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susceptibility1.tex
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@ -566,12 +566,11 @@ Theoretical work \citep{Voronenko2017} derived analytical expressions for weakly
\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: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 \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 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.
\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{Linearization by white-noise stimulation}
Not only the intrinsic noise but also the stimulaion with external white-noise linearizes the cells. This applies for both, the stimulation with AMs in P-units (\subfigrefb{fig:data_overview}{E}) and direct stimulation in ampullary cells (\subfigrefb{fig:data_overview}{F}). The stronger the effective stimulus, the less pronounced are the peaks in second-order susceptibility. In order to characterize the non-linearity 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 models driven with pairs of 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.