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susceptibility1.tex.bak
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\ead{jan.greqe@uni-tuebingen.de} \ead{jan.greqe@uni-tuebingen.de}
% \ead[url]{home page} % \ead[url]{home page}
\author[1]{Alexandra Barayeu\corref{fnd1}}
%\ead{alexandra.rudnaya@uni-tuebingen.de}
\author[2,3]{Maria Schlungbaum}
\author[2,3]{Benjamin Lindner}
\author[1,4,5]{Jan Benda}
\author[1]{Jan Grewe\corref{cor1}}
\ead{jan.greqe@uni-tuebingen.de}
% \ead[url]{home page}
\cortext[cor1]{Corresponding author} \cortext[cor1]{Corresponding author}
\affiliation[1]{organization={Neuroethology, Institute for Neurobiology, Eberhard Karls University}, \affiliation[1]{organization={Neuroethology, Institute for Neurobiology, Eberhard Karls University},
city={T\"ubingen}, postcode={72076}, country={Germany}} city={T\"ubingen}, postcode={72076}, country={Germany}}
\affiliation[2]{organization={Bernstein Center for Computational Neuroscience T\"ubingen}, \affiliation[4]{organization={Bernstein Center for Computational Neuroscience T\"ubingen},
city={T\"ubingen}, postcode={72076}, country={Germany}} city={T\"ubingen}, postcode={72076}, country={Germany}}
\affiliation[3]{organization={Werner Reichardt Centre for Integrative Neuroscience}, \affiliation[5]{organization={Werner Reichardt Centre for Integrative Neuroscience},
city={T\"ubingen}, postcode={72076}, country={Germany}} city={T\"ubingen}, postcode={72076}, country={Germany}}
\affiliation[4]{organization={Bernstein Center for Computational Neuroscience Berlin}, city={Berlin}, country={Germany}} \affiliation[2]{organization={Bernstein Center for Computational Neuroscience Berlin}, city={Berlin}, country={Germany}}
\affiliation[5]{organization={Physics Department of Humboldt University Berlin},city={Berlin}, country={Germany}} \affiliation[3]{organization={Physics Department of Humboldt University Berlin},city={Berlin}, country={Germany}}
%Nonlinearities contribute to the encoding of the full behaviorally relevant signal range in primary electrosensory afferents. %Nonlinearities contribute to the encoding of the full behaviorally relevant signal range in primary electrosensory afferents.

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susceptibility2.aux
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\newlabel{burst_add}{{3}{5}{Adding spikes to a burst package increases the second-order susceptibility in low-CV models with 1 million stimulus realizations (see table~\ref {modelparams} for model parameters of 2013-01-08-aa). A model with an intrinsic noise split (see methods section \ref {intrinsicsplit_methods}). \figitem {A} Carrier (gray) with a RAM stimulus (red). \figitem {B} Spike trains with increasing number of spikes per bust package. The colors connect the spikes trains and the resulting analysis of the model in \panel {C--F}. \figitem {C--F} Top: Absolute value of the second-order susceptibility. Bottom: Projected diagonal (see methods section \ref {projected_method}). \figitem {C} Original model 2013-01-08-aa (see table~\ref {modelparams} for model parameters). \figitem {D} Two-spikes burst packages: A spike was added after an EOD period to each spike in the original model in \panel {C} (see methods section \ref {modelburstadd_method}). Pink lines -- edges of the structure that occur when \fone {}, \ftwo {} and \fsum {} are equal to \fbase {} or its harmonic. Orange line -- part of the structure when \fsum {} is equal to half \feod . \figitem {E} Three-spikes burst packages: A spike was added after one and two EOD cycles to each spike in the original model in \panel {C}. \figitem {F} Four-spikes burst packages: A spike was added after one, two and three EOD cycles to each spike in the original model in \panel {C}. \relax }{figure.caption.6}{}} \newlabel{burst_add}{{3}{5}{Adding spikes to a burst package increases the second-order susceptibility in low-CV models with 1 million stimulus realizations (see table~\ref {modelparams} for model parameters of 2013-01-08-aa). A model with an intrinsic noise split (see methods section \ref {intrinsicsplit_methods}). \figitem {A} Carrier (gray) with a RAM stimulus (red). \figitem {B} Spike trains with increasing number of spikes per bust package. The colors connect the