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% Title must be 250 characters or less.
\begin{flushleft}
{\Large
\textbf{Second-order susceptibility in electrosensory primary afferents in a three-fish setting}\\[2ex]
\textbf{Estimating and interpreting non-linear encoding in electrosensory primary afferents (in a cocktail party problem)}
\textbf{Weakly nonlinear responses at low intrinsic noise in two types of electrosensory primary afferents}}
}
\newline
% Insert author names, affiliations and corresponding author email (do not include titles, positions, or degrees).
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\section*{Introduction}
%with nonlinearities being observed in all sensory modalities
Nonlinear processes are key to neuronal information processing. Decision making is a fundamentally nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a threshold\cite{Hodgkin1952,Koch1995}. In nonlinear systems noise may even facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000, Neiman2011fish}.
Nonlinear processes are key to neuronal information processing. Decision making is a fundamentally nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a threshold\cite{Hodgkin1952,Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task.
At the heart of nonlinear system identification is the Volterra series\cite{Rieke}. Second-order kernels have been used to predict firing rate responses of catfish retinal ganglion cells \cite{Marmarelis1972}.
In the frequency domain second-order kernels are known as second-order response functions or susceptibilities. They quantify the amplitude of the response at the sum and difference of two stimulus frequencies. Adding also third-order kernels, spike trains of a spider mechanorecptors have beend predicted from sensory stimuli \cite{French2001}. The nonlinear nature of Y cells in contrast to the more linear responses of X cells in cat retinal ganglion cells has been demonstrated by means of second-order kernels\cite{Victor1977}. Interactions between different frequencies in the response of neurons in visual cortices of cats and monkeys have been studied using bispectra, the crucial constituent of the second-order susceptibliity \cite{Schanze1997}. Locking of chinchilla auditory nerve fibres to pure tone stimuli is captured by second-order kernels\cite{Temchin2005}. In paddlefish ampullary afferents, bursting in response to strong, natural sensory stimuli boost nonlinear responses in the bicoherence, the bispectrum normalized by stimulus and response spectra \cite{Neimann2011}.
\notejb{der Abschnitt kann auch weg oder in Discussion}Estimating the infinite Volterra series from limited experimental data is usually limited to the first two or three kernels, that then might not be sufficient for a proper prediction of the neuronal response \cite{French2001}. Making assumptions about the nonlinearities in a system reduces the amount of data needed for parameter estimation. In particular, models combining linear filtering with static nonlinearities\cite{Chichilnisky2001}, have been successful in capturing functionally relevant neuronal computations in visual \cite{Gollisch2009} as well as auditory systems\cite{Clemens2013}. On the other hand, linear methods based on backward models for estimating the stimulus from neuronal responses, have been extensively used to quantify information transmission in neural systems \cite{Theunissen1996,Borst1996,Wessel1996,Chacron?}, because backward models do not need to generate action potentials \cite{Rieke}.
Noise linearizes nonlinear systems \notejb{Check references, add Lindner papers}\cite{Longtin1993, Chialvo1997, Roddey2000} and therefore noisy systems can be well described by linear response theory in the limit of vanishing stimulus amplitudes \notejb{what else to cite?} \cite{Doiron2004}. When increasing stimulus amplitude, first the contribution of the second-order kernel of the Volterra series becomes relevant in addition to the linear one. For these weakly nonlinear responses of leaky-integrate-and fire (LIF) neurons an analytical expression for the second-order susceptibility has been derived \cite{Voronenko2017} in addition to its linear response function \notejb{Lindner B and Schimansky-Geier L 2001 Phys. Rev. Lett. 86 2934–7}. In the superthreshold regime, where the LIF generates a baseline firing rate in the absense of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data such strucures in the second-order susceptibility have not been reported yet.
Here we study weakly nonlinear repsonses
Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, the P-units, encode such AMs of the underlying EOD carrier in their time-dependent firing rates \cite{Bastian1981a,Walz2014}. P-units are heterogeneous in their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \cite{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. We start with exploring the influence of intrinsic noise on nonlinear encoding in P-units.
\notejb{in Voronenko they talk about second-order response functions and not of susceptibilities}
We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}.
In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.
While the encoding of signals can often be well described by linear models in the sensory periphery\cite{Machens2001}, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Adelson1985,Gabbiani1996,Olshausen1996,Gollisch2009}.
In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.
