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nonlinearbaseline.tex

@@ -2,8 +2,6 @@
 
 \title{Spike generation in electroreceptor afferents introduces additional spectral response components by weakly nonlinear interactions}
 
-%\title{Weakly nonlinear interactions of the spike generator introduce additional spectral response components in electroreceptor afferents}
-
 \author{Alexandra Barayeu\textsuperscript{1},
   Maria Schlungbaum\textsuperscript{2,3},
   Benjamin Lindner\textsuperscript{2,3},
@@ -22,13 +20,13 @@
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 \newcommand{\showkeywords}{true}
 
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 \begin{document}
 
-\begin{comment}
-
 \maketitle
 
-\thispagestyle{empty}
-
-\noindent
-\begin{tabular}{@{}lr}
-  Number of figures & 10 \\
-  Number of tables & 0 \\
-  Number of multimedia & 0 \\
-  Number of words in abstract & 201 \\
-  Number of words in introduction & 659 \\
-  Number of words in discussion & 2041
-\end{tabular}
-
-\paragraph{Acknowledgements}
-We thank Tim Hladnik, Henriette Walz, Franziska Kuempfbeck, Fabian Sinz, Laura Seidler, Eva Vennemann, and Ibrahim Tunc for the data they recorded over the years in our lab.
-
-\paragraph{Conflict of interest:}The authors declare no conflict of interest.
-
-%\paragraph{Author Contributions:} All authors designed the study and discussed the results. AB performed the data analyses and modeling, AB and JG drafted the paper, all authors discussed and revised the manuscript.
-
-\paragraph{Founding sources}
-Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by the DFG, project number 430157666.
-  
-\begin{keywords}
-\item Volterra series
-\item second-order susceptibility
-\item electric fish
-\item nonlinear coding
-\end{keywords}
-
-
-\newpage
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-
-\end{comment}
-
-\begin{center}
-  \sffamily\bfseries\LARGE Spike generation in electroreceptor afferents
-  introduces\\ additional spectral response components by\\ weakly
-  nonlinear interactions
-\end{center}
-
 % 250 words
 \section{Abstract}
 Spiking thresholds in neurons or rectification at synapses are essential for neuronal computations rendering neuronal processing inherently nonlinear. Nevertheless, linear response theory has been instrumental for understanding, for example, the impact of noise or neuronal synchrony on signal transmission, or the emergence of oscillatory activity, but is valid only at low stimulus amplitudes or large levels of intrinsic noise. At higher signal-to-noise ratios, however, nonlinear response components become relevant. Theoretical results for leaky integrate-and-fire neurons in the weakly nonlinear regime suggest strong responses at the sum of two input frequencies if one of these frequencies or their sum match the neuron's baseline firing rate.
@@ -288,21 +243,28 @@ We here analyze nonlinear responses in two types of primary electroreceptor affe
 % max 120 words
 \section{Significance statement}
 The generation of action potentials involves a strong threshold nonlinearity. Nevertheless, the encoding of stimuli with small amplitudes by neurons with sufficient intrinsic noise can be well described as a linear system. As the stimulus amplitude is increased, new spectral components start to appear in the so called weakly nonlinear regime. Theory predicts nonlinear interactions whenever one or the sum of two stimulus frequencies matches the neuron's baseline firing rate. Indeed, we find these interactions in a large set of electrophysiological recordings from primary electroreceptive afferents of a weakly electric fish. The nonlinear response components could boost sensory responses to weak signals emitted, for example, by distant conspecifics.
+  
+\begin{keywords}
+\item Volterra series
+\item second-order susceptibility
+\item electric fish
+\item nonlinear coding
+\end{keywords}
 
 
 \section{Introduction}
 
-\begin{figure*}[t]
-  \includegraphics[width=\columnwidth]{lifsuscept}
-  \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. $r$ and $2r$ mark the neuron's baseline firing rate and its harmonic (dashed vertical lines). \figitem{B} Magnitude of the second-order (nonlinear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because Fourier spectra have positive and negative frequencies, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.}
-\end{figure*}
-
 We like to think about signal encoding in terms of linear relations with unique mapping of a given input value to a certain output of the system under consideration. Indeed, such linear methods, for example the transfer function or first-order susceptibility shown in \subfigref{fig:lifsuscept}{A}, have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tools to characterize neural systems \citep{Eggermont1983,Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. Deciding for one action over another is a 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 \citep{Hodgkin1952, Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task.
 
 The transfer function that describes the linear properties of a system, is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. Including higher-order terms of the Volterra series, the nonlinear nature of mammalian visual systems \citep{Victor1977, Schanze1997}, auditory responses in the Torus semicircularis of frogs \citep{Aertsen1981}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, spider mechanoreceptors \citep{French2001}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated.
 
 Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Also, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, in which the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifsuscept}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where one of two stimulus frequencies equals or both frequencies add up to the neuron's baseline firing rate (\subfigrefb{fig:lifsuscept}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
 
+\begin{figure*}[t]
+  \includegraphics[width=\columnwidth]{lifsuscept}
+  \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. $r$ and $2r$ mark the neuron's baseline firing rate and its harmonic (dashed vertical lines). \figitem{B} Magnitude of the second-order (nonlinear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because Fourier spectra have positive and negative frequencies, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.}
+\end{figure*}
+
 Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The p-type electroreceptor afferents of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly nonlinear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the receiver's EOD. Nonlinear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the superpositions of the EODs induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation nonlinearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}.
 