spikes trains and the resulting analysis of the model in \panel {C--F}. \figitem {C--F} Top: Absolute value of the second-order susceptibility. Bottom: Projected diagonal (see methods section \ref {projected_method}). \figitem {C} Original model 2013-01-08-aa (see table~\ref {modelparams} for model parameters). \figitem {D} Two-spikes burst packages: A spike was added after an EOD period to each spike in the original model in \panel {C} (see methods section \ref {modelburstadd_method}). Pink lines -- edges of the structure that occur when \fone {}, \ftwo {} and \fsum {} are equal to \fbase {} or its harmonic. Orange line -- part of the structure when \fsum {} is equal to half \feod . \figitem {E} Three-spikes burst packages: A spike was added after one and two EOD cycles to each spike in the original model in \panel {C}. \figitem {F} Four-spikes burst packages: A spike was added after one, two and three EOD cycles to each spike in the original model in \panel {C}. \relax }{figure.caption.6}{}}
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@ -82,9 +84,6 @@
\newlabel{data_overview}{{4}{7}{Population statistics of ampullary cells (green) and P-units (purple) in \lepto {} (\panel {A}) and \eigen {} (\panel {B}). If only P-units are presented in the subplots (\panel [iii-vi]{A}), the marginal distributions are purple and the scatter is color-coded, as indicated in the color bar below the subplot. \fbase {} -- mean baseline firing rate before burst correction. \fbasecorr {} -- mean baseline firing rate after burst correction. \cv {} -- baseline CV before burst correction. \cvbasecorr {} -- baseline CV after burst correction. \nli {} \nlicorr {} -- see methods section \ref {projected_method}. Burst fraction -- see methods section \ref {burstfraction}. Response modulation -- see methods section \ref {response_modulation}. \figitem [i]{A},\textbf {\,\panel [ii]{A},\,\panel [i]{B}} Baseline statistics. Burst correction leads to a reduction of baseline CVs and mean firing rates in \lepto {} (\panel [i, ii]{A}). \eigen {} has lower CVs (\panel [i]{B}). \figitem [iv]{A} Gray circles -- the lower marker corresponds to \subfigrefb {burst_cells_suscept}{B} and the higher marker to \subfigrefb {burst_cells_suscept}{A}. \figitem [v]{A} Cells with a high burst fraction deviate from the equality line. \figitem [vi]{A} High bust fraction cells do not increase their bursting during stimulation ($_{Stim}$) and are close to the gray equality line. The example cells in \figrefb {cells_suscept} are marked with gray circles. \relax }{figure.caption.10}{}} \newlabel{data_overview}{{4}{7}{Population statistics of ampullary cells (green) and P-units (purple) in \lepto {} (\panel {A}) and \eigen {} (\panel {B}). If only P-units are presented in the subplots (\panel [iii-vi]{A}), the marginal distributions are purple and the scatter is color-coded, as indicated in the color bar below the subplot. \fbase {} -- mean baseline firing rate before burst correction. \fbasecorr {} -- mean baseline firing rate after burst correction. \cv {} -- baseline CV before burst correction. \cvbasecorr {} -- baseline CV after burst correction. \nli {} \nlicorr {} -- see methods section \ref {projected_method}. Burst fraction -- see methods section \ref {burstfraction}. Response modulation -- see methods section \ref {response_modulation}. \figitem [i]{A},\textbf {\,\panel [ii]{A},\,\panel [i]{B}} Baseline statistics. Burst correction leads to a reduction of baseline CVs and mean firing rates in \lepto {} (\panel [i, ii]{A}). \eigen {} has lower CVs (\panel [i]{B}). \figitem [iv]{A} Gray circles -- the lower marker corresponds to \subfigrefb {burst_cells_suscept}{B} and the higher marker to \subfigrefb {burst_cells_suscept}{A}. \figitem [v]{A} Cells with a high burst fraction deviate from the equality line. \figitem [vi]{A} High bust fraction cells do not increase their bursting during stimulation ($_{Stim}$) and are close to the gray equality line. The example cells in \figrefb {cells_suscept} are marked with gray circles. \relax }{figure.caption.10}{}}
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@ -584,7 +584,7 @@ The baseline firing rate power spectrum of P-units of \eigen{} is also mirrored
\begin{figure*}[hp] \begin{figure*}[hp!]
