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susceptibility1.tex
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@ -449,10 +449,9 @@ When the receiver fish with EOD frequency \feod{} is alone, a peak at \fbase{} i
%$\Delta f_{1}$ and $\Delta f_{2}$ \bone{} and \btwo{}
If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}.
If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}. In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD.
In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD and the white noise and whether it is possible to access the second-order susceptibility of P-units will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
Whether it is possible to access the second-order susceptibility of P-units with white noise stimulation will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
%is not the direct input to the neurons. with RAM stimuli

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susceptibility1.tex.bak
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@ -449,10 +449,9 @@ When the receiver fish with EOD frequency \feod{} is alone, a peak at \fbase{} i
%$\Delta f_{1}$ and $\Delta f_{2}$ \bone{} and \btwo{}
If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}.
If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}. In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD.
In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD and the white noise and whether it is possible to access the second-order susceptibility of P-units will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
Whether it is possible to access the second-order susceptibility of P-units with white noise stimulation will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
%is not the direct input to the neurons. with RAM stimuli
@ -516,16 +515,15 @@ In a high-CV P-unit we could not find such nonlinear structures, neither in the
\end{figure*}
\subsection{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
We calculated the second-order susceptibility surfaces at \fsumb{} 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 \fone{} and \ftwo{} 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 $\Delta f_{1}$ was similar to \fbase{}, corresponding to the vertical line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{B}. 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[iv]{B}, in which only the nonlinearity at \fsum{} in the response is addressed (\eqnref{eq:crosshigh}).
We calculated the second-order susceptibility surfaces at \fsumb{} 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 \fone{} and \ftwo{} 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]{B}. 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[iv]{B}, 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. Mean 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}.}
\end{figure*}
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 the \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 \subfigref{model_full}{B} for the nonlinearity at \fsum{} and extend into the lower-right quadrant (representing \fdiff)fading out towards more negative $f_2$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in \figref{motivation}.
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 the \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 \subfigref{model_full}{B} 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 vertical line in the lower-right quadrant (\subfigrefb{model_full}{B}, \citealp{Schlungbaum2023}). Even though the second-order susceptibilities were here 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}.
\notejg{Only show the model?}
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 (\subfigrefb{model_full}{B}, \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}.
\notejg{Why do we see peaks at the vertical lines in the three fish setting but not in the RAM situation? SNR? Discussion?}
\begin{figure*}[ht!]