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references.bib
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@ -675,6 +675,24 @@ pages = {811--824}
VOLUME = {104},
PAGES = {2806-2820} }
@incollection{Benda2013,
author = {Benda, Jan and Grewe, Jan and Krahe, R{\~A}ƒ{\^A}ƒ{\~A}‚{\^A}ƒ{\~A}ƒ{\^A}‚{\~A}‚{\^A}¼diger},
booktitle = {Animal Communication and Noise},
doi = {10.1007/978-3-642-41494-7_12},
editor = {Brumm, Henrik},
isbn = {978-3-642-41493-0},
language = {English},
pages = {331-372},
publisher = {Springer Berlin Heidelberg},
refid = {876},
series = {Animal Signals and Communication},
timestamp = {2014.02.03},
title = {Neural Noise in Electrocommunication: From Burden to Benefits},
url = {http://dx.doi.org/10.1007/978-3-642-41494-7_12},
volume = {2},
year = {2013},
bdsk-url-1 = {http://dx.doi.org/10.1007/978-3-642-41494-7_12}}
@incollection{Benda2020,
title={The physics of electrosensory worlds.},
author={Benda, Jan},
@ -6455,6 +6473,18 @@ groups and electrogenic mechanisms.},
PAGES = {341--354}
}
@article{Wiesenfeld1995,
author = {Wiesenfeld, K. and Moss, F.},
journal = {Nature},
keywords = {noise, statistics},
pages = {33--36},
refid = {440},
timestamp = {2008.09.26},
title = {Stochastic resonance and the benefits of noise: from ice ages to crayfish and squids},
volume = {373},
year = {1995}
}
@Article{Wilkens2002,
Title = {The electric sense of the paddlefish: a passive system for the detection and capture of zooplankton prey.},
Author = {Lon A. Wilkens and Michael H. Hofmann and Winfried Wojtenek},

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39
susceptibility1.tex
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@ -406,39 +406,9 @@ In this work, the influence of nonlinearities on stimulus encoding in the primar
%with nonlinearities being observed in all sensory modalities
Neuronal systems are inherently nonlinear \citealp{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. A prominent example of a nonlinearity is rectification that is assumed to occur through the transduction machinery of inner hair cells \citealp{Peterson2019}, signal rectification in receptor cells \citealp{Chacron2000, Chacron2001} or the rheobase of action-potential generation \citealp{Middleton2007, Longtin2008}. Nonlinearity can be necessary to explain the behavior of complex cells in the visual system \citealp{Adelson1985}, to extract information about the stimulus \citealp{Barayeu2023} and to encode stimulus features as up- and down-strokes \citealp{Gabbiani1996}.
Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\citealp{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\citealp{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \citealp{Peterson2019}, signal rectification in electroreceptor cells \citealp{Chacron2000, Chacron2001}, or in complex cells of the visual system \citealp{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\citealp{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citealp{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \citealp{Engler2001, Hupe2008, Henninger2018, Benda2020}.
The time-resolved firing rate of a neuron can be described by the Volterra series where the first-order term describes the linear contribution and all higher-order terms the nonlinear contributions \citealp{Voronenko2017}. In previous work, the second-order response of the Volterra series, the second-order susceptibility, was analytically retrieved based on leaky integrate-and-fire (LIF) models, where the input were two sine waves \citealp{Voronenko2017}. There second-order susceptibility was demonstrated to be very pronounced at specific input frequencies. A triangular nonlinear shape, with nonlinearities appearing if one of the input frequencies, the sum or the difference of the frequencies was equal to the baseline firing rate \fbase{} of the cell, was predicted. Since these nonlinear effects were especially pronounced if one signal was weak, they might influence the faint signal detection as it was observed in the field in the framework of the electrosensory cocktail party \citealp{Henninger2018}.