\includegraphics[width=\columnwidth]{cells_eigen} % {ampullary}{burst_cells_suscept}, {cells_eigen} , calculated as the gain of the transfer function \includegraphics[width=\columnwidth]{cells_eigen} % {ampullary}{burst_cells_suscept}, {cells_eigen} , calculated as the gain of the transfer function
\caption{\label{cells_eigen} Response of experimentally measured P-units of \eigen{} with two low-CV P-unit examples. \figitem{A} Low-CV P-unit with strong second-order susceptibility when \fsumb{}. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate.\figitem[ii]{A} Top: EOD carrier (gray) with a RAM (red). Bottom: Spike trains. \figitem[iii]{A} First-order susceptibility (\eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. Red line -- edges of the structure when \fsum{} is equal to \feod{}. \figitem[v]{A} Projected diagonal, calculated as the mean of the anti-diagonals of the matrix in \panel[iv]{A}. \figitem{B} Low-CV P-unit with strong second-order susceptibility when \fsumb{} (pink) and \fsumehalf{} (orange, in \panel[iv]{A}).} \caption{\label{cells_eigen} Response of experimentally measured P-units of \eigen{} with two low-CV P-unit examples. \figitem{A} Low-CV P-unit with strong second-order susceptibility when \fsumb{}. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate.\figitem[ii]{A} Top: EOD carrier (gray) with a RAM (red). Bottom: Spike trains. \figitem[iii]{A} First-order susceptibility (\eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. Red line -- edges of the structure when \fsum{} is equal to \feod{}. \figitem[v]{A} Projected diagonal, calculated as the mean of the anti-diagonals of the matrix in \panel[iv]{A}. \figitem{B} Low-CV P-unit with strong second-order susceptibility when \fsumb{} (pink) and \fsumehalf{} (orange, in \panel[iv]{A}).}
\end{figure*} \end{figure*}
@ -597,7 +597,7 @@ The baseline firing rate power spectrum of P-units of \eigen{} is also mirrored
%\subsection{Summary}These nonlinear effects might have an influence on the encoding of conspecifics in P-units. when \fone{}, \ftwo{}, \fsum{} or \fdiff{} %\subsection{Summary}These nonlinear effects might have an influence on the encoding of conspecifics in P-units. when \fone{}, \ftwo{}, \fsum{} or \fdiff{}
\section{Discussion} \section{Discussion}
In this chapter, the second-order susceptibility in spiking responses of P-units In this chapter, the second-order susceptibility in spiking responses of P-units
was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions, except for bursting high-CV P-units that processed very strong nonlinearities in relation to the burst-corrected mean baseline firing rate \fbasecorr{}. Bursting was identified as a factor to increase the already present nonlinearities with a higher burst fraction, namely more spikes in a burst package. P-units of \eigen, that do not burst but have very low CVs, exhibited strong nonlinearities, similar to the nonlinearities of bursty P-units of \lepto. These two implementations highlighted that nonlinearity might be a critical feature necessary to sustain in these fish. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present. A three-fish setting might be especially interesting when it becomes the electrosensory cocktail party \citep{Henninger2018}, where a faint signal has to be detected despite the presence of a strong signal. The here presented second-order susceptibility might provide the boost necessary for improved faint signal detection. How this might be implemented will be discussed in the chapter \ref{chapter4}. was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions, except for bursting high-CV P-units that processed very strong nonlinearities in relation to the burst-corrected mean baseline firing rate \fbasecorr{}. Bursting was identified as a factor to increase the already present nonlinearities with a higher burst fraction, namely more spikes in a burst package. P-units of \eigen, that do not burst but have very low CVs, exhibited strong nonlinearities, similar to the nonlinearities of bursty P-units of \lepto. These two implementations highlighted that nonlinearity might be a critical feature necessary to sustain in these fish.
@ -712,9 +712,7 @@ The EOD of the fish with the stimulus was termed local EOD and was measured betw
wires located next to the left gill of the fish and orthogonal to its longitudinal wires located next to the left gill of the fish and orthogonal to its longitudinal
body axis (amplification 200--500 times, band-pass filtered with 3 to body axis (amplification 200--500 times, band-pass filtered with 3 to
1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm, 1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm,
Germany, \figrefb{Settup}, red markers). Unfortunately, these filter settings were too narrow for the Germany, \figrefb{Settup}, red markers).
high stimulus frequencies that were used in chapter \ref{chapter1}. For \subfigref{toblerone}{A--F} the local EOD waveforms were recreated by adding the recorded stimulus
output to the global EOD, to avoid unwanted phase shifts.
