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results section on amplitude dependence

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susceptibility1.tex
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\newcommand{\bone}{$\Delta f_{1}$}
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\end{figure*}
Theoretical work on leaky integrate-and-fire and conductance-based models suggests a distinct structure of the second-order response function for neurons with low levels of intrinsic noise driven in the super-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we re-analyze a large set of recordings of P-units and ampullary cells of the active and passive electrosensory systems of the brown ghost knifefish \textit{Apteronotus leptorhynchus} together with simulations of LIF-based models of P-unit spiking to search for such weakly nonlinear responses in real neurons. We start with a few example P-units to demonstrate the basic concepts.
Theoretical work on leaky integrate-and-fire and conductance-based models suggests a distinct structure of the second-order response function for neurons with low levels of intrinsic noise driven in the super-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we re-analyze a large set of recordings of P-units and ampullary cells of the active and passive electrosensory systems of the brown ghost knifefish \textit{Apteronotus leptorhynchus} together with simulations of LIF-based models of P-unit spiking to search for such weakly nonlinear responses in real neurons. We start with a few example P-units and models to demonstrate the basic concepts.
\subsection{Nonlinear responses in P-units stimulated with two beat frequencies}
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\caption{\label{fig:nonlin_regime} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone = 30$\,Hz and $\btwo{}=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum o fthe stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-units as a function of beat contrast (contrasts increase equally for both beats). }
\end{figure*}
Without any external stimulation, a P-unit fires action potentials at a spontaneous baseline rate \fbase{} to the fish's own EOD of frequency \feod{}. Accordingly, a peak at \fbase{} is present in the power spectrum of this baseline activity (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently, a peak at this beat frequency appears the the power spectrum of the response (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic indicates a nonlinear response that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish with a higher beat frequency $\Delta f_2 = f_2 - \feod$ results in a weaker response with a single peak in the response power spectrum (\subfigrefb{fig:motivation}{C}). Note that $\Delta f_2$ has been chosen to match the P-unit's baseline firing rate.
When stimulating the fish with both frequencies, additional peaks appear in the response power spectrum at the sum \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}). Thus, the response is not equal to the sum of the responses to the two beats presented separately. These peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that by definition are absent in linear systems.
The beat stimuli in the example were strong and partially revealed saturating nonlinearities of the P-units. For weakly nonlinear responses we need to use stimuli of much lower amplitudes.
\subsection{Linear and weakly nonlinear regimes}
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{nonlin_regime.pdf}
\caption{\label{fig:nonlin_regime} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone=30$\,Hz and $\btwo=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum o fthe stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-units as a function of beat contrast (contrasts increase equally for both beats).}
\end{figure*}
The response of a P-unit to varying beat amplitudes can be modeled by a leaky-integrate-and-fire (LIF) model, fitted to the baseline firing properties an electrophysiologically measured P-unit. In the chosen P-unit model nonlinear peaks (orange marker) appear for intermediate beat contrasts (\subfigrefb{fig:motivation}{B}), decrease for stronger contrasts (\subfigrefb{fig:motivation}{C}) and again emerges for very strong beat contrasts (\subfigrefb{fig:motivation}{D}). Thus two regimes of nonlinearity can be observed for intermediate and strong beat amplitudes (\subfigrefb{fig:motivation}{E}).
The beat stimuli in the example shown in \figref{fig:motivation} had just one amplitude that was not small. Whether this falls into the weakly nonlinear regime is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use an stochastic leaky-integrate-and-fire (LIF) model \citep{Barayeu2023}, that faithfully reproduces baseline firing properties and responses to step stimuli of a specific electrophysiologically measured P-unit.
For this example, we have chosen two specific stimulus (beat) frequencies. For a full characterization of the nonlinear responses, we need to measure the response of the P-units to many different combinations of stimulus frequencies.
At very low stimulus contrasts smaller than about 0.5\,\% only peaks at the stimulating beat frequencies appear in the response spectrum (\subfigref{fig:nonlin_regime}{A}). The amplitude of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}). This is the linear response at low stimulus amplitudes.
The linear regime is followed by the weakly non-linear regime. In addition to peaks at the stimulus frequencies, peaks at the sum and the difference of the stimulus frequencies appear in the response spectrum (\subfigref{fig:nonlin_regime}{B}). THe amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}).
At higher stimulus amplitudes (\subfigref{fig:nonlin_regime}{C \& D}) additional peaks appear in the response. The linear response and the weakly-nonlinear response start to deviate from their linear and waudratic dependence on amplitude (\subfigref{fig:nonlin_regime}{E}). The responses may even decrease for intermediate stimulus contrasts (\subfigref{fig:nonlin_regime}{D}).
For this example, we have chosen two specific stimulus (beat) frequencies. In particular one matching the P-unit's baseline firing rate. In the following, however, we are interested on how the non-linear responses depend on different combinations of stimulus frequencies in the weakly-nonlinear regime.
\subsection{Nonlinear signal transmission in low-CV P-units}