From 06c29fed2cb4e817310359ae1494a02413cdbb07 Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Tue, 25 Mar 2025 11:30:42 +0100
Subject: [PATCH] removed absolute values from motivation

---
 motivation.pdf      | Bin 198014 -> 198710 bytes
 motivation.py       |   4 ++--
 susceptibility1.tex |  42 ++++++++++++++++++++----------------------
 3 files changed, 22 insertions(+), 24 deletions(-)

diff --git a/motivation.pdf b/motivation.pdf
index f2b2e4a098502ee1180a0c6d9dca7bf1066197da..c54abd0de70a8b6adacba25bfb7237624595e8c0 100644
GIT binary patch
delta 20364
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diff --git a/motivation.py b/motivation.py
index bac6148..bd8f48c 100644
--- a/motivation.py
+++ b/motivation.py
@@ -106,7 +106,7 @@ def motivation_all_small(dev_desired = '1', ylim=[-1.25, 1.25], c1=10, devs=['2'
                      arrays_original[3]], spikes_pure, cell, grid0, mean_type,
                     group_mean, xlim=xlim, row=1,
                     array_chosen=array_chosen,
-                    color0_burst=color0_burst, color01=color01, color02=color02,ylim_log=(-15, 3),
+                    color0_burst=color0_burst, color01=color01, color02=color02,ylim_log=(-22, 3),
                     color012=color012,color012_minus = color01_2,color0=color0)
 
                 ##########################################################################
@@ -285,4 +285,4 @@ def plot_pictograms(ax_s, grid00):
 
 if __name__ == '__main__':#2.5
     motivation_all_small(dev_desired = '1', c1=10, devs=['05'], save=True, end='all',
-                         sorted_on='LocalReconst0.2NormAm')#
\ No newline at end of file
+                         sorted_on='LocalReconst0.2NormAm')#
diff --git a/susceptibility1.tex b/susceptibility1.tex
index 29f296e..821122f 100644
--- a/susceptibility1.tex
+++ b/susceptibility1.tex
@@ -295,20 +295,19 @@
 \newcommand{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies
 
 
-\newcommand{\signalnoise}{$s_\xi(t)$}%su\right m 
-
-\newcommand{\bsumb}{$\bsum{}=\fbase{}$}%su\right m 
-\newcommand{\btwob}{$\Delta f_{2}=\fbase{}$}%sum 
-\newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}%sum 
-\newcommand{\bsumbtwo}{$\bsum{}=2 \fbase{}$}%sum 
-\newcommand{\bsumbc}{$\bsum{}=\fbasecorr{}$}%sum 
-\newcommand{\bsume}{$\bsum{}=\feod{}$}%sum 
-\newcommand{\bsumehalf}{$\bsum{}=\feod{}/2$}%sum 
+\newcommand{\signalnoise}{$s_\xi(t)$}
+
+\newcommand{\bsumb}{$\bsum{}=\fbase{}$}
+\newcommand{\btwob}{$\Delta f_{2}=\fbase{}$}
+\newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}
+\newcommand{\bsumbtwo}{$\bsum{}=2 \fbase{}$}
+\newcommand{\bsumbc}{$\bsum{}=\fbasecorr{}$}
+\newcommand{\bsume}{$\bsum{}=\feod{}$}
+\newcommand{\bsumehalf}{$\bsum{}=\feod{}/2$}
 \newcommand{\bdiffb}{$\bdiff{}=\fbase{}$}%diff of both beat frequencies
 \newcommand{\bdiffbc}{$\bdiff{}=\fbasecorr{}$}%diff of both beat frequencies
 \newcommand{\bdiffe}{$\bdiff{}=\feod{}$}%diff of both 
-\newcommand{\bdiffehalf}{$\bdiff{}=\feod{}/2$}%diff of both
-%beat frequencies
+\newcommand{\bdiffehalf}{$\bdiff{}=\feod{}/2$}%diff of both beat frequencies
 
 %%%%%%%%%%%%%%%%%%%%%%% Frequency combinations
 \newcommand{\fsum}{\ensuremath{f_{1} + f_{2}}}%sum 
@@ -425,7 +424,7 @@ We like to think about signal encoding in terms of linear relations with unique
 
 The transfer function used to describe linear properties of a system is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. Including higher-order terms of the Volterra series, the nonlinear nature of spider mechanoreceptors \citep{French2001}, mammalian visual systems \citep{Victor1977, Schanze1997}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated.
 
-Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Vice versa, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
+Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Vice versa, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where one of two stimulus frequencies equals or both add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
 
 Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The p-type electroreceptors of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly nonlinear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the receiver's EOD. Nonlinear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation nonlinearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}.
 