@ -737,7 +735,7 @@ gills. Gray horizontal bars -- electrodes for the stimulation. Green vertical ba
\subsection{White noise stimuli}\label{rammethods} \subsection{White noise stimuli}\label{rammethods}
For chapter \ref{chapter2} the fish were stimulated with band-pass limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The standard deviation of the white noise was expressed in relation to the EOD size of the fish in the experimental set-up and termed contrast. The contrast varied between 1 and 20\,$\%$ for \lepto{} and between 2.5 and 40\,$\%$ for \eigen. Only cell recordings with at least 10\,s of white noise stimulation were included for the analysis in chapter \ref{chapter2}. When ampullary cells were recorded the white noise was directly applied as the stimulus. For P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation (RAM, MXS-01M; npi electronics). For chapter \ref{chapter2} the fish were stimulated with band-pass limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The standard deviation of the white noise was expressed in relation to the EOD size of the fish in the experimental set-up and termed contrast. The contrast varied between 1 and 20\,$\%$ for \lepto{} and between 2.5 and 40\,$\%$ for \eigen. Only cell recordings with at least 10\,s of white noise stimulation were included for the analysis. When ampullary cells were recorded the white noise was directly applied as the stimulus. For P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation (RAM, MXS-01M; npi electronics).
@ -769,7 +767,7 @@ using the packages matplotlib, numpy, scipy, sklearn, pandas, nixio
(\url{https://github.com/bendalab/thunderfish}). (\url{https://github.com/bendalab/thunderfish}).
\paragraph{Baseline calculation}\label{baselinemethods}% chapter 2 in chapter \ref{chapter2} \paragraph{Baseline calculation}\label{baselinemethods}%
The mean baseline firing rate \fbase{} was calculated as the number The mean baseline firing rate \fbase{} was calculated as the number
of spikes divided by the duration of the baseline recording (on of spikes divided by the duration of the baseline recording (on
average 18\,s). The coefficient of variation (CV) was calculated as the standard deviation of the interspike intervals ISI divided by the mean ISI: $\rm{CV} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle}\langle ISI \rangle$. If the baseline was recorded several times in a cell, the mean \fbase{} and mean CV were calculated. average 18\,s). The coefficient of variation (CV) was calculated as the standard deviation of the interspike intervals ISI divided by the mean ISI: $\rm{CV} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle}\langle ISI \rangle$. If the baseline was recorded several times in a cell, the mean \fbase{} and mean CV were calculated.
@ -814,7 +812,7 @@ The firing rate of a cell is modulated around an average firing rate similar to
%, therefore the stimulusSecond-order susceptibility %, therefore the stimulusSecond-order susceptibility
\paragraph{Spectral analysis}\label{susceptibility_methods}%chapter 2In this score the stimulus $s(t)$ and the response $r(t)$ were set into relation. \paragraph{Spectral analysis}\label{susceptibility_methods}%chapter 2In this score the stimulus $s(t)$ and the response $r(t)$ were set into relation.
In chapter \ref{chapter2} the first-order and second-order susceptibility between the stimulus $s(t)$ and the response $r(t)$ were calculated (\eqnref{linearencoding_methods}, \eqnref{susceptibility}). The response $r(t)$ was the firing rate of the neuron, calculated as the binary spike train representation, with the sampling rate value when a spike occurred, zero everywhere else and the unit Hz. The noise stimulus $s(t)$ (see section \ref{rammethods}), was expressed in relation to the EOD and had no unit. A stimulus had a duration of 10\,s that were subdivided into $\n{}=20$ segments with no overlap, each with the duration of $T=0.5$\,s. Since the sampling rate $S_{r}$ varied between cell recordings (20\,kHz, 40\,kHz or 100\,kHz) $n_{\rm fft}$ was set to $S_{r}\cdot0.5$\,s, resulting in a frequency resolution of 2\,Hz for each cell. The first-order and second-order susceptibility between the stimulus $s(t)$ and the response $r(t)$ were calculated (\eqnref{linearencoding_methods}, \eqnref{susceptibility}). The response $r(t)$ was the firing rate of the neuron, calculated as the binary spike train representation, with the sampling rate value when a spike occurred, zero everywhere else and the unit Hz. The noise stimulus $s(t)$ (see section \ref{rammethods}), was expressed in relation to the EOD and had no unit. A stimulus had a duration of 10\,s that were subdivided into $\n{}=20$ segments with no overlap, each with the duration of $T=0.5$\,s. Since the sampling rate $S_{r}$ varied between cell recordings (20\,kHz, 40\,kHz or 100\,kHz) $n_{\rm fft}$ was set to $S_{r}\cdot0.5$\,s, resulting in a frequency resolution of 2\,Hz for each cell.
