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susceptibility1.tex
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@ -425,7 +425,7 @@ Noise linearizes nonlinear systems \citep{Yu1989, Chialvo1997} and therefore noi
Here we study weakly nonlinear responses in two electrosensory systems in the wave-type electric fish \textit{Apteronotus leptorhynchus}. These fish generate a quasi-sinusoidal dipolar electric field (electric organ discharge, EOD). In communication contexts \citep{Walz2014, Henninger2018} the EODs of close-by fish superimpose and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \citep{Yu2005, Fotowat2013}. Therefore, stimuli with multiple distinct frequencies are part of the everyday life of wave-type electric fish \citep{Benda2020} and interactions of these frequencies in the electrosensory periphery are to be expected. P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, use nonlinearities to extract and encode these AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. 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) \citep{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. On the other hand, ampullary cells of the passive electrosensory system are homogeneous in their response properties and have very low CVs \citep{Grewe2017}.
Field observations have shown that courting males were able to react to distant intruder males despite the strong EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear responses are of direct relevance in this setting since they could boost responses to the faint signal of the distant intruder and thus improve detection \citep{Mascha}.
Field observations have shown that courting males were able to react to distant intruder males despite the strong EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear responses are of direct relevance in this setting since they could boost responses to the faint signal of the distant intruder and thus improve detection \citep{Schlungbaum2023}.
@ -447,16 +447,22 @@ Theoretical work on leaky integrate-and-fire and conductance-based models sugges
\end{figure*}
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\caption{\label{fig:nonlin_regime} Nonlinear response of a model P-unit in response to increasing beat amplitudes in a three-fish or two-beat setting. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D}. Top -- stimulus consisting of two beats. The beat frequencies are 30\,Hz (\bone{}) and 130\,Hz (\btwo{}). \btwo{} is equal to the baseline firing rate \fbase{}. The beat contrasts are equal in the respective example, and increase from \panel{A} to \panel{D}. Middle -- spike response of the model P-unit to the stimulus above. Bottom -- power spectrum of the firing rate of this P-unit. Nonlinear effects at \bsum{} (orange marker) and at \bdiff{} (red marker) increase for intermediate contrasts (\panel{B}), decrease for stronger contrasts (\panel{C}) and again increase for very strong contrasts (\panel{D}). \figitem{E} Amplitude of the linear (\bone{} and \btwo{}) and nonlinear (\bdiff{} and \bsum{}) responses of the model P-units plotted for increasing beat contrasts (contrasts equal for both beats). }
\end{figure*}
Without any external stimulation, a P-unit fires action potentials at a spontaneous baseline rate \fbase{} to the fish's own EOD of frequency \feod{}. Accordingly, a peak at \fbase{} is present in the power spectrum of this baseline activity (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently, a peak at this beat frequency appears the the power spectrum of the response (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic indicates a nonlinear response that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish with a higher beat frequency $\Delta f_2 = f_2 - \feod$ results in a weaker response with a single peak in the response power spectrum (\subfigrefb{fig:motivation}{C}). Note that $\Delta f_2$ has been chosen to match the P-unit's baseline firing rate.
When stimulating the fish with both frequencies, additional peaks appear in the response power spectrum at the sum \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}). Thus, the response is not equal to the sum of the responses to the two beats presented separately. These peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that by definition are absent in linear systems.
For this example, we have chosen two specific stimulus (beat) frequencies. However, for a full characterization of the nonlinear responses, we would need to measure the response of the P-units to many different combinations of stimulus frequencies. In addition, the beat stimuli in the example were strong and partially revealed saturating nonlinearities of the P-units. For weakly nonlinear responses we need to use stimuli of much lower amplitudes.
The beat stimuli in the example were strong and partially revealed saturating nonlinearities of the P-units. For weakly nonlinear responses we need to use stimuli of much lower amplitudes.
The response of a P-unit to varying beat amplitudes has been estimated by leaky-integrate-and-fire (LIF) models, fitted to the baseline firing properties of electrophysiologically measured P-units. In the chosen P-units model nonlinear peaks (red and orange markers) appear for intermediate beat stimuli (\subfigrefb{fig:motivation}{B}), decreases for stronger stimuli (\subfigrefb{fig:motivation}{C}) and again emerges for very strong stimuli (\subfigrefb{fig:motivation}{D}). Thus two regimes of nonlinearity can be observed for intermediate and strong beat amplitudes (\subfigrefb{fig:motivation}{E}).
For this example, we have chosen two specific stimulus (beat) frequencies. For a full characterization of the nonlinear responses, we need to measure the response of the P-units to many different combinations of stimulus frequencies.
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\caption{\label{fig:nonlin_regime} Nonlinear response of a model P-unit in response to increasing beat amplitudes in a three-fish or two-beat setting. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D}. Top -- stimulus consisting of two beats. The beat frequencies are 30\,Hz (\bone{}) and 130\,Hz (\btwo{}). \btwo{} is equal to the baseline firing rate \fbase{}. The beat contrasts are equal in the respective example, and increase from \panel{A} to \panel{D}. Middle -- spike response of the model P-unit to the stimulus above. Bottom -- power spectrum of the firing rate of this P-unit. Nonlinear effects at \bsum{} (orange marker) and at \bdiff{} (red marker) increase for intermediate contrasts (\panel{B}), decrease for stronger contrasts (\panel{C}) and again increase for very strong contrasts (\panel{D}). \figitem{E} Amplitude of the linear (\bone{} and \btwo{}) and nonlinear (\bdiff{} and \bsum{}) responses of the model P-units plotted for increasing beat contrasts (contrasts equal for both beats). }
\end{figure*}
\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}).

