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18
references.bib
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@ -7032,3 +7032,21 @@ microelectrode recordings from visual cortex and functional implications.},
volume={51},
pages={107–-128},
}
@Article{Pikovsky1997,
title={Coherence resonance in a noise-driven excitable system.},
author={Arkady S. Pikovsky and J\"urgen Kurths},
year={1997},
journal={Phys Rev Lett},
volume={78},
pages={775},
}
@Article{Lindner2004,
title={Effects of noise in excitable systems.},
author={B. Lindner and J. Garc\'ia-Ojalvo and A. Neiman and L. Schimansky-Geier},
year={2004},
journal={Phys Rep},
volume={392},
pages={321–-424},
}

41
susceptibility1.tex
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@ -398,7 +398,7 @@
% Please keep the abstract below 300 words
\section{Abstract}
Neuronal processing is inherently nonlinear --- spiking thresholds or rectification in synapses are central to neuronal computations. Nevertheless, linear response theory has been instrumental in understanding, for example, the impact of noise or synchronous spikes on signal transmission, or the emergence of oscillatory activity. At higher signal-to-noise ratios, however, the third term in the Volterra series becomes relevant. This second-order susceptibility captures nonlinear interactions between pairs of stimulus frequencies. Theoretical results for leaky integrate-and-fire neurons suggest strong responses at the sum of two input frequencies only when these frequencies or their sum match the neuron's baseline firing rate. We here analyze second-order susceptibilities in two types of primary electroreceptor afferents, P-units of the active and ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we find the predicted weakly nonlinear responses in some P-units with very low baseline interspike-interval variability and much stronger in all ampullary cells, which are less noisy than P-units. Such nonlinear responses boost responses to weak sinusoidal stimuli and are therefore of immediate relevance for wave-type electric fish that are exposed to superpositions of many frequencies in social contexts.
Neuronal processing is inherently nonlinear --- spiking thresholds or rectification in synapses are central to neuronal computations. Nevertheless, linear response theory has been instrumental in understanding, for example, the impact of noise or synchronous spikes on signal transmission, or the emergence of oscillatory activity. At higher signal-to-noise ratios, however, the second order in the Volterra \notebl{Volterra wurde noch nicht erwaehnt} series becomes relevant. This second-order susceptibility captures nonlinear interactions between pairs of stimulus frequencies. Theoretical results for leaky integrate-and-fire neurons suggest strong responses at the sum of two input frequencies only when these frequencies or their sum match the neuron's baseline firing rate. We here analyze second-order susceptibilities in two types of primary electroreceptor afferents, P-units of the active and ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we find the predicted weakly nonlinear responses in some P-units with very low baseline interspike-interval variability and much stronger in all ampullary cells, which are less noisy than P-units. Such nonlinear responses boost responses to weak sinusoidal stimuli and are therefore of immediate relevance for wave-type electric fish that are exposed to superpositions of many frequencies in social contexts.
% Please keep the Author Summary between 150 and 200 words
% Use the first person. PLOS ONE authors please skip this step.
@ -421,7 +421,7 @@ We like to think about signal encoding in terms of linear relations with unique
The transfer function used to describe 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 spider mechanoreceptors \citep{French2001}, mammalian visual systems \citep{Victor1977, Schanze1997}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated.
Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Vice versa, 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{Lindner2001, Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, where 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:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Vice versa, 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, where 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:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
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 electroreceptors 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 non-linear 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. Non-linear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation non-linearities 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}.
@ -449,6 +449,7 @@ When stimulating with both foreign signals simultaneously, additional peaks appe
\subsection{Linear and weakly nonlinear regimes}
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\notebl{E: add linear and quadratic functions}
\caption{\label{fig:nonlin_regime} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone=30$\,Hz and $\btwo=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum of the stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-units as a function of beat contrast (contrasts increase equally for both beats).}
\end{figure*}
@ -465,21 +466,21 @@ For this example, we have chosen two specific stimulus (beat) frequencies. One o
\subsection{Nonlinear signal transmission in low-CV P-units}
Weakly nonlinear responses are expected in cells with sufficiently low intrinsic noise levels, i.e. low baseline CVs \citep{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD \citep{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 \citep{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}).