\notejb{What are the main findings of the manuscript? (i) We find patterns that match the theoretical predictions (Voronekov) for weak stimulation. No saturation regime! (ii) we see a bit of nonlinearity in low-CV P-units and strong one in ampullary cells. Nonlinearities show up at very specific frequencies. (iii) AMs with carrier (auditory) (iv) There is an estimation problem because of low N and RAM stimulus introducing linearizing noise. (v) noise split gives a good estimate that translates well to pure sine wave stimulation. (vi) motivation by functional not mechanistic characterization in a behaviorally relevant context. (vii) 2nd order susceptibility can be measured using RAM stimuli (not new, see Victor papers)}
\notejb{For the estimation problem we need to cite work that also measured higher order Wiener kernels and filters (we need to find the Andrew French paper, who else? Gabbiani?). For nonlinear encoding we need to talk about linear-nonlinear models (Chinchilinsky, Gollisch, Jan Clemens) versus Wiener series (French). And we find nonlinear responses in neurons that have been considered as quite linear, but only at specific frequency combinations and low signal amplitudes.}
Old system idendification: forward models Wiener VOlterra series both in time and frequency domain, mechanotransduction in spiders, visual systems.
Old system idendification: forward models Wiener Volterra series both in time and frequency domain, mechanotransduction in spiders, visual systems. Descriptive.
Later reverse correlations, stimulus estimation, mutual information. Of course forward prediction of the spike train is horribly nonlinear, but predicting stimulus from spikes is much more linear.
Later reverse correlations, stimulus estimation. Of course forward prediction of the spike train is horribly nonlinear, but predicting stimulus from spikes is much more linear.
Mostly linear characterization by transfer functions and coherences, lower MI bound. Rarely bispectra/bicoherences. But then mostly at higher stimulus amplitudes where saturation of fI curve gets relevant.
Shift away from series to linear-nonlinear models, very successfull in forward modelling of sensory neurons.
Shift away from Wiener series to linear-nonlinear models, very successfull in forward modelling of sensory neurons.
Mostly linear characterization by transfer functions and coherences. Rarely bispectra/bicoherences. But then mostly at higher stimulus amplitudes where saturation of fI curve gets relevant.
Linear response theory (complex valued transfer function or 1st order suscepibility) has been successful in predicting neuronal firing rates at low signal to noise ratios (small stimulus amplitudes). Cite Benji papers.
Recent theoretical work makes predictions for bispectra in the limit to small stimulus amplitudes. Very specific combinations of stimulus frequencies at f1 + f2 = fb evoke nonlinear responses.
For larger stimulus amplitudes, the nonlinear components emerge, the first one is the third term in the volterra series (cite Rieke book), in Fourier the bispectrum. Recent theoretical work makes predictions for bispectra in the limit to small stimulus amplitudes. Very specific combinations of stimulus frequencies at f1 + f2 = fb evoke nonlinear responses.
We set out to find these nonlinear responses in electroreceptor afferents. Advantage high baseline rate, good for estimation problem.
@ -545,31 +561,31 @@ We set out to find these nonlinear responses in electroreceptor afferents. Advan
\item \notejb{\cite{French1973} Derivation of the Fourier transformed kernels measured with white noise.}
\item \notejb{\cite{French1976} Technical issues and tests of Fourier transformed kernels measured with white noise.}
\item \notejb{\cite{Victor1977} Cat retinal ganglion cells, gratings with sum of 6 or 8 sinusoids. X - versus Y cells. Peak at f1 == f2 in Y cells. X-cells rather linear. Discussion of mechanism, where a nonlinearity comes in along the pathway}
\item \notejb{\cite{Marmarelis1972} Temporal 2nd order kernels, how well do kernels predict responses}
\item \notejb{\cite{Marmarelis1972} Temporal 2nd order kernels, how well do kernels predict responses, catfish retinal ganglion cells}
\item \notejb{\cite{Marmarelis1973} Temporal 2nd order kernels, how well do kernels predict responses}
\item \notejb{\cite{Victor1988} Cat retinal ganglion cells, sum of sinusoids, very technical, one measurement similar to \cite{Victor1977}.}
\item \notejb{\citep{Nikias1993} Third order spectra or bispectra. Very technical overview to higher order spectra}
\item \notejb{\cite{Mitsis2007} Spider mechanorecptor. Linear filterss, multivariate nonlinearity, and threshold. Second order kernel needed for this. Gausian noise stimuli.}