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susceptibility1.tex.bak
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@ -425,7 +425,7 @@ Noise linearizes nonlinear systems \citep{Yu1989, Chialvo1997} and therefore noi
Here we study weakly nonlinear responses in two electrosensory systems in the wave-type electric fish \textit{Apteronotus leptorhynchus}. These fish generate a quasi-sinusoidal dipolar electric field (electric organ discharge, EOD). In communication contexts \citep{Walz2014, Henninger2018} the EODs of close-by fish superimpose and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \citep{Yu2005, Fotowat2013}. Therefore, stimuli with multiple distinct frequencies are part of the everyday life of wave-type electric fish \citep{Benda2020} and interactions of these frequencies in the electrosensory periphery are to be expected. P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, use nonlinearities to extract and encode these AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. 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) \citep{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. On the other hand, ampullary cells of the passive electrosensory system are homogeneous in their response properties and have very low CVs \citep{Grewe2017}.
Field observations have shown that courting males were able to react to distant intruder males despite the strong EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear responses are of direct relevance in this setting since they could boost responses to the faint signal of the distant intruder and thus improve detection \citep{Mascha}.
Field observations have shown that courting males were able to react to distant intruder males despite the strong EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear responses are of direct relevance in this setting since they could boost responses to the faint signal of the distant intruder and thus improve detection \citep{Schlungbaum2023}.
@ -447,16 +447,20 @@ Theoretical work on leaky integrate-and-fire and conductance-based models sugges
\end{figure*}
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\caption{\label{fig:nonlin_regime} Nonlinear response of a model P-unit in response to increasing beat amplitudes in a three-fish or two-beat setting. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D}. Top -- stimulus consisting of two beats. The beat frequencies are 30\,Hz (\bone{}) and 130\,Hz (\btwo{}). \btwo{} is equal to the baseline firing rate \fbase{}. The beat contrasts are equal in the respective example, and increase from \panel{A} to \panel{D}. Middle -- spike response of the model P-unit to the stimulus above. Bottom -- power spectrum of the firing rate of this P-unit. Nonlinear effects at \bsum{} (orange marker) and at \bdiff{} (red marker) increase for intermediate contrasts (\panel{B}), decrease for stronger contrasts (\panel{C}) and again increase for very strong contrasts (\panel{D}). \figitem{E} Amplitude of the linear (\bone{} and \btwo{}) and nonlinear (\bdiff{} and \bsum{}) responses of the model P-units plotted for increasing beat contrasts (contrasts equal for both beats). }
\end{figure*}
Without any external stimulation, a P-unit fires action potentials at a spontaneous baseline rate \fbase{} to the fish's own EOD of frequency \feod{}. Accordingly, a peak at \fbase{} is present in the power spectrum of this baseline activity (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently, a peak at this beat frequency appears the the power spectrum of the response (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic indicates a nonlinear response that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish with a higher beat frequency $\Delta f_2 = f_2 - \feod$ results in a weaker response with a single peak in the response power spectrum (\subfigrefb{fig:motivation}{C}). Note that $\Delta f_2$ has been chosen to match the P-unit's baseline firing rate.
When stimulating the fish with both frequencies, additional peaks appear in the response power spectrum at the sum \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}). Thus, the response is not equal to the sum of the responses to the two beats presented separately. These peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that by definition are absent in linear systems.
For this example, we have chosen two specific stimulus (beat) frequencies. However, for a full characterization of the nonlinear responses, we would need to measure the response of the P-units to many different combinations of stimulus frequencies. In addition, the beat stimuli in the example were strong and partially revealed saturating nonlinearities of the P-units. For weakly nonlinear responses we need to use stimuli of much lower amplitudes.
For this example, we have chosen two specific stimulus (beat) frequencies. In addition, the beat stimuli in the example were strong and partially revealed saturating nonlinearities of the P-units. For weakly nonlinear responses we need to use stimuli of much lower amplitudes.
For a full characterization of the nonlinear responses, we need to measure the response of the P-units to many different combinations of stimulus frequencies. Such a P-unit response can be estimated by leaky-integrate-and-fire (LIF) models, fitted to the baseline firing properties of electrophysiologically measured P-units. In the chosen P-units model a nonlinear peak appears for intermediate beat stimuli (\subfigrefb{fig:motivation}{B}) decreases for stronger stimuli (\subfigrefb{fig:motivation}{C}) and again emerges for very strong stimuli (\subfigrefb{fig:motivation}{D}).
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\caption{\label{fig:nonlin_regime} Nonlinear response of a model P-unit in response to increasing beat amplitudes in a three-fish or two-beat setting. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D}. Top -- stimulus consisting of two beats. The beat frequencies are 30\,Hz (\bone{}) and 130\,Hz (\btwo{}). \btwo{} is equal to the baseline firing rate \fbase{}. The beat contrasts are equal in the respective example, and increase from \panel{A} to \panel{D}. Middle -- spike response of the model P-unit to the stimulus above. Bottom -- power spectrum of the firing rate of this P-unit. Nonlinear effects at \bsum{} (orange marker) and at \bdiff{} (red marker) increase for intermediate contrasts (\panel{B}), decrease for stronger contrasts (\panel{C}) and again increase for very strong contrasts (\panel{D}). \figitem{E} Amplitude of the linear (\bone{} and \btwo{}) and nonlinear (\bdiff{} and \bsum{}) responses of the model P-units plotted for increasing beat contrasts (contrasts equal for both beats). }
\end{figure*}
\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}).