Weakly nonlinear responses are expected in cells with sufficiently low intrinsic noise levels, i.e. low baseline CVs \citep{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD \citep{Bastian1981a}. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:punit}{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 \citep{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:punit}{B}).
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{cells_suscept.pdf}
\caption{\label{fig:cells_suscept} 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) measures the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation of 10\,s duration. \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.}
\caption{\label{fig:punit} 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) measures the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation of 10\,s duration. \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.}
\end{figure*}
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 using 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 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 purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). 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 \citep{Benda2005}.
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:punit}{C}, top trace, red line), are 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 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 purple for low and high contrast stimuli, \subfigrefb{fig:punit}{C}). 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: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{eq:susceptibility}, quantifies the amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. 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:cells_suscept}{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 \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \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:lifresponse}{B}, pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
The second-order susceptibility, \eqnref{eq:susceptibility}, quantifies for each combination of two stimulus frequencies \fone{} and \ftwo{} the amplitude and phase of the stimulus-evoked response at the sum \fsum{} \notebl{but the response spectrum is also multiplied with the stimulus spectra!}\notebl{Also add that this includes responses at f1-f2}. 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 \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \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:lifresponse}{B}, pink triangle in \subfigsref{fig:punit}{E, F}).
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 is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected the susceptibility values onto the diagonal by averaging over the anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). At 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 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.
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:punit}{E}). This structure vanishes 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 by averaging over the anti-diagonals (\subfigrefb{fig:punit}{G}). At low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:punit}{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: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.
In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounced nonlinearities even at low stimulus contrasts (\figrefb{fig:cells_suscept_high_CV}). Irrespective of the CV, neither of the two example P-units shows the complete expected structure of nonlinear interactions.
In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounced nonlinearities even at low stimulus contrasts (\figrefb{fig:punithighcv}). Irrespective of the CV, neither of the two example P-units shows the complete expected structure of nonlinear interactions.
\subsection{Ampullary afferents exhibit strong nonlinear interactions}
@ -493,7 +494,7 @@ Electric fish possess an additional electrosensory system, the passive or ampull
\subsection{Model-based estimation of the nonlinear structure}
In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fig:ampullary}), we only observe nonlinear responses on the anti-diagonal 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 \citep{Voronenko2017,Franzen2023}. However, a pronounced nonlinear response at frequencies \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. In the following, we investigate how these discrepancies can be understood.
In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampullary}), we only observe nonlinear responses on the anti-diagonal 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 \citep{Voronenko2017,Franzen2023}. However, a pronounced nonlinear response at frequencies \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. In the following, we investigate how these discrepancies can be understood.
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{model_and_data.pdf}
@ -527,15 +528,15 @@ Is it possible to predict nonlinear responses in a three-fish setting based on s
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{data_overview_mod.pdf}
\caption{\label{fig:dataoverview} Nonlinear responses in P-units and ampullary cells. The second-order susceptibility is condensed into the peakedness of the nonlinearity, \nli{} \eqnref{eq:nli_equation}, that relates the amplitude of the projected susceptibility at a cell's baseline firing rate to its median (see \subfigrefb{fig:cells_suscept}{G}). Each of the recorded neurons contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
% The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:dataoverview}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles -- \figrefb{fig:cells_suscept_high_CV}).
\caption{\label{fig:dataoverview} Nonlinear responses in P-units and ampullary cells. The second-order susceptibility is condensed into the peakedness of the nonlinearity, \nli{} \eqnref{eq:nli_equation}, that relates the amplitude of the projected susceptibility at a cell's baseline firing rate to its median (see \subfigrefb{fig:punit}{G}). Each of the recorded neurons contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:punit} and circles to the cell in \figrefb{fig:punithighcv}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
% The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:dataoverview}{A, C, E} (squares -- \figrefb{fig:punit}, circles -- \figrefb{fig:punithighcv}).
