1147 lines
88 KiB
TeX
1147 lines
88 KiB
TeX
%\documentclass[12pt,a4paper]{book}
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%date{June 2022}
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%Two ways to be linear: Susceptibility in P-unit models and experimentally recorded P-units and ampullary cells
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%intro
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%\newcommand{\signalcomp}{$s$}
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\newcommand{\aeod}{\ensuremath{A(f_{EOD})}}
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\newcommand{\fbasecorrsolid}{\ensuremath{f_{\rm{BaseCorrected}}}}
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\newcommand{\fbasecorr}{\ensuremath{f_{BaseCorrected}}}
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\newcommand{\ffall}{$f_{EOD}$\&$f_{1}$\&$f_{2}$}
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\newcommand{\ffvary}{$f_{EOD}$\&$f_{1}$}%sum
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\newcommand{\ffstable}{$f_{EOD}$\&$f_{2}$}%sum
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%\newcommand{\colstableone}{dark blue}%sum
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Beat combinations
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\newcommand{\boneabs}{\ensuremath{|\Delta f_{1}|}}%sum
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\newcommand{\btwoabs}{\ensuremath{|\Delta f_{2}|}}%sum
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\newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum
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\newcommand{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies
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\newcommand{\bdiffb}{$\bdiff{}=\fbase{}$}%diff of both beat frequencies
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\newcommand{\bdiffbc}{$\bdiff{}=\fbasecorr{}$}%diff of both beat frequencies
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\newcommand{\bdiffe}{$\bdiff{}=f_{EOD}$}%diff of both
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\newcommand{\bdiffehalf}{$\bdiff{}=f_{EOD}/2$}%diff of both
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%beat frequencies
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%%%%%%%%%%%%%%%%%%%%%%
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% Frequency combinations
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\newcommand{\fsum}{\ensuremath{f_{1} + f_{2}}}%sum
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\newcommand{\fdiff}{\ensuremath{|f_{1}-f_{2}|}}%diff of
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%%%%%%%%%%%%%%%%%%%%%%
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%Cocktailparty combinations
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%\newcommand{\bctwo}{$\Delta f_{Female}$}%sum
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%\newcommand{\bcdiff}{$|\Delta f_{Female} - \Delta f_{Intruder}|$}%diff of both beat frequencies
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%\newcommand{\bcdiffb}{$|\Delta f_{Female} - \Delta f_{Intruder}|=f_{Base}$}%diff of both beat frequencies
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%\newcommand{\bcdiffbc}{$|\Delta f_{Female} - \Delta f_{Intruder}|=\fbasecorr{}$}%diff of both beat frequencies
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%\newcommand{\bcdiffe}{$|\Delta f_{Female} - \Delta f_{Intruder}|=f_{EOD}$}%diff of both
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\newcommand{\fctwo}{\ensuremath{f_{\rm{Female}}}}%sum
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\newcommand{\bcsumshort}{\ensuremath{|\bctwoshort{} + \bconeshort{}|}}%sum
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\newcommand{\abcsumb}{\ensuremath{A(\bcsum{})=\fbasesolid{}}}%sum
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\newcommand{\bcdiff}{\ensuremath{||\bctwo{}| - |\bcone{}||}}%diff of both beat frequencies
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\newcommand{\bcsumb}{\ensuremath{\bcsum{} =\fbasesolid{}}}%su\right m
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\newcommand{\bcsumbn}{\ensuremath{\bcsum{} \neq \fbasesolid{}}}%su\right m
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\newcommand{\bctwob}{\ensuremath{\bctwo{} =\fbasesolid{}}}%sum
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\newcommand{\bconeb}{\ensuremath{\bcone{} =\fbasesolid{}}}%sum
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\newcommand{\bcsumbtwo}{\ensuremath{\bcsum{}=2 \fbasesolid{}}}%sum
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\newcommand{\bcsumbc}{\ensuremath{\bcsum{}=\fbasecorr{}}}%sum
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\newcommand{\bcsume}{\ensuremath{\bcsum{}=f_{\rm{EOD}}}}%sum
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\newcommand{\bcsumehalf}{\ensuremath{\bcsum{}=f_{\rm{EOD}}/2}}%sum
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\newcommand{\bcdiffb}{\ensuremath{\bcdiff{}=\fbasesolid{}}}%diff of both beat frequencies
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\newcommand{\bcdiffbc}{\ensuremath{\bcdiff{}=\fbasecorr{}}}%diff of both beat frequencies