The Fourier transform of a time signal was calculated as $\tilde s(\omega) = \int_{0}^{T} \, s(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. The power spectrum was calculated as The Fourier transform of a time signal was calculated as $\tilde s(\omega) = \int_{0}^{T} \, s(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. The power spectrum was calculated as
\begin{equation} \begin{equation}
@ -860,7 +858,7 @@ The absolute value of a second-order susceptibility matrix is visualized in \fig
%\subsection{First-order susceptibility} \label{linearencoding_methods} %\subsection{First-order susceptibility} \label{linearencoding_methods}
% In chapter \ref{chapter2} the segments had no overlap, no smoothing of the segments was applied and each segment had a duration of $T=0.5$\,s.
% The lower right and upper left quadrants in the susceptibility matrix in \figrefb{model_full} were calculated as % The lower right and upper left quadrants in the susceptibility matrix in \figrefb{model_full} were calculated as
%\begin{equation} %\begin{equation}
% \label{susceptibility2} % \label{susceptibility2}
@ -913,7 +911,7 @@ The resulting signal was then low-pass filtered with a time constant $\tau_{d}$
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}^{p} \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}^{p}
\end{equation} \end{equation}
Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high
sensitivity to small amplitude modulations. Because the input was unitless, the dendritic voltage was unitless, too. In chapter \ref{chapter1} the rectified stimulus was optionally taken to a power of $p$. If not stated otherwise the exponent $p$ was set to one resulting in a pure threshold. This thresholding and low-pass filtering extracted the amplitude modulation of the input $x(t)$. sensitivity to small amplitude modulations. Because the input was unitless, the dendritic voltage was unitless, too. The exponent $p$ was set to one resulting in a pure threshold. This thresholding and low-pass filtering extracted the amplitude modulation of the input $x(t)$.
The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
@ -1030,7 +1028,7 @@ Based on the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} the total
\end{equation} \end{equation}
% das stimmt so, das c kommt unter die Wurzel! % das stimmt so, das c kommt unter die Wurzel!
In chapter \ref{chapter2} a big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{D}). Both components have to add up to the initial 100\,$\%$ of the total noise, otherwise the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} would not be applicable. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied \citep{Egerland2020}. In the here used LIF model with EOD carrier, this is more complicated since the noise stimulus $RAM(t)$ is first multiplied with the carrier (\eqnref{ram_split}), the signal is then subjected to rectification and subsequent dendritic low-pass filtering and becomes colored (\eqnref{Noise_split_intrinsic_dendrite}). This is the component that is added to the noise component in \eqnref{Noise_split_intrinsic} and should in sum lead to a total noise of 100\,\%. A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{D}). Both components have to add up to the initial 100\,$\%$ of the total noise, otherwise the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} would not be applicable. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied \citep{Egerland2020}. In the here used LIF model with EOD carrier, this is more complicated since the noise stimulus $RAM(t)$ is first multiplied with the carrier (\eqnref{ram_split}), the signal is then subjected to rectification and subsequent dendritic low-pass filtering and becomes colored (\eqnref{Noise_split_intrinsic_dendrite}). This is the component that is added to the noise component in \eqnref{Noise_split_intrinsic} and should in sum lead to a total noise of 100\,\%.
To compensate for these transformations, the generated noise $RAM(t)$ was scaled up by a factor $\rho$ (\eqnref{ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). The $\rho$ scaling factor was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier present) and the CV during stimulation (total noise split with $c_{signal}$ and $c_{noise}$). The assumption behind this approach was that as long the CV stays the same between baseline and stimulation both components have added up to 100\,$\%$ of the total noise and the noise split is valid. To compensate for these transformations, the generated noise $RAM(t)$ was scaled up by a factor $\rho$ (\eqnref{ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). The $\rho$ scaling factor was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier present) and the CV during stimulation (total noise split with $c_{signal}$ and $c_{noise}$). The assumption behind this approach was that as long the CV stays the same between baseline and stimulation both components have added up to 100\,$\%$ of the total noise and the noise split is valid.