% Several of the recorded neurons contribute with two samples to the population analysis as their responses have been recorded to two different contrasts of the same RAM stimulus. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:dataoverview}{A}, see methods).
% The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{} values are highlighted with black squares.
}
\end{figure*}
\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single-cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the projected nonlinearity at \fbase{} (\nli{}) \eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:cells_suscept}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:dataoverview}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:dataoverview}{C}).
All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single-cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the projected nonlinearity at \fbase{} (\nli{}) \eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:punit}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:dataoverview}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:dataoverview}{C}).
The effective stimulus strength also plays an important role. We quantify the effect of stimulus strength on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. P-units are heterogeneous in their sensitivity, their response modulations to the same stimulus contrast vary a lot \citep{Grewe2017}. Cells with weak responses to a stimulus, be it an insensitive cell or a weak stimulus, have higher \nli{} values and thus a more pronounced ridge in the second-order susceptibility at \fsumb{} in comparison to strongly responding cells that basically have flat second-order susceptibilities (\subfigrefb{fig:dataoverview}{E}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and both the CV and response strength during stimulation (effective output noise).
%(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
@ -550,17 +551,17 @@ The population of ampullary cells is generally more homogeneous, with lower base
Theoretical work \citep{Voronenko2017,Franzen2023} derived analytical expressions for weakly-nonlinear responses in LIF and theta model neurons driven by two sine waves with distinct frequencies. We here looked for such nonlinear responses in two types of electroreceptor afferents that differ in their intrinsic noise levels \citep{Grewe2017} using white-noise stimuli to estimate second-order susceptibilities. Following \citet{Voronenko2017} we expected to observe increased levels of second-order susceptibility where either of the stimulus frequencies alone or the sum of the stimulus frequencies matches the baseline firing rate ($f_1=\fbase{}$, $f_2=\fbase{}$ or \fsumb{}). We find traces of these nonlinear responses in most of the low-noise ampullary afferents and only those P-units with very low intrinsic noise levels. Complementary model simulations demonstrate, in the limit to vanishing stimulus amplitudes and extremely high number of repetitions, that the second order susceptibilities estimated from the electrophysiological data are indeed indicative of the theoretically expected weakly nonlinear responses.
% EOD locking:
% 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}) \citep{Sinz2020}.
% Phase-locking to the own field also leads to a representation of \feod{} in the P-unit firing rate (see \figref{fig:punit}\panel{B}) \citep{Sinz2020}.
% Weakly nonlinear responses versus saturation regime
\subsection{Intrinsic noise limits nonlinear responses}
The pattern of elevated second-order susceptibility found in the experimental data matches the theoretical expectations only partially. Only P-units with low coefficients of variation (CV $<$ 0.25) of the interspike-interval distribution in their baseline response show the expected nonlinearities (\figref{fig:cells_suscept}, \figref{fig:model_full}, \subfigref{fig:dataoverview}{A}). Such low-CV cells are rare among the 221 P-units used in this study. On the other hand, the majority of the ampullary cells have generally lower CVs (median of 0.12) and have an approximately ten-fold higher level of second-order susceptibilities where \fsumb{} (\figref{fig:ampullary}, \subfigrefb{fig:dataoverview}{B}).
The pattern of elevated second-order susceptibility found in the experimental data matches the theoretical expectations only partially \notebl{too negative. The theory also predicts vanishing patterns for too much noise in the system}. Only P-units with low coefficients of variation (CV $<$ 0.25) of the interspike-interval distribution in their baseline response show the expected nonlinearities (\figref{fig:punit}, \figref{fig:model_full}, \subfigref{fig:dataoverview}{A}). Such low-CV cells are rare among the 221 P-units used in this study. On the other hand, the majority of the ampullary cells have generally lower CVs (median of 0.12) and have an approximately ten-fold higher level of second-order susceptibilities where \fsumb{} (\figref{fig:ampullary}, \subfigrefb{fig:dataoverview}{B}).