%. The EOD
In this work, nonlinearities are studied in the gymnotiform weakly electric fish of the species \lepto. These fish are equipped with an electric organ \citealp{Salazar2013}, that is constantly active and produces a quasi-sinusoidal electric organ discharge (EOD), with a long-term stable and fish-specific frequency (\feod, \citealp{Knudsen1975, Henninger2020}), that is used for electrolocation \citealp{Fotowat2013, Nelson1999} and communication \citealp{Fotowat2013, Walz2014, Henninger2018}. If two fish meet, their EODs interfere and result in a new signal with a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD of the encountered fish and is expressed in relation to the receiver EOD as a contrast. The beat frequency is defined as the difference between the two EOD frequencies ($\Delta f_{1} = f_{1}-\feod{}$). Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citealp{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing the context for electrocommunication in weakly electric fish \citealp{Engler2001, Hupe2008, Henninger2018, Benda2020}. Whereas the predictions of the nonlinear response theory \citealp{Voronenko2017} are applicable if the main driving force for a neuron are beats and not the whole signal, will be addressed in this work.
%apply to P-units,
Cutaneous tuberous organs, that are distributed all over the body of these fish\citealp{Carr1982}, sense the actively generated electric field and its modulations.
Within a single tuberous electroreceptor organ about 30 primary electroreceptors form ribbon synapses onto the dendrites of the same afferent fiber
\citealp{Szabo1965, Wachtel1966} that projects via the lateral line nerve to the hindbrain. There, the afferent fiber synapses onto pyramidal neurons in the
electrosensory lateral line lobe (ELL, \citealp{Maler2009a}). The EOD is encoded by the p-type electroreceptor afferents, the P-units. These fire phase-locked to the EOD of the fish with a spike at a similar phase in the EOD cycle, with the phase depending on the receptor location \citealp{Hladnik2023}. P-units fire in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at integer multiples of the EOD period. In previous works, P-units have mainly been considered to be linear encoders \citealp{Xu1996, Benda2005, Gussin2007, Grewe2017}. Still, P-units exhibit nonlinear effects, especially for strong beat amplitudes \citealp{Nelson1997}. Although some P-units can be described by a linear decoder, other P-units require a nonlinear decoder \citealp{Chacron2004}.
When the receiver fish with EOD frequency \feod{} is alone, a peak at \fbase{} is present in the power spectrum of the firing rate of a recorded P-unit (\subfigrefb{motivation}{A}, bottom). P-units also represent \feod{} in their firing rate, but this peak is beyond the range of frequencies addressed in this figure. The beat frequency in a two fish setting is represented in the spike trains, firing rate and the corresponding power spectrum of P-units (\subfigrefb{motivation}{B, C}). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{motivation}{D}, top). The power spectrum of the P-unit's firing rate contains both beat frequencies, but also nonlinear peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). This difference of the two beat frequencies is known as the social envelope \citealp{Stamper2012Envelope, Savard2011} that often emerges as the modulation of two beats in the superimposed signal. The question whereas P-units encode envelopes has been the subject of controversy, with some works not considering P-units as envelope encoders \citealp{Middleton2006}, while others identify some P-unit populations as successful envelope encoders \citealp{Savard2011}. In this work the second-order susceptibility at the sum and difference frequency will be characterized for the three fish setting, addressing whereas P-units encode envelopes and how this encoding is influenced by their baseline firing properties.
%encoding of envelopes in
%$\Delta f_{1}$ and $\Delta f_{2}$ \bone{} and \btwo{}
If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}. In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD.
Whether it is possible to access the second-order susceptibility of P-units with white noise stimulation will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
%is not the direct input to the neurons. with RAM stimuli
%In this work, the second-order susceptibility in the spiking responses of P-units will be accessed with white noise stimulation.
%White noise stimulation will be confirmed as a method to access the second-order susceptibility in P-units.
%The influence of the baseline firing properties, such as the CV, on nonlinear interactions will be investigated.
While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\citealp{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\citealp{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\citealp{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\citealp{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$)\citealp{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \citealp{Carr1982} are tuned to the own field\citealp{Hopkins1976,Viancour1979}. P-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\citealp{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\citealp{Krahe2014}. The P-units ot the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\citealp{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike interval distribution. Even though the extraction of the AM itself requires a nonlinearity\citealp{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\citealp{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\citealp{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\citealp{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\citealp{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\citealp{Nelson1997,Chacron2004}.
\begin{figure*}[h!]