@@ -437,29 +436,28 @@ Here we search for such weakly nonlinear responses in electroreceptors of the tw
   }
 \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 supra-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we explored 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} to search for such weakly nonlinear responses in real neurons. Our work is supported by simulations of LIF-based models of P-unit spiking. We start with demonstrating the basic concepts using example P-units and models.
+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 supra-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we explored 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} to search for such weakly nonlinear responses in real neurons. Additional simulations of LIF-based models of P-unit spiking put the experimental findings into context. We start with demonstrating the basic concepts using example P-units and models.
 
 \subsection{Nonlinear responses in P-units stimulated with two frequencies}
 Without external stimulation, a P-unit is driven by the fish's own EOD alone (with EOD frequency \feod{}) and spontaneously fires action potentials at the baseline rate \fbase{}. Accordingly, the power spectrum of the baseline activity has a peak at \fbase{} (\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 the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - \feod > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note that $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate.
 
-When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}). Thus, the cellular 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.
+When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum $\Delta f_1 + \Delta f_2$ and the difference frequency $\Delta f_2 - \Delta f_1$ (\subfigrefb{fig:motivation}{D}). Thus, the cellular 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.
 
 
 \subsection{Linear and weakly nonlinear regimes}
-\begin{figure*}[tp]
+
+\begin{figure*}[p]
   \includegraphics[width=\columnwidth]{nonlin_regime.pdf}
-  %\includegraphics[width=\columnwidth]{regimes/regimes.pdf}
-  \notejb{Update caption to new figure.}\\
-  \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 of the stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively.}
+  \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 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) and their amplitudes increase linearly with stimulus contrast (thin lines). \figitem{B} At intermediate stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). Their amplitudes grow quadraticlaly with stimulus constrast (thin lines). \figitem{C} At stronger stimulation the amplitudes of these nonlinear repsonses deviate from the quadratic dependency on stimulus contrast. \figitem{D} At higher stimulus contrasts additional peaks appear in the power spectrum. \figitem{E} Amplitude of the linear (at  $\Delta f_1$ and  $\Delta f_2$) and nonlinear (at $\Delta f_2 - \Delta f_1$ and $\Delta f_1 + \Delta f_2$) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively.}
 \end{figure*}
 
-The stimuli used in \figref{fig:motivation} had the same not-small amplitude. Whether this stimulus conditions falls into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}.
+The stimuli used in \figref{fig:motivation} had the same not-small amplitude. Whether this stimulus condition falls into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}.
 
-At very low stimulus contrasts (in the example cell less than approximately 0.5\,\% relative to the receiver's EOD amplitude) the spectrum has small peaks only at the beat frequencies (note, $\Delta f_2 = f_{base}$, \subfigref{fig:nonlin_regime}{A}, green and purple). The amplitudes of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}, thin lines), an indication of the linear response at lowest stimulus amplitudes.
+At very low stimulus contrasts (in the example cell less than approximately 1.5\,\% relative to the receiver's EOD amplitude) the spectrum has small peaks only at the beat frequencies (note, $\Delta f_2 = f_{base}$, \subfigref{fig:nonlin_regime}{A}, green and purple). The amplitudes of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}, thin lines), an indication of the linear response at lowest stimulus amplitudes.
 
-This linear regime is followed by the weakly nonlinear 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}, orange and red). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}, thin lines).
+This linear regime is followed by the weakly nonlinear regime (in the example cell approximately between 1.5\,\% and 4\,\% stimulus contrasts). 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}, orange and red). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}, thin lines).
 
-At higher stimulus amplitudes (\subfigref{fig:nonlin_regime}{C \& D}) additional peaks appear in the response spectrum. The linear response and the weakly-nonlinear response start to deviate from their linear and quadratic dependence on amplitude (\subfigref{fig:nonlin_regime}{E}). The responses may even decrease for intermediate stimulus contrasts (\subfigref{fig:nonlin_regime}{D}). At high stimulus contrasts, additional nonlinearities in the system, in particular clipping of the firing rate, shape the responses.
+At higher stimulus amplitudes the linear response and the weakly-nonlinear response start to deviate from their linear and quadratic dependence on amplitude (\subfigrefb{fig:nonlin_regime}{C \& E}) and additional peaks appear in the response spectrum (\subfigrefb{fig:nonlin_regime}{D}). At high stimulus contrasts, additional nonlinearities in the system, in particular clipping of the firing rate, shape the responses.
 
 For this example, we have chosen two specific stimulus (beat) frequencies. One of these matching the P-unit's baseline firing rate. In the following, however, we are interested in how the nonlinear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime. For the sake of simplicity we will drop the $\Delta$ notation event though P-unit stimuli are beats.