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@ -488,7 +488,7 @@ In this work, the second-order susceptibility in the spiking responses of P-unit
\subsection{Bursting boosts second-order susceptibility} \subsection{Bursting boosts second-order susceptibility}
%Although most high-CV P-units have overall low \nli{} values, some high-CV P-units have high \nli{} values (upper right corner in \subfigrefb{data_overview_mod}{B}). To understand what makes these cells different, in the following bursting of P-units will be considered. %Although most high-CV P-units have overall low \nli{} values, some high-CV P-units have high \nli{} values (upper right corner in \subfigrefb{data_overview_mod}{B}). To understand what makes these cells different, in the following bursting of P-units will be considered.
The example cells discussed so far did not burst, mostly firing isolated spikes interleaved with quiescence (\figrefb{cells_suscept}). Bursting P-units fire a burst package of spikes interleaved with quiescence (\subfigrefb{burst_cells_suscept}\,\panel[ii]{A}). When cells burst, an overall bimodal ISI distribution occurs, with the first distribution containing the intra-burst intervals and the second the inter-burst intervals (\subfigrefb{burst_cells_suscept}\,\panel[i]{A}, left). In most cells the intra-burst distribution is below 1.5\,EOD periods, meaning that during the burst the cell fires after each EOD cycle. In some cells, the ISI distribution is better separated by a threshold of 2.5 or even more EOD periods. To get a burst-corrected spike train all spikes after the first spike in a burst package were removed (\subfigrefb{burst_cells_suscept}\,\panel[ii]{A}, dark purple spikes). Based on the baseline burst-corrected spike train the mean firing rate \fbasecorr{} and \cvbasecorr{} were calculated (see methods sections \ref{burstfraction}). \fbasecorr{} is represented with a peak in the baseline power spectrum of the firing rate before burst correction (\subfigrefb{burst_cells_suscept}\,\panel[i]{A}, right). Burst correction during RAM stimulation leads to a reduction of the linear encoding when comparing the linear encoding of all spikes (light purple) and of the burst-corrected spike train (dark purple, \subfigrefb{burst_cells_suscept}\,\panel[iii]{A}). Bursting induces wide bands in the absolute value of the second-order susceptibility (\subfigrefb{burst_cells_suscept}\,\panel[iv]{A}) that disappear after burst correction (\subfigrefb{burst_cells_suscept}\,\panel[v]{A}). In the second-order susceptibility after burst correction nonlinearity bands appear at \fsumbc{} (pink triangle edges, \subfigrefb{burst_cells_suscept}\,\panel[v]{A}). Burst correction leads to a reduced CV (compare the title of \subfigrefb{burst_cells_suscept}\,\panel[iv]{A} and \subfigrefb{burst_cells_suscept}\,\panel[v]{A}) and reduces the projection diagonal values and the \fbasecorr{} peak size in it (\subfigrefb{burst_cells_suscept}\,\panel[vi]{A}, gray marker on dark purple line). Based on this bursts seem to increase the second-order susceptibility at the frequencies related to \fbasecorr{} (\subfigrefb{burst_cells_suscept}\,\panel[vi]{A}).%meaningful for the cocktail party problem. Often one can find not only the \fbasecorr{} peak but also a peak at \feod{} and a peak at \feod{} - \fbasecorr{} (\subfigrefb{burst_cells_suscept}\,\panel[vi]{B}). Some P-units do not burst, mostly firing isolated spikes interleaved with quiescence (\figrefb{cells_suscept}). Bursting P-units fire a burst package of spikes interleaved with quiescence (\subfigrefb{burst_cells_suscept}\,\panel[ii]{A}). When cells burst, an overall bimodal ISI distribution occurs, with the first distribution containing the intra-burst intervals and the second the inter-burst intervals (\subfigrefb{burst_cells_suscept}\,\panel[i]{A}, left). In most cells the intra-burst distribution is below 1.5\,EOD periods, meaning that during the burst the cell fires after each EOD cycle. In some cells, the ISI distribution is better separated by a threshold of 2.5 or even more EOD periods. To get a burst-corrected spike train all spikes after the first spike in a burst package were removed (\subfigrefb{burst_cells_suscept}\,\panel[ii]{A}, dark purple spikes). Based on the baseline burst-corrected spike train the mean firing rate \fbasecorr{} and \cvbasecorr{} were calculated (see methods sections \ref{burstfraction}). \fbasecorr{} is