\includegraphics[width=\columnwidth]{motivation}
@ -446,6 +416,11 @@ Whether it is possible to access the second-order susceptibility of P-units with
}
\end{figure*}
Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\citealp{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\citealp{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{motivation}{A}, bottom).\notejg{more baseline trials, fake it, if needed}. Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{cells_suscept}\panel{B})\citealp{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{motivation}{D}, top). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). The latter is known as the social envelope\citealp{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\citealp{Middleton2006}, while others identify some P-units as envelope encoders\citealp{Savard2011}.
The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\citealp{Voronenko2017,Neiman2011fish,Nikias1993}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding.
\section{Results}
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.

39
susceptibility1.tex.bak
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@ -406,39 +406,9 @@ In this work, the influence of nonlinearities on stimulus encoding in the primar
%with nonlinearities being observed in all sensory modalities
Neuronal systems are inherently nonlinear \citealp{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. A prominent example of a nonlinearity is rectification that is assumed to occur through the transduction machinery of inner hair cells \citealp{Peterson2019}, signal rectification in receptor cells \citealp{Chacron2000, Chacron2001} or the rheobase of action-potential generation \citealp{Middleton2007, Longtin2008}. Nonlinearity can be necessary to explain the behavior of complex cells in the visual system \citealp{Adelson1985}, to extract information about the stimulus \citealp{Barayeu2023} and to encode stimulus features as up- and down-strokes \citealp{Gabbiani1996}.
Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\citealp{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\citealp{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \citealp{Peterson2019}, signal rectification in electroreceptor cells \citealp{Chacron2000, Chacron2001}, or in complex cells of the visual system \citealp{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\citealp{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citealp{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \citealp{Engler2001, Hupe2008, Henninger2018, Benda2020}.
The time-resolved firing rate of a neuron can be described by the Volterra series where the first-order term describes the linear contribution and all higher-order terms the nonlinear contributions \citealp{Voronenko2017}. In previous work, the second-order response of the Volterra series, the second-order susceptibility, was analytically retrieved based on leaky integrate-and-fire (LIF) models, where the input were two sine waves \citealp{Voronenko2017}. There second-order susceptibility was demonstrated to be very pronounced at specific input frequencies. A triangular nonlinear shape, with nonlinearities appearing if one of the input frequencies, the sum or the difference of the frequencies was equal to the baseline firing rate \fbase{} of the cell, was predicted. Since these nonlinear effects were especially pronounced if one signal was weak, they might influence the faint signal detection as it was observed in the field in the framework of the electrosensory cocktail party \citealp{Henninger2018}.
%. The EOD
In this work, nonlinearities are studied in the gymnotiform weakly electric fish of the species \lepto. These fish are equipped with an electric organ \citealp{Salazar2013}, that is constantly active and produces a quasi-sinusoidal electric organ discharge (EOD), with a long-term stable and fish-specific frequency (\feod, \citealp{Knudsen1975, Henninger2020}), that is used for electrolocation \citealp{Fotowat2013, Nelson1999} and communication \citealp{Fotowat2013, Walz2014, Henninger2018}. If two fish meet, their EODs interfere and result in a new signal with a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD of the encountered fish and is expressed in relation to the receiver EOD as a contrast. The beat frequency is defined as the difference between the two EOD frequencies ($\Delta f_{1} = f_{1}-\feod{}$). Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citealp{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing the context for electrocommunication in weakly electric fish \citealp{Engler2001, Hupe2008, Henninger2018, Benda2020}. Whereas the predictions of the nonlinear response theory \citealp{Voronenko2017} are applicable if the main driving force for a neuron are beats and not the whole signal, will be addressed in this work.
%apply to P-units,
Cutaneous tuberous organs, that are distributed all over the body of these fish\citealp{Carr1982}, sense the actively generated electric field and its modulations.
