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@ -520,9 +520,11 @@ In a high-CV P-unit we could not find such nonlinear structures, neither in the
We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral component of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{motivation}{D}) and at the difference between the two beat frequencies (rosa marker, \subfigrefb{motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (rosa circle, \subfigrefb{motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\eqnref{eq:crosshigh}).
\begin{figure*}[h]
\includegraphics[width=\columnwidth]{model_full}
\caption{\label{model_full} Second-order susceptibility of an electrophysiologically recorded P-unit and of the corresponding model (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. \figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B} Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods section \ref{intrinsicsplit_methods}). The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.}
\caption{\label{model_full} Second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods). The colored markers highlight the nonlinear effects found in \subfigrefb{motivation}{D}.}
\end{figure*}
%section \ref{intrinsicsplit_methods}).\figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B} The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.
The second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \citealp{Voronenko2017} (\figref{model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \figref{model_full} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in \figref{motivation}.
The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{motivation}{D}) can be explained by the fading out horizontal line in the upper-left quadrant (\figrefb{model_full}, \citealp{Schlungbaum2023}). Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\citealp{Voronenko2017, Schlungbaum2023}.
@ -530,11 +532,12 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
\begin{figure*}[ht!]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity indec (\nli{}) is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} The \nli{} is plotted against the response modulation, \eqnref{response_modulation}, 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 222 for P-units and 45 for ampullary cells.
\notejg{switch order of P-units and ampullaries.}
\caption{\label{data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity index \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} The \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 222 for P-units and 45 for ampullary cells.
}
\end{figure*}
%\eqnref{response_modulation}
\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three that depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's $r=-0.16$, $p<0.01$). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{A, C}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
%(Pearson's $r=-0.35$, $p<0.001$)