represented with a peak in the baseline power spectrum of the firing rate before burst correction (\subfigrefb{burst_cells_suscept}\,\panel[i]{A}, right). Burst correction during RAM stimulation leads to a reduction of the linear encoding when comparing the linear encoding of all spikes (light purple) and of the burst-corrected spike train (dark purple, \subfigrefb{burst_cells_suscept}\,\panel[iii]{A}). Bursting induces wide bands in the absolute value of the second-order susceptibility (\subfigrefb{burst_cells_suscept}\,\panel[iv]{A}) that disappear after burst correction (\subfigrefb{burst_cells_suscept}\,\panel[v]{A}). In the second-order susceptibility after burst correction nonlinearity bands appear at \fsumbc{} (pink triangle edges, \subfigrefb{burst_cells_suscept}\,\panel[v]{A}). Burst correction leads to a reduced CV (compare the title of \subfigrefb{burst_cells_suscept}\,\panel[iv]{A} and \subfigrefb{burst_cells_suscept}\,\panel[v]{A}) and reduces the projection diagonal values and the \fbasecorr{} peak size in it (\subfigrefb{burst_cells_suscept}\,\panel[vi]{A}, gray marker on dark purple line). Based on this bursts seem to increase the second-order susceptibility at the frequencies related to \fbasecorr{} (\subfigrefb{burst_cells_suscept}\,\panel[vi]{A}).%meaningful for the cocktail party problem. Often one can find not only the \fbasecorr{} peak but also a peak at \feod{} and a peak at \feod{} - \fbasecorr{} (\subfigrefb{burst_cells_suscept}\,\panel[vi]{B}).
\begin{figure*}[h!]%p! \begin{figure*}[h!]%p!
\includegraphics[width=\columnwidth]{burst_cells_suscept}% \includegraphics[width=\columnwidth]{burst_cells_suscept}%
@ -584,7 +584,7 @@ The baseline firing rate power spectrum of P-units of \eigen{} is also mirrored
\begin{figure*}[hp] \begin{figure*}[hp!]
\includegraphics[width=\columnwidth]{cells_eigen} % {ampullary}{burst_cells_suscept}, {cells_eigen} , calculated as the gain of the transfer function \includegraphics[width=\columnwidth]{cells_eigen} % {ampullary}{burst_cells_suscept}, {cells_eigen} , calculated as the gain of the transfer function
\caption{\label{cells_eigen} Response of experimentally measured P-units of \eigen{} with two low-CV P-unit examples. \figitem{A} Low-CV P-unit with strong second-order susceptibility when \fsumb{}. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate.\figitem[ii]{A} Top: EOD carrier (gray) with a RAM (red). Bottom: Spike trains. \figitem[iii]{A} First-order susceptibility (\eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. Red line -- edges of the structure when \fsum{} is equal to \feod{}. \figitem[v]{A} Projected diagonal, calculated as the mean of the anti-diagonals of the matrix in \panel[iv]{A}. \figitem{B} Low-CV P-unit with strong second-order susceptibility when \fsumb{} (pink) and \fsumehalf{} (orange, in \panel[iv]{A}).} \caption{\label{cells_eigen} Response of experimentally measured P-units of \eigen{} with two low-CV P-unit examples. \figitem{A} Low-CV P-unit with strong second-order susceptibility when \fsumb{}. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate.\figitem[ii]{A} Top: EOD carrier (gray) with a RAM (red). Bottom: Spike trains. \figitem[iii]{A} First-order susceptibility (\eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. Red line -- edges of the structure when \fsum{} is equal to \feod{}. \figitem[v]{A} Projected diagonal, calculated as the mean of the anti-diagonals of the matrix in \panel[iv]{A}. \figitem{B} Low-CV P-unit with strong second-order susceptibility when \fsumb{} (pink) and \fsumehalf{} (orange, in \panel[iv]{A}).}
\end{figure*} \end{figure*}
@ -597,7 +597,7 @@ The baseline firing rate power spectrum of P-units of \eigen{} is also mirrored
%\subsection{Summary}These nonlinear effects might have an influence on the encoding of conspecifics in P-units. when \fone{}, \ftwo{}, \fsum{} or \fdiff{} %\subsection{Summary}These nonlinear effects might have an influence on the encoding of conspecifics in P-units. when \fone{}, \ftwo{}, \fsum{} or \fdiff{}
\section{Discussion} \section{Discussion}
In this chapter, the second-order susceptibility in spiking responses of P-units In this chapter, the second-order susceptibility in spiking responses of P-units
