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@ -11,7 +11,7 @@ PT=$(wildcard *.py)
PYTHONFILES=$(filter-out plotstyle.py myfunctions.py numerical_compar_both.py, $(PT))
PYTHONPDFFILES=$(PYTHONFILES:.py=.pdf)
REVISION=
REVISION=fa4f64c144b4fd95b04bd9d631bd8844bb9c820a
ifdef REBUTTALBASE
REBUTTALTEXFILE=$(REBUTTALBASE).tex

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susceptibility1.tex
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@ -311,7 +311,8 @@
% Beat combinations
\newcommand{\boneabs}{\ensuremath{|\Delta f_{1}|}}%sum
\newcommand{\btwoabs}{\ensuremath{|\Delta f_{2}|}}%sum
\newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum
%\newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum
\newcommand{\bsum}{\ensuremath{\boneabs{} + \btwoabs{}}}%sum
\newcommand{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies
@ -526,35 +527,38 @@ While the sensory periphery can often be well described by linear models, this i
Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{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\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials as a baseline response $r_{0}$ at a spontaneous baseline rate \fbase{}. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom). Phase-locking to the own field also leads to a representation of \feod{} in the P-unit firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{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{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, second row). The power spectrum of the neuronal activity contains both previously seen beat peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom), thus the response of the system is not equal to the input (\subfigrefb{fig:motivation}{D}, top). The appearing difference peak is known as the social envelope\cite{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\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{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\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. 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.
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\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. 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 enhance nonlinear encoding.
\section*{Results}
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) that are called beats (two-fish interaction) and envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units encode such AMs of the underlying EOD carrier in their time-dependent firing rates \cite{Bastian1981a,Walz2014}. 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 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.
%P-units are heterogeneous in their baseline firing properties \cite{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness.
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 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 the P-unit population.
\subsection*{Nonlinear signal transmission in low-CV P-units} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
\subsection*{Nonlinear signal transmission in low-CV P-units}
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}).
Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\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 of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
\begin{figure*}[!ht]
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{cells_suscept}
\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) of the same P-unit. The stimulus contrast reflects the strength of the AM. \figitem{D} 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 the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.}
\end{figure*}
Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \Eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus driven responses of sensory neurons by means of transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly to 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 purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{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 \notejb{Cite Moe paper?}.
Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize signal
transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}.
The second-order susceptibility, \Eqnref{eq:susceptibility}, measures for each combination of two stimulus frequencies \fone{} and \ftwo{} the amplitude of the response spectrum at the sum \fsum{} of these two frequencies. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that simulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). Large values of the second-order susceptibility indicate peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. For a LIF driven in the super-threshold regime with two sinewave stimuli, theory predicts non-linear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} excatly match the neuron's baseline firing rate \fbase{}. Only then additional peaks appear in the power spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
% DAS GEHOERT IN DIE DISKUSSION:
% To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}).
\begin{figure*}[!ht]
For the low-CV P-unit we observe a band of slightly elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}). This structure vanishes for the stronger stimulus (\subfigref{fig:cells_suscept}{F}). Further, the overall level of the second-order susceptibility changes with the stimulus strength. To quantify the structural changes in the susceptibility matrices we projected all susceptibility values onto the diagonal by taking the mean over all anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). For the low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:cells_suscept}{G}, dot on top line). For the higher RAM contrast, however, this peak vanishes and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}). The reason behind this reduction is that the higher RAM 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 \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease.
In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{ampullary}
\caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
}
@ -562,7 +566,7 @@ High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus co
\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
Irrespective of the CV, neither P-unit shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green).
Irrespective of the CV, neither of the two example P-units shows the complete expected structure of nonlinear interactions. Electric fish posses an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs (CV$_{\text{base}}=0.08$--$0.22$)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a low-contrast noise stimulus (note: this is not an AM but a stimulus that is added to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}). The projected diagonal looses its distinct peak at \fsum{} and its overal level is reduced (\subfigrefb{fig:ampullary}{G}, bottom).