Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. In true white noise, each frequency component contributes with equal amplitude. This has the methodological advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\citealp{Borst1999}\notejg{better references?}. In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation \figsref{fig:cells_suscept},\,\ref{ampullary} while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
%\subsection{Methodological implications}%implying that the
\subsection{Full nonlinear structure present in LIF models with and without carrier}%implying that the
%occurring nonlinearities could only be explained based on a (\foneb{}, \ftwob{}, \fsumb{}, \fdiffb{},
In this chapter, it was demonstrated that in small homogeneous populations of electrophysiological recorded cells, a diagonal structure appeared when the sum of the input frequencies was equal to \fbase{} in the second-order susceptibility (\figrefb{cells_suscept}, \figrefb{ampullary}). The size of a population is limited by the possible recording duration in an experiment, but there are no such limitations in P-unit models, where an extended nonlinear structure with diagonals, vertical and horizontal lines at \fsumb{}, \fdiffb{}, \foneb{} or \ftwob{} appeared with increased stimulus realizations (\figrefb{model_full}). This nonlinear structure is in line with the one predicted in the framework of the nonlinear response theory, which was developed based on LIF models without carrier \citealp{Voronenko2017} and was here confirmed to be valid in models with carrier (\figrefb{model_and_data}).
% The nonlinearity of ampullary cells in paddlefish \citealp{Neiman2011fish} has been previously accessed with bandpass limited white noise.
% Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells.
\subsection{Heterogeneity of P-units might influence nonlinearity}\notejg{Better title...}
The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\citealp{Henninger2018, Henninger2020}\notejg{more refs on eod freq distributions?}. To be behaviorally relevant the faint signal detection would require reliable neuronal signalling irrespective of the individual EOD frequencies.
P-units, however are very heterogeneous in their baseline firing properties\citealp{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)\citealp{Barayeu2023}. It is thus likely that there are P-units which match approximately to the specificities of the different encounters.
\subsection{Noise stimulation approximates the nonlinearity in a three-fish setting}%implying that the
In this chapter, the nonlinearity of P-units and ampullary cells was retrieved based on white noise stimulation, where all behaviorally relevant frequencies were simultaneously present in the stimulus. The nonlinearity of ampullary cells in paddlefish \citealp{Neiman2011fish} has been previously accessed with bandpass limited white noise.
On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.5\notejg{CVs aus Abbildung einfuegen}, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\citealp{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\citealp{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish surface \citealp{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citealp{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\citealp{Hladnik2023} and other systems\citealp{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
A possible readout mechanism should be the topic of future studies that need to take into account that the nonlinearities are stronger in pure sine-wave stimulation and the fraction of cells that show it under naturalistic stimulation might be larger than expected from the distribution of CVs.
Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells.
\subsection{Behavioral relevance of nonlinear interactions}
The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\citealp{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\citealp{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units responses but would arise thorough nonlinear processing downstream in the ELL \citealp{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\citealp{Savard2011}. These findings are in contrast to the previously mentioned work\citealp{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\citealp{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\citealp{Savard2011}. How bursts influence the second-order susceptibility of will be addressed in following works (in preparation).
Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, strong nonlinearities were demonstrated for weak stimuli but to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \lepto{} usually prefers small groups of fish\citealp{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons.
%\notejg{a problem, that we use a new noise for each trial?}\notesr{Using a new noise for each trial, is the way this method is defined. When using the same noise for one million repetiotions we will not see the triangular shape at any time. I tried this and Benjamin confirmed that this would be not possible.}
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview_mod}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\citealp{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation might is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\citealp{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\citealp{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\citealp{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\citealp{Grewe2017} while it is within the linear coding range in paddlefish\citealp{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\figref{fig:data_overview_mod}\panel{D}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\citealp{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies.
%the power spectrum of the firing rate
%The existence of the extended nonlinearity structure (\figrefb{model_full}) was
%\bsum{} and \bdiff{}
%, where nonlinearities in the second-order susceptibility matrix are predicted to appear at frequency combinations at the diagonals \fsumb{}, \fdiffb{}, at the vertical line \foneb{} and the horizontal line at \ftwob.