was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions, except for bursting high-CV P-units that processed very strong nonlinearities in relation to the burst-corrected mean baseline firing rate \fbasecorr{}. Bursting was identified as a factor to increase the already present nonlinearities with a higher burst fraction, namely more spikes in a burst package. P-units of \eigen, that do not burst but have very low CVs, exhibited strong nonlinearities, similar to the nonlinearities of bursty P-units of \lepto. These two implementations highlighted that nonlinearity might be a critical feature necessary to sustain in these fish. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present. A three-fish setting might be especially interesting when it becomes the electrosensory cocktail party \citep{Henninger2018}, where a faint signal has to be detected despite the presence of a strong signal. The here presented second-order susceptibility might provide the boost necessary for improved faint signal detection. How this might be implemented will be discussed in the chapter \ref{chapter4}. was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions, except for bursting high-CV P-units that processed very strong nonlinearities in relation to the burst-corrected mean baseline firing rate \fbasecorr{}. Bursting was identified as a factor to increase the already present nonlinearities with a higher burst fraction, namely more spikes in a burst package. P-units of \eigen, that do not burst but have very low CVs, exhibited strong nonlinearities, similar to the nonlinearities of bursty P-units of \lepto. These two implementations highlighted that nonlinearity might be a critical feature necessary to sustain in these fish.
@ -1069,10 +1069,12 @@ For the analysis in \figrefb{burst_add} the spikes in the non-busty model with i
\appendix \appendix
\setcounter{secnumdepth}{2}
\section{Appendix} \section{Appendix}
\begin{figure*}[h!]%p! \begin{figure*}[h!]%p!
\includegraphics[width=\columnwidth]{burst_cells_suscept_appendix}% \includegraphics[width=\columnwidth]{burst_cells_suscept_appendix}%
\caption{\label{burst_cells_suscept} Response of two experimentally measured bursty P-units before and after burst correction (\panel{A}, \panel{B}). Burst correction has three effects: 1) It lowers the baseline frequency from \fbase{} to \fbasecorr{}. 2) It decreases the overall second-order susceptibility. 3) It decreases the size of the \fbasecorr{} peak in the projected diagonal. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Bottom: Dark purple -- spikes after burst correction. \figitem[iii]{A} First-order susceptibility (\eqnref{linearencoding_methods}). Light purple -- based on all spikes (before burst correction). Dark purple -- based only on the first spike of a burst package (after burst correction). \figitem[iv]{A} Absolute value of the second-order susceptibility without burst correction. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolut value of the second-order susceptibility after burst correction. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbasecorr{}. Orange line as in \panel[iv]{A}. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals in the matrices in \panel[iv,v]{A}. Colors as in \,\panel[iii]{A}. Gray marker -- \fbasecorr{}. Dashed lines -- median of the projected diagonals.} \caption{\label{burst_cells_suscept_appendix} Response of two experimentally measured bursty P-units before and after burst correction (\panel{A}, \panel{B}). Burst correction has three effects: 1) It lowers the baseline frequency from \fbase{} to \fbasecorr{}. 2) It decreases the overall second-order susceptibility. 3) It decreases the size of the \fbasecorr{} peak in the projected diagonal. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Bottom: Dark purple -- spikes after burst correction. \figitem[iii]{A} First-order susceptibility (\eqnref{linearencoding_methods}). Light purple -- based on all spikes (before burst correction). Dark purple -- based only on the first spike of a burst package (after burst correction). \figitem[iv]{A} Absolute value of the second-order susceptibility without burst correction. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolut value of the second-order susceptibility after burst correction. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbasecorr{}. Orange line as in \panel[iv]{A}. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals in the matrices in \panel[iv,v]{A}. Colors as in \,\panel[iii]{A}. Gray marker -- \fbasecorr{}. Dashed lines -- median of the projected diagonals.}
\end{figure*} \end{figure*}
\newpage \newpage