The CV is a proxy for the intrinsic noise in the cells \citep{Vilela2009}. In both cell types, we observe a negative correlation between the second-order susceptibility at \fsumb{} and the CV, indicating that it is the level of intrinsic noise that shapes nonlinear responses. These findings are in line with previous theoretical and experimental studies showing the linearizing effects of noise in nonlinear systems \citep{Roddey2000, Chialvo1997, Voronenko2017}. Increased intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citep{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL \citep{Chacron2006}. Only in cells with sufficiently low levels of intrinsic noise, weakly nonlinear responses can be observed.
The CV is a proxy for the intrinsic noise in the cells (\citealp{Vilela2009}, note however the effect of coherence resonance for excitable systems close to a bifurcation, \citealp{Pikovsky1997, Lindner2004}). In both cell types, we observe a negative correlation between the second-order susceptibility at \fsumb{} and the CV, indicating that it is the level of intrinsic noise that shapes nonlinear responses. These findings are in line with previous theoretical and experimental studies showing the linearizing effects of noise in nonlinear systems \citep{Roddey2000, Chialvo1997, Voronenko2017}. Increased intrinsic noise has been demonstrated to increase the CV and to reduce nonlinear phase-locking in vestibular afferents \citep{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL \citep{Chacron2006}. Only in cells with sufficiently low levels of intrinsic noise, weakly nonlinear responses can be observed.
\subsection{Linearization by white-noise stimulation}
Not only the intrinsic noise but also the stimulaion with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\subfigrefb{fig:dataoverview}{E}) and direct stimulation in ampullary cells (\subfigrefb{fig:dataoverview}{F}). The stronger the effective stimulus, the less pronounced are the peaks in second-order susceptibility. This linearizing effect of noise stimuli limits the weakly nonlinear regime to small stimulus amplitudes. At higher stimulus amplitudes, however, other non-linearities of the system eventually show up in the second-order susceptibility.
Not only the intrinsic noise but also the stimulation with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\subfigrefb{fig:dataoverview}{E}) and direct stimulation in ampullary cells (\subfigrefb{fig:dataoverview}{F}). The stronger the effective stimulus, the less pronounced are the peaks in second-order susceptibility (see \subfigref{fig:punit}{E\&F} for a P-unit example and \subfigref{fig:ampullary}{E\&F} for an ampullary cell). This linearizing effect of noise stimuli limits the weakly nonlinear regime to small stimulus amplitudes. At higher stimulus amplitudes, however, other non-linearities of the system eventually show up in the second-order susceptibility.
In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. The model of a low-CV P-unit then shows the full nonlinear structure (\figref{model_and_data}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}.
@ -583,7 +584,7 @@ Noise stimuli have the advantage that a range of frequencies can be measured wit
We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. The total signal power in noise stimuli is uniformly distributed over a wide frequency band while it is spectrally focused in pure sinewave stimuli.
% If both stimuli have the same total power, i.e. are presented with the same amplitude (contrast), then at the frequency of the sinewave stimulus the power of the noise stimulus is much lower than the one of the sinewave stimulus.
The linearizing effect of such narrow-band signals is much weaker in comparison to a broad-band noise stimulus of the same total power \notejb{Benjamin: can we say it like that? Is there a reference for it?}. This explains why we can observe nonlinear interactions between sinewave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{model_and_data}). As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals.
The linearizing effect of such narrow-band signals is much weaker in comparison to a broad-band noise stimulus of the same total power \notejb{Benjamin: can we say it like that? No. Is there a reference for it? No.}. \notejb{Periodische SIgnale vor dem Hintergrund des intrinsischen Zellrauschens, oder: Breitbandrauschstimuli (mit Bestimmung von Spektren hoeherer Ordnung) tragen zusaetzliches Rauschen bei, die die Dynamik linearisieren.}. This explains why we can observe nonlinear interactions between sinewave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{model_and_data}). As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals.
%We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here, in accordance with the the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies.