%llowing for data amounts that could not be acquired experimentally
%. It could be shown that if the nonlinearity at \fsumb{} could be found with a low stimulus repeat number in electrophysiologically recorded P-units,
%\subsection{Nonlinearity and CV}%
\subsection{Nonlinearity and CV}%
In this chapter, the CV has been identified as an important factor influencing nonlinearity in the spiking response, with strong nonlinearity in low-CV cells (\subfigrefb{data_overview_mod}{A--B}). These findings of strong nonlinearity in low-CV cells are in line with previous literature, where it has been proposed that noise linearizes the system \citealp{Roddey2000, Chialvo1997, Voronenko2017}. More noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citealp{Schneider2011}. Reduced noise of low-CV cells has been associated with stronger nonlinearity in pyramidal cells of the ELL \citealp{Chacron2006}.
\subsection{Ampullary cells}%
In this chapter strong nonlinear interactions were found in a subpopulation of low-CV P-units and low-CV ampullary cells (\subfigrefb{data_overview_mod}{A, B}). Ampullary cells have lower CVs than P-units, do not phase-lock to the EOD, and have unimodal ISI distributions. With this, ampullary cells have very similar properties as the LIF models, where the theory of weakly nonlinear interactions was developed on \citealp{Voronenko2017}. Almost all here investigated ampullary cells had pronounced nonlinearity bands in the second-order susceptibility for small stimulus amplitudes (\figrefb{ampullary}). With this ampullary cells are an experimentally recorded cell population close to the LIF models that fully meets the theoretical predictions of low-CV cells having very strong nonlinearities \citealp{Voronenko2017}. Ampullary cells encode low-frequency changes in the environment e.g. prey movement \citealp{Engelmann2010, Neiman2011fish}. The activity of a prey swarm of \textit{Daphnia} strongly resembles Gaussian white noise \citealp{Neiman2011fish}, as it has been used in this chapter. How such nonlinear effects in ampullary cells might influence prey detection could be addressed in further studies.
%\notejg{dumped here since not strictly result...}
%These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
%and here this could be confirmed experimentally.
%Nonlinear effects only for specific frequency combinations
\subsection{Heterogeneity of P-units might influence nonlinearity}
%is required to cover all female-intruder combinations} %gain_via_mean das ist der Frequenz plane faktor
%associated with nonlinear effects as \fdiffb{} and \fsumb{} were with the same firing rate low-CV P-units
%and \figrefb{ROC_with_nonlin}
P-units are very heterogeneous in their baseline firing properties considering their baseline firing rates (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}), their CV and also their bursting, the repeating firing of spikes after an EOD cycle \citealp{Chacron2004}. How this heterogeneity influences nonlinear effects when an population of P-units is integrated by the next connected pyramidal cells in the electrosensory lateral line lobe (ELL) should be addressed in further studies.
The nonlinearity in this work were found in low-CV P-units. A selective readout from a homogeneous population of low-CV cells might be required for this nonlinear effects to sustain on a population level. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citealp{Maler2009a} and P-units have heterogeneous baseline properties that do not depend on the location of the receptor on the fish body \citealp{Hladnik2023} as a result each pyramidal cell in the ELL will integrate over a heterogeneous and not homogeneous P-unit population. The center-surround receptive field organization of superficial and intermediate layer pyramidal cells in addition increases this heterogeneity of the integrated population. Heterogeneous populations with different baseline properties can be more advantageous for the encoding compared to homogeneous populations, as has been shown in P-units and models \citealp{Beiran2018, Hladnik2023}.
%\subsubsection{A heterogeneous readout is required to cover all female-intruder combinations}
Nonlinear effects might facilitate the encoding of faint signals during a three fish setting, the electrosensory cocktail party. The EOD frequencies of the encountered three fish would be drawn from the EOD frequency distribution and a stable faint signal detection would require a response irrespective of the individual EOD frequencies. Here nonlinear effects, that might influence the detection of faint signals, was found only at specific frequencies in relation to the baseline firing rate \fbase{}. Weather integrating from a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) would cover the behaviorally relevant range in the electrosensory cocktail party could be addressed in further studies.
%Thus, a heterogeneous readout might be not only physiologically plausible but also required since nonlinear effects depending on cell property, as the baseline firing rate \fbase{}, and not on stimulus properties, might be not behaviorally relevant.
%In this work, nonlinear effects were always found only for specific frequencies in relation to \fbase{}, corresponding to findings from previous literature \citealp{Voronenko2017}.
%Only a heterogeneous population could cover the whole stimulus space required during the electrosensory cocktail party.