 
@@ -331,8 +293,8 @@ Electric stimuli were attenuated (ATN-01M, \secret{npi-electronics, Tamm, German
 
 \subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, nixio \citep{Stoewer2014}, and thunderlab (\secret{\url{https://github.com/bendalab/thunderlab}}).
 
-\paragraph{Code accessibility}
-The P-unit model parameters and spectral analysis algorithms are available at \secret{\url{https://github.com/bendalab/punitmodel/tree/v1}}.
+\paragraph{Code and data accessibility}
+The P-unit model parameters and spectral analysis algorithms are available at \secret{\url{https://github.com/bendalab/punitmodel/tree/v1}}. The electrophysiological data of all the P-units and Ampullary cells analyzed here, the analysis scripts, and the scripts simulating the P-unit model responses are accessible at \secret{\url{https://doi.org/10.12751/g-node.4w8ebq}}.
 
 \paragraph{Baseline analysis}\label{baselinemethods}
 The baseline firing rate $r$ was calculated as the number of spikes divided by the duration of the baseline recording (median 32\,s). The coefficient of variation (CV) of the interspike intervals (ISI) is their standard deviation relative to their mean: $\rm{CV}_{\rm base} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle} / \langle ISI \rangle$. If the baseline was recorded several times in a recording, the measures from the longest recording were taken.
@@ -531,7 +493,7 @@ P-units fire action potentials probabilistically phase-locked to the self-genera
 
 Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:punit}{C}, top trace, red line), have been commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate from existing recordings the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark blue for low and high contrast stimuli, respectively, \subfigrefb{fig:punit}{C}). Linear encoding, quantified by the first-order susceptibility or transfer function, \eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:punit}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \citep{Benda2005}.
 
-The second-order susceptibility, \eqnref{chi2},  quantifies for each combination of two stimulus frequencies $f_1$ and $f_2$ the amplitude and phase of the stimulus-evoked response at the sum $f_1 + f_2$ (and also the difference, \subfigrefb{fig:lifsuscept}{A}). Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:punit}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies $f_1$ and $f_2$ or their sum $f_1 + f_2$ exactly match the neuron's baseline firing rate $r$ \citep{Voronenko2017,Franzen2023}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifsuscept}{B}).
+The second-order susceptibility, \eqnref{chi2},  quantifies for each combination of two stimulus frequencies $f_1$ and $f_2$ the amplitude and phase of the stimulus-evoked response at the sum $f_1 + f_2$ (and also the difference, \subfigrefb{fig:lifsuscept}{B}). Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:punit}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies $f_1$ and $f_2$ or their sum $f_1 + f_2$ exactly match the neuron's baseline firing rate $r$ \citep{Voronenko2017,Franzen2023}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifsuscept}{B}).
 
 For the example P-unit, we observe a ridge of elevated second-order susceptibility for the low RAM contrast at $f_1 + f_2 = r$ (yellowish anti-diagonal, \subfigrefb{fig:punit}{E}). This structure is less prominent for the stronger stimulus (\subfigref{fig:punit}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected the susceptibility values onto the diagonal (white dashed line) by averaging over the anti-diagonals (\subfigrefb{fig:punit}{G}). At low RAM contrast this projection indeed has a distinct peak close to the neuron's baseline firing rate (\subfigrefb{fig:punit}{G}, dot on top line). For the higher RAM contrast this peak is much smaller and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:punit}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude, but also increases the total noise in the system. Increased noise is known to linearize signal transmission \citep{Longtin1993, Chialvo1997,  Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease.
 
@@ -707,6 +669,13 @@ Electric fish are able to slowly modulate their EOD frequency, as for example du
 
 We have demonstrated pronounced nonlinear responses in primary electrosensory afferents at weak stimulus amplitudes and sufficiently low intrinsic noise levels. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. The resulting nonlinear components introduce spectral components not present in the original stimulus, that may provide an edge in the context of signal detection problems at stimulus amplitudes close to threshold \citep{Schlungbaum2023}. Electrosensory afferents share an evolutionary history with hair cells \citep{Baker2019} and share many response properties with mammalian auditory nerve fibers \citep{Barayeu2023, Joris2004}. Thus, we expect weakly nonlinear responses for near-threshold stimulation in auditory nerve fibers as well. These could boost or distort responses to two simultaneously presented tones and thus might play a role in forming perception of music.
 
+\paragraph{Acknowledgements}
+We thank Tim Hladnik, Henriette Walz, Franziska Kuempfbeck, Fabian Sinz, Laura Seidler, Eva Vennemann, and Ibrahim Tunc for the data they recorded over the years in our lab.
+
+\paragraph{Founding sources}
+Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by the DFG, project number 430157666,
+and by the Open Access Publication Fund of the University of T\"ubingen.
+
 
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