Within a single tuberous electroreceptor organ about 30 primary electroreceptors form ribbon synapses onto the dendrites of the same afferent fiber
\citealp{Szabo1965, Wachtel1966} that projects via the lateral line nerve to the hindbrain. There, the afferent fiber synapses onto pyramidal neurons in the
electrosensory lateral line lobe (ELL, \citealp{Maler2009a}). The EOD is encoded by the p-type electroreceptor afferents, the P-units. These fire phase-locked to the EOD of the fish with a spike at a similar phase in the EOD cycle, with the phase depending on the receptor location \citealp{Hladnik2023}. P-units fire in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at integer multiples of the EOD period. In previous works, P-units have mainly been considered to be linear encoders \citealp{Xu1996, Benda2005, Gussin2007, Grewe2017}. Still, P-units exhibit nonlinear effects, especially for strong beat amplitudes \citealp{Nelson1997}. Although some P-units can be described by a linear decoder, other P-units require a nonlinear decoder \citealp{Chacron2004}.
When the receiver fish with EOD frequency \feod{} is alone, a peak at \fbase{} is present in the power spectrum of the firing rate of a recorded P-unit (\subfigrefb{motivation}{A}, bottom). P-units also represent \feod{} in their firing rate, but this peak is beyond the range of frequencies addressed in this figure. The beat frequency in a two fish setting is represented in the spike trains, firing rate and the corresponding power spectrum of P-units (\subfigrefb{motivation}{B, C}). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{motivation}{D}, top). The power spectrum of the P-unit's firing rate contains both beat frequencies, but also nonlinear peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). This difference of the two beat frequencies is known as the social envelope \citealp{Stamper2012Envelope, Savard2011} that often emerges as the modulation of two beats in the superimposed signal. The question whereas P-units encode envelopes has been the subject of controversy, with some works not considering P-units as envelope encoders \citealp{Middleton2006}, while others identify some P-unit populations as successful envelope encoders \citealp{Savard2011}. In this work the second-order susceptibility at the sum and difference frequency will be characterized for the three fish setting, addressing whereas P-units encode envelopes and how this encoding is influenced by their baseline firing properties.
%encoding of envelopes in
%$\Delta f_{1}$ and $\Delta f_{2}$ \bone{} and \btwo{}
If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}. In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD.
Whether it is possible to access the second-order susceptibility of P-units with white noise stimulation will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
%is not the direct input to the neurons. with RAM stimuli
%In this work, the second-order susceptibility in the spiking responses of P-units will be accessed with white noise stimulation.
%White noise stimulation will be confirmed as a method to access the second-order susceptibility in P-units.
%The influence of the baseline firing properties, such as the CV, on nonlinear interactions will be investigated.
While the sensory periphery can often be well described by linear models, this is not valid for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\citealp{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\citealp{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\citealp{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\citealp{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < feod{}/2$\citealp{Barayeu2023}). Cutaneous electroreceptor organs that are distributed over the bodies of these fish \citealp{Carr1982} are tuned to the own field\citealp{Hopkins1976,Viancour1979}. P-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\citealp{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\citealp{Krahe2014}. The P-units ot the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\citealp{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike interval distribution. Even though the extraction of the AM itself requires a nonlinearity\citealp{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\citealp{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\citealp{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\citealp{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\citealp{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\citealp{Nelson1997,Chacron2004}.
\begin{figure*}[h!]
\includegraphics[width=\columnwidth]{motivation}
@ -446,6 +416,11 @@ Whether it is possible to access the second-order susceptibility of P-units with
}
\end{figure*}
Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\citealp{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\citealp{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials at a spontaneous baseline rate. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{motivation}{A}, bottom).\notejg{more baseline trials, fake it, if needed}. Phase-locking to the own field also leads to a representation of \feod{} in their firing rate (see \figref{cells_suscept}\panel{B})\citealp{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{motivation}{D}, top). The power spectrum of the neuronal activity contains both previously seen peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). The latter is known as the social envelope\citealp{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\citealp{Middleton2006}, while others identify some P-units as envelope encoders\citealp{Savard2011}.
The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\citealp{Voronenko2017,Neiman2011fish,Nikias1993}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding.
\section{Results}
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.