Extracting the AM from the stimulus already requires a (threshold) nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. This nonlinearity, however, does not show up in our estimates of the susceptibilities, because in our analysis we directly relate the AM waveform to the recorded cellular responses. Encoding the time-course of the AM, however, has been shown to be linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. In contrast, we here have demonstrated nonlinear interactions originating from the spike generator for broad-band noise stimuli in the limit of vanishing amplitudes and for stimulation with two distinct frequencies. Both settings have not been studied yet.
@ -718,7 +719,7 @@ We expected to see elevated values in the second-order susceptibility at $\omega
\label{eq:nli_equation}
\nli{} = \frac{ \max D(\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz})}{\mathrm{median}(D(f))}
\end{equation}
where $D(f)$ is the projection of the absolute values of the second-order susceptibility matrix onto the diagonal, calculated by taking the mean of the anti-diagonal elements. The peak of $D(f)$ at the neuron's baseline firing rate $\fbase$ was found by finding the maximum of $D(f)$ in the range $\fbase \pm 5$\,Hz. This peak was then normalized by the median of $D(f)$ (\subfigrefb{fig:cells_suscept}{G}).
where $D(f)$ is the projection of the absolute values of the second-order susceptibility matrix onto the diagonal, calculated by taking the mean of the anti-diagonal elements. The peak of $D(f)$ at the neuron's baseline firing rate $\fbase$ was found by finding the maximum of $D(f)$ in the range $\fbase \pm 5$\,Hz. This peak was then normalized by the median of $D(f)$ (\subfigrefb{fig:punit}{G}).
If the same RAM was recorded several times in a cell, each trial resulted in a separate second-order susceptibility matrix. For the population statistics in \figref{fig:dataoverview} the mean of the resulting \nli{} values is used.
@ -833,12 +834,12 @@ Both, the reduced intrinsic noise and the RAM stimulus, need to replace the orig
\section{Supporting information}
\subsection{S1 Second-order susceptibility of high-CV P-unit}
CVs in P-units can range up to 1.5 \citep{Grewe2017, Hladnik2023}. We show the same analysis as in \figrefb{fig:cells_suscept} for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-units, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however, shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
CVs in P-units can range up to 1.5 \citep{Grewe2017, Hladnik2023}. We show the same analysis as in \figrefb{fig:punit} for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:punithighcv}{A}). In contrast to low-CV P-units, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:punithighcv}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:punithighcv}{E--F}), nor in the projected diagonals (\subfigrefb{fig:punithighcv}{G}). The overall level of nonlinearity, however, shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:punithighcv}{F}).
\label{S1:highcvpunit}
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept_high_CV.pdf}
\caption{\label{fig:cells_suscept_high_CV} Response of a noisy P-units (cell ``2018-08-24-af") with a relatively high baseline CV of 0.34 to RAM stimuli with two different contrasts. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum. \figitem{C} Top: EOD carrier (gray) with RAM (red). Center: Spike trains in response to the 5\,\% RAM contrast. Bottom: Spike trains in response to the 10\,\% RAM contrast. \figitem{D} First-order susceptibility (\eqnref{linearencoding_methods}). \figitem{E} Absolute value $|\chi_2(f_1, f_2)|$ of the second-order susceptibility, \eqnref{eq:susceptibility}, for the 5\,\% RAM contrast. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{F} $|\chi_2(f_1, f_2)|$ for the 10\,\% RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Dashed lines: Medians of the projected diagonals.}
\caption{\label{fig:punithighcv} Response of a noisy P-units (cell ``2018-08-24-af") with a relatively high baseline CV of 0.34 to RAM stimuli with two different contrasts. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum. \figitem{C} Top: EOD carrier (gray) with RAM (red). Center: Spike trains in response to the 5\,\% RAM contrast. Bottom: Spike trains in response to the 10\,\% RAM contrast. \figitem{D} First-order susceptibility (\eqnref{linearencoding_methods}). \figitem{E} Absolute value $|\chi_2(f_1, f_2)|$ of the second-order susceptibility, \eqnref{eq:susceptibility}, for the 5\,\% RAM contrast. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{F} $|\chi_2(f_1, f_2)|$ for the 10\,\% RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Dashed lines: Medians of the projected diagonals.}
\end{figure*}