%If pyramidal cells would integrate only from P-units with the same baseline firing rate \fbasesolid{} not all fish encounters, relevant for the context of the electrosensory cocktail party, could be covered.
%
%Second-order susceptibility was strongest the lower the CV of the use LIF model \citealp{Voronenko2017}.
% These nonlinear effects characterized In this chapter, (\bone{}, \btwo{}, \bsum{} or \bdiff{}) are specific for a three-fish setting. could be identified at \fsumb{} in small ow-CV ampullary cells and low-CV P-units
%\subsection{Full nonlinear structure corresponds to theoretical predictions}
%where second-order susceptibility could be explored systematically,
%These low-frequency modulations of the amplitude modulation are
\subsection{Encoding of secondary envelopes}%($\n{}=49$) Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study
The RAM stimulus used in this work is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citealp{Stamper2012Envelope}.
In previous works it was demonstrated that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citealp{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citealp{Middleton2007}. Based on our work we would predict that only a small class of cells, with very low CVs, should encode the social envelope at the difference frequency. If the sample in that previous work \citealp{Middleton2007} did not contain low CVs cells, this could explain the conclusion that P-units were not identified as envelope encoders.
On the other hand in previous literature the encoding of social envelopes was attributed to a subpopulation of P-units with strong nonlinearities, low firing rates and high CVs \citealp{Savard2011}. These findings are in contrast to the findings in the previously mentioned work \citealp{Middleton2007} and on first glance also to our findings. The missing link, that has not been considered in this work, might be bursting of P-units, the repeated firing of spikes after one EOD period interleaved with quiescence (unpublished work). Bursting was not explicitly addressed in the previous work, still the high CVs of the envelope encoding P-units indicate a higher rate of bursting \citealp{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
%This small percentage of the low-CV cells would be in line with no P-units found in the work.
\subsection{More fish would decrease second-order susceptibility}%
When using noise stimulation strong nonlinearity was demonstrated to appear for small noise stimuli but to decrease for stronger noise stimuli (\figrefb{cells_suscept}). A white noise stimulus is a proxy of many fish being present simultaneously. When the noise amplitude is small, those fish are distant and the nonlinearity is maintained. When the stimulus amplitude increases, many fish close to the receiver are mimicked and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \eigen{} can usually be found in groups of three to four fish \citealp{Tan2005} and \lepto{} in groups of two \citealp{Stamper2010}. Thus the here described second-order susceptibility might still be behaviorally relevant for both species. The decline of nonlinear effects when several fish are present might be adaptive for the receiver, reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons. How nonlinear effects might influence the three-fish setting known as the electrosensory cocktail party (\citealp{Henninger2018}) will be addressed in the next chapter.
\notejg{Why do we see peaks at the vertical lines in the three fish setting but not in the RAM situation? SNR? Discussion?}
\subsection{Conclusion} In this work, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. It was demonstrated that the theory of weakly nonlinear interactions \citealp{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinear effects were identified in experimentally recorded low-CV cells primary sensory afferents, the P-units and the ampullary cells. P-units share several features with mammalian
\subsection{Conclusion}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferences of the weakly electric fish \lepto{}, systems that are very often characterized using linear methods. The observed nonlinearities match the expectations from previous theoretical studies\citealp{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-oder susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citealp{Joris2004}.
@ -709,9 +651,9 @@ We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$.
\label{eq:nli_equation}
NLI(\fbase{}) = \frac{\max_{\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
\end{equation}
For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $\fbase{}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $\fbase{} \pm 5$\,Hz (\subfigrefb{cells_suscept}{G}) and dividing it by the median of $D(f)$.
For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $\fbase{}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $\fbase{} \pm 5$\,Hz (\subfigrefb{fig:cells_suscept}{G}) and dividing it by the median of $D(f)$.
If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} values was used for the population statistics in \figref{data_overview_mod}.
If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} values was used for the population statistics in \figref{fig:data_overview_mod}.
\subsection{Leaky integrate-and-fire models}\label{lifmethods}
@ -877,7 +819,7 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
\begin{figure*}[hp]%hp!
\includegraphics{cells_suscept_high_CV}
\caption{\label{cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low RAM contrast.
\caption{\label{fig:cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low RAM contrast.
Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Dashed lines: Medians of the projected diagonals.
}
\end{figure*}