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@ -561,26 +561,25 @@ The population of ampullary cells is generally more homogeneous, with lower base
% TO DISCUSSION:
%Even though the second-order susceptibilities here were estimated from data and models with a modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}.
\notejb{Insert short summary here}
\subsection{Theory applies to systems with and without carrier}
Theoretical work \citep{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \citep{Voronenko2017, Egerland2020, Neiman2011fish, Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=\fbase{}$ or $f_2=\fbase{}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} (e.g.\,\subfigrefb{model_and_data}\,\panel[iii]{C}). Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches the expectations only partially and only in a subset of neurons (\figsref{fig:cells_suscept} and \ref{fig:ampullary}). Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
Theoretical work \citep{Voronenko2017} derived analytical expressions for weakly-nonlinear responses in a LIF model neuron driven by two sine waves with distinct frequencies. We here search 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 observed traces are indeed indicative of the expected nonlinear responses.
\subsection{Intrinsic noise limits nonlinear responses}
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\subfigref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinearities (\subfigref{fig:data_overview}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however, have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are about 10-fold higher than in P-units.
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:data_overview}{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:data_overview}{B}).
The CV is a proxy for the intrinsic noise in the cells \cite{}\notejb{Cite Lindner IF by rate and CV}. 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 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 \notejb{Cite Lindner IF by rate and CV}. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system \citep{Roddey2000, Chialvo1997, Voronenko2017}. More 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}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity \citep{Barayeu2023}. We can use this model and apply a noise-split \citep{Lindner2022} based on the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}, to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\,\% of the total noise as a signal and simulating large numbers of trials uncovers the full nonlinear structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli \citep{Voronenko2017}.
\subsection{Linearizing effects of white-noise stimulation}
Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model \citep{Barayeu2023}. We can use this model and apply a noise-split \citep{Lindner2022} based on the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}, to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\,\% of the total noise as a signal and simulating large numbers of trials uncovers the full nonlinear structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli \citep{Voronenko2017}.
%
\subsection{Noise stimulation approximates the real three-fish interaction}
Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission. In our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \citep{Voronenko2017}.
In the natural situation, the stimuli are periodic signals defined by the difference frequencies. How well can we extrapolate from the white noise analysis to the pure sinewave situation? %\notejg{Predictions from the X2 matrix and the equations in Voronekov}
In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) during noise stimulation, the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}, \subfigref{fig:model_full}{B--E}). With the noise-splitting trick, we could show that in low-CV cells, that have a low-CV and are not subject to strong stimulation, the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
Our analysis is based on the neuronal responses to white-noise stimulus sequences. %For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells.
These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission.
%In our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \citep{Voronenko2017}.
In the real-world situation, the natural stimuli are periodic signals defined by the EODs of the interacting fish and the resulting AMs. How well can we extrapolate from the white-noise analysis to the pure sinewave situation?
In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total signal power of the noise stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}, \subfigref{fig:model_full}{B--E}). With the noise-splitting trick, we could show \notejg{too hard?} that low-CV cells do have the full nonlinearity pattern that is, however, covered by the intrinsic noise. 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. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies.
% The nonlinearity of ampullary cells in paddlefish \citep{Neiman2011fish} has been previously accessed with bandpass limited white noise.
@ -603,7 +602,7 @@ The afferents of the passive electrosensory system, the ampullary cells, were fo
\subsection{Conclusion}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferents of the weakly electric fish \lepto{}, systems that are very often characterized using linear methods. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-order susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citep{Joris2004}.
auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citep{Joris2004}. Nevertheless, the theory holds for systems that are directly driven by a sinusoidal input (ampullary cells) and systems that are driven by amplitude modulations of a carrier (P-units) and is thus widely applicable.
\section{Methods}