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shortened comparison to beat stimulation

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@ -607,13 +607,12 @@ The population of ampullary cells is generally more homogeneous, with lower base
\caption{\label{fig:model_full} Second-order susceptibility qualitatively predicts responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ FFT segments of model simulations in the noise-split condition (cell ``2013-01-08-aa'', table~\ref{modelparams}). Dashed white lines mark zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies. \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 2\,\%. Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
\end{figure*}
\subsection{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
Using the RAM stimulation we found pronounced nonlinear responses in the limit to weak stimulus amplitudes. How do these findings relate to the situation of pure sinewave stimuli with finite amplitudes that approximate the interference of EODs of real animals? For the P-units, the relevant signals are the beat frequencies \fone{} and \ftwo{} that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, \ftwo{} was similar to \fbase{}, corresponding to the horizontal line of the second-order susceptibility estimated for a vanishing external RAM stimulus (\subfigrefb{fig:noisesplit}\,\panel[iii]{C}). In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red dot, \subfigrefb{fig:motivation}{D}), that is not covered by the so-far shown part of the second-order susceptibility (\subfigrefb{fig:noisesplit}\,\panel[iii]{C}), in which only the response at the sum of the two stimulus frequencies is addressed. % less prominent,
\subsection{Second-order susceptibility explains nonlinear responses to pure sinewave stimuli}
Using the RAM stimulation we found pronounced nonlinear responses, in particular in the limit to weak stimulus amplitudes. How do these findings relate to the situation of pure sinewave stimuli with finite amplitudes that approximate the interference of EODs of real animals? For the P-units, the relevant signals are the beat frequencies $f_1$ and $f_2$ that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, $f_2$ was similar to the baseline firing rate, corresponding to the horizontal ridge of the second-order susceptibility (\subfigrefb{fig:lifresponse}{B}). The difference frequency is not covered by the so-far shown part of the second-order susceptibility, in which only the response at the sum of the two stimulus frequencies is addressed.
However, the second-order susceptibility \eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff), where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}).
However, the second-order susceptibility \eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full $\chi_2$ matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at the sum $f_1 + f_2$ are shown, whereas in the lower-right and upper-left quadrants responses at the difference $f_2 - f_1$ are shown (\figref{fig:model_full}). The vertical and horizontal lines at $f_1 = r$ and $f_2 = r$ are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} and extend into the upper-left quadrant, where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at $f_2 - f_1$ evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}).
Is it possible to predict nonlinear responses in a three-fish setting based on second-order susceptibility matrices estimated from RAM stimulation (\subfigrefb{fig:model_full}{A})? We test this by stimulating the same model with two weak beats (\subfigrefb{fig:model_full}{B--E}). If we choose a frequency combination in which the sum of the two beat frequencies is equal to the model's baseline firing rate \fbase{}, a peak at the sum of the two beat frequencies appears in the power spectrum of the response (\subfigrefb{fig:model_full}{B}), as expected from \suscept. If instead we choose two beat frequencies that differ from \fbase{}, a peak is present at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one of the beat frequency matches \fbase{}, both, a peak at the sum and at the difference frequency are present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met, neither a peak at the sum nor at the difference of the two beat frequencies appears (\subfigrefb{fig:model_full}{E}).
\notejb{This works only qualitatively. In fact, the beat matrix looks quite a bit different and also heavily depends on contrast. This will be addressed in a future paper.}
Is it possible to predict nonlinear responses in a three-fish setting based on second-order susceptibility matrices estimated from RAM stimulation in the noise-split configuration (\subfigrefb{fig:model_full}{A})? We test this by stimulating the same model with two weak beats (\subfigrefb{fig:model_full}{B--E}). Qualitatively, the second-order susceptibility predicts the presence and absence of peaks in the response spectrum at the sums and differences of the two beat frequencies. However, a quantitative prediction fails. This is because pure sine waves influence a nonlinear system in different ways than a white-noise stimulus. This will be addressed in detail in a future manuscript.
\section{Discussion}