\item \notejb{\cite{French2001} Time kernels up to 3rd order for predicting spider mechanorecptor responses (spikes!)}
\item \notejb{\cite{French1999} Review on time domoin nonlinear systems identification}
\item \notejb{\cite{French1999} Review on time domain nonlinear systems identification}
\item \notejb{\cite{Temchin2005,RecioSpinosa2005} 2nd order Wiener kernel for predicting chinchilla auditory nerve fibre firing rate responses. Strong 2nd order blob at characteristic frequency}
\item \notejb{\cite{Schanze1997} lots of bispectra, visual cortex MUA recordings}
\item \notejb{\cite{Theunissen1996} Linear backward stimulus reconstruction in the context of information theory/signal-to-noise ratios}
\item \notejb{\cite{Wessel1996} Same as Theunissen1996 but for P-units}
\item \notejb{\cite{Schanze1997} lots of bispectra, visual cortex MUA recordings}
\item \notejb{\cite{Neimann2011} cross bispectrum, bicoherence, mutual information, saturating nonlinearities, `` ampullary electroreceptors of paddlefish are perfectly suited to linearly encode weak low-frequency stimuli.''}
\item \notejb{\cite{Chichilnisky2001} Linear Nonlinear Poisson model}
\item \notejb{\cite{Gollisch2009} Linear Nonlinear models in retina}
\item \notejb{\cite{Clemens2013} Grasshoppper model for female preferences}
\item \notejb{\cite{Neimann2011} cross bispectrum, bicoherence, mutual information, saturating nonlinearities, `` ampullary electroreceptors of paddlefish are perfectly suited to linearly encode weak low-frequency stimuli.''}
\end{itemize}
\noteab{The nonlinearity of a system has been accessed with the use of wiener kernels \cite{French1973,French1976}, measuring the system response to white noise stimulation. Besides that the nonlinearity of a system has been addressed by pure sinewave simulation, considering the Fourier transform of the Volterra series \cite{Victor1977,Victor1980,Shapley1979}. The estimates of the nonlinearity with both methods, white noise and sinewave stimulation, was shown to yield similar results \cite{Vitor1979}. Nonlinearity was investigated, not addressing the system properties, but focusing on the quadratic phase coupling of the two input frequencies \cite{Nikias1993, Neiman2011fish}. With these approaches nonlinearity at the sum of two input frequencies was quantified in retinal cells \cite{Shapley1979} for stimuli with small amplitudes, in ampullary cells \cite{Neiman2011fish}, in the EEG of sleep \cite{Barnett1971,Bullock1997} and in mechanorecetors \cite{French1976}. Second-order responses have been quantified in not amplitude modulated \cite{Neiman2011fish} and amplitude modulated systems \cite{Victor1977,Victor1980,Shapley1979}.}
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{plot_chi2}
\caption{\label{fig:plot_chi2} Nonlinearity predicted based on the analytic results in \cite{Voronenko2017}. \figitem{A} Second-order susceptibility. \figitem{B} First-order susceptibility.
}
\caption{\label{fig:lifresponse} First- (linear) and second-order response functions of the leaky integrate-and-fire model \cite{Voronenko2017}. \figitem{A} First-order susceptibility. \figitem{B} Second-order susceptibility. The plots show the analytical solutions from \cite{Lindner} and \cite{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$.}
\end{figure*}
While the encoding of signals can often be well described by linear models in the sensory periphery\cite{Machens2001}, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Adelson1985,Gabbiani1996,Olshausen1996,Gollisch2009}.
In active electrosensation, the self-generated electric field (electric organ discharge, EOD) that is quasi sinusoidal in wavetype electric fish acts as the carrier signal that is amplitude modulated in the context of communication\cite{Walz2014, Henninger2018, Benda2020}, object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < \feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a \notejb{nonlinearity}\cite{Middleton2006,Stamper2012Envelope,Savard2011,Barayeu2023} encoding the time-course of the AM is linear over a wide range of AM amplitudes and frequencies\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}.
\begin{figure*}[t]
@ -585,7 +601,6 @@ The P-unit responses can be partially explained by simple linear filters. The li
\section*{Results}
\notejb{Diese Absatz ist eigentlich eine schoene Einleitung!}
Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, the P-units, encode such AMs of the underlying EOD carrier in their time-dependent firing rates \cite{Bastian1981a,Walz2014}. P-units are heterogeneous in their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \cite{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. We start with exploring the influence of intrinsic noise on nonlinear encoding in P-units.