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PYTHONFILES=$(filter-out plotstyle.py myfunctions.py numerical_compar_both.py, $(PT))
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ifdef REBUTTALBASE
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REBUTTALTEXFILE=$(REBUTTALBASE).tex
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%%%%% notes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Title must be 250 characters or less.
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\begin{flushleft}
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{\Large
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\textbf\newline{Second-order susceptibility in electrosensory primary afferents in a three-fish setting}
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\textbf{Second-order susceptibility in electrosensory primary afferents in a three-fish setting}\\[2ex]
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\textbf{Estimating and interpreting non-linear encoding in electrosensory primary afferents (in a cocktail party problem)}
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}
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\newline
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% Insert author names, affiliations and corresponding author email (do not include titles, positions, or degrees).
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@ -516,11 +518,20 @@
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\section*{Introduction}
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%with nonlinearities being observed in all sensory modalities
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Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985,
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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 neuronal noise may facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000, Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.
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While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < \feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a nonlinearity\cite{Middleton2006,Stamper2012Envelope,Savard2011} encoding the time-course of the AM is linear over a wide range\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}.
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\begin{figure*}[!ht]
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\notejb{Wo genau soll die Einleitung hinzielen? Das ist bis jetzt eine Ansammlung von Aussagen zum Thema nonlinearities und irgendwie schleicht sich da der Fisch noch rein.}
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\notejb{What are the main findings of the manuscript? (i) 2nd order susceptibility can be measured using RAM stimuli, (ii) we see a bit of nonlinearity in low-CV P-units and strong one in ampullary cells. (iii) There is an estimation problem because of low N and RAM stimulus introducing linearizing noise. (iii) noise split gives a good estimate that translates well to pure sine wave stimulation.}
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\notejb{Strong aspects are (i) how to estimate 2nd order susceptibilities, (ii) what do they tell us about relevant stimuli, (iii) nonlinearities show up only for very specific frequencies}
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\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? John Miller?). 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.}
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While the encoding of signals can often be well described by linear models in the sensory periphery\citet{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 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}.
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\begin{figure*}[t]
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\includegraphics[width=\columnwidth]{motivation}
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\caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting (cell identifier ``2021-08-03-ac"). Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both encountered fish lead to a beat contrast of 10\,\%. Top: Sheme of a nonlinear system. Second row: Interference of the receiver EOD with the EODs of other fish. Third row: Spike trains of the P-unit. Forth row: Firing rate, retrieved as the convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: Only the receiver is present. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate. \figitem{B} The receiver and the fish with EOD frequency $f_{1}=631$\,Hz are present. \figitem{C} The receiver and the fish with EOD frequency $f_{2}=797$\,Hz are present. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate.
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}
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@ -533,6 +544,7 @@ The P-unit responses can be partially explained by simple linear filters. The li
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\section*{Results}
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\notejb{Diese Absatz ist eigentlich eine schoene Einleitung!}
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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.
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@ -541,7 +553,7 @@ Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-
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Nonlinear encoding as quantified by the second-order susceptibility is expected to be especially pronounced in cells with weak intrinsic noise, i.e. low baseline CVs \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD \cite{Bastian1981a}. 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 baseline ISI distribution has a CV$_{\text{base}}$ of 0.2, which is at the lower end of the P-unit population \cite{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 \feod{} of the electric organ and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
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\begin{figure*}[t]
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\begin{figure*}[tp]
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\includegraphics[width=\columnwidth]{cells_suscept}
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\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit (cell identifier ``2010-06-21-ai"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) reflects the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, of the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities. \notejb{``Calculated based on the first frozen noise repeat.'' Really? Only one trial has been used for computing the susceptibilities?} \noteab{Yes. In the population statistics of the last 20 years I was trying to maximize the number cells and maximize the duration. In the end the susceptibility is calculated based on 10 seconds. The susceptibility has to be calculated with newly initiated noise, with 10 seconds and 0.5 nfft durations this results in 20 trials. For the overview figure the mean of all NLI values for each frozen noise was calculated. It is not possible to take the mean over several frozen noise matrices since the baseline properties sometimes change and this disrupts the shape of the second order susceptibility.}}
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\end{figure*}
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@ -575,7 +587,7 @@ Irrespective of the CV, neither of the two example P-units shows the complete ex
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%Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents.
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In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fig:ampullary}), we only observe nonlinear responses on the anti-diagnal of the second-order susceptibility, where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response at frequencies \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. Here we investigate how these discrepancies can be understood.
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\begin{figure*}[t]
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\begin{figure*}[tp]
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\includegraphics[width=\columnwidth]{model_and_data}
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\caption{\label{model_and_data} Estimating second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of the\notejb{same as in Fig 2?} \noteab{No it is a different cell. In this cell we do not have several contrasts recorded, but a model.} low-CV P-unit (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters) driven with a weak RAM stimulus. The stimulus strength was with 2.5\,\% contrast in the electrophysiologically recorded P-unit. The contrast for the P-unit model was with 0.009\,\%, thus reproducing the CV of the electrophysiolgoical recorded P-unit during the 2.5\,\% contrast stimulation. Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ stimulus repetitions. \figitem[i-iii]{C} Same as in \panel[i-iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility. \notejb{to methods: ``Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).''}
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}
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\subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
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We estimated the second-order susceptibility of P-unit responses using RAM stimuli. In particular, we found pronounced nonlinear responses in the limit of weak stimulus amplitudes. How do these findings relate to the situation of two pure sinewave stimuli with finite amplitudes that approximates the interference of EODs of real animals? For the P-units the relevant signals are the beat frequencies \bone{} and \btwo{} that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line of the second-order susceptibility estimated for a vanishing external RAM stimulus (\subfigrefb{model_and_data}\,\panel[iii]{C}). In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red dot, \subfigrefb{fig:motivation}{D}), that is not covered by the so-far shown part of the second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iii]{C}), in which only the response at the sum of the two stimulus frequencies is addressed. % less prominent,
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\begin{figure*}[t]
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\begin{figure*}[tp]
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\includegraphics[width=\columnwidth]{model_full}
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\caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. Baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters correspond to the beat frequencies used for the stimulation with pure sine-waves in the subsequent panels and indicate the sum/difference of those beat frequencies. \figitem[]{B} Absolute value of the first-order susceptibility. \figitem{C--F} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three fish scenario). The contrasts of beat beats is 0.0065. Colored circles highlight the height of selected peaks in the power spectrum. Black circles highlight the peak height that can be predicted from \panel{A, B}. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{C} None of the two beat frequencies matches \fbase{}. \noteab{Für die Transfer Funktion habe ich jetzt einen Faktor 1, für die Nichtlinearität einen Faktor 30, aber vielleicht wenn ich über mehrere Punkte mitteln muss und das alles so noisy ist das eben noch keine Gute Abschätzung in der Stauration?}}
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\end{figure*}
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