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\newcommand{\bcdiffe}{\ensuremath{\bcdif{}f=f_{\rm{EOD}}}}%diff of both
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\newcommand{\bcdiffehalf}{\ensuremath{\bcdiff{}=f_{\rm{EOD}}/2}}%diff of both
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%\newcommand{\spnr}{$\langle spikes_{burst}\rangle$}%diff of
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\newcommand{\burstcorr}{\ensuremath{{Corrected}}}
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\newcommand{\cvbasecorr}{CV\ensuremath{_{BaseCorrected}}}
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\newcommand{\cv}{CV\ensuremath{_{Base}}}%\cvbasecorr{}
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\newcommand{\nli}{NLI\ensuremath{(\fbase{})}}%Fr$_{Burst}$
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\newcommand{\nlicorr}{NLI\ensuremath{(\fbasecorr{}})}%Fr$_{Burst}$
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\newcommand{\rostra}{\textit{Apteronotus rostratus}}
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\newcommand{\lepto}{\textit{Apteronotus leptorhynchus}}
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\newcommand{\eigen}{\textit{Eigenmannia virescens}}
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\newcommand{\suscept}{$|\chi_{2}|$}
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\newcommand{\frcolor}{pink lines}
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%\newcommand{\rec}{\ensuremath{f_{Receiver}}}
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%\newcommand{\intruder}{\ensuremath{f_{Intruder}}}
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%\newcommand{\female}{\ensuremath{f_{Female}}}
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%\newcommand{\rif}{\ensuremath{\rec{} & \intruder{} & \female{}}}
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%\newcommand{\ri}{\ensuremath{\rec{} & \intruder{}}}%sum
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%\newcommand{\rf}{\ensuremath{\rec{} & \female{}}}%sum
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\newcommand{\rec}{\ensuremath{\rm{R}}}%{\ensuremath{con_{R}}}
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\newcommand{\ri}{\ensuremath{\rm{RI}}}%sum
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\newcommand{\rf}{\ensuremath{\rm{RF}}}%sum
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\newcommand{\withfemale}{\rm{ROC}\ensuremath{\rm{_{Female}}}}%sum \textit{
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\newcommand{\wofemale}{\rm{ROC}\ensuremath{\rm{_{NoFemale}}}}%sum
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\newcommand{\dwithfemale}{\rm{\ensuremath{\auc_{Female}}}}%sum CV\ensuremath{_{BaseCorrected}}
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\newcommand{\dwofemale}{\rm{\ensuremath{\auc_{NoFemale}}}}%sum
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%\newcommand{\withfemale}{\rm{ROC}\ensuremath{_{Female}}}%sum \textit{
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%\newcommand{\wofemale}{\rm{ROC}\ensuremath{_{NoFemale}}}%sum
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%\newcommand{\dwithfemale}{\auc\ensuremath{_{Female}}}%sum CV\ensuremath{_{BaseCorrected}}
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%\newcommand{\dwofemale}{\auc\ensuremath{_{NoFemale}}}%sum
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\newcommand{\fp}{\ensuremath{\rm{FP}}}%sum
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\newcommand{\cd}{\ensuremath{\rm{CD}}}%sum
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%\dwithfemale
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%%%%% tables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% notes
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\journal{iScience}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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%\maketitle
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%Nonlinearities contribute to the encoding of the full behaviorally relevant signal range in primary electrosensory afferents.
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%Nonlinear effects identified as mechanisms that contribute
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%Nonlinearities in primary electrosensory afferents, the P-units, of \textit{Apteronotus leptorhynchus} enables the encoding of a wide dynamic range of behavioral-relevant beat frequencies and amplitudes
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%Nonlinearities in primary electrosensory afferents contribute to the representation of a wide range of beat frequencies and amplitudes
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%Nonlinearities contribute to the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents
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%The role of nonlinearities in the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents
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%Nonlinearities facilitate the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents
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%\newpage
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%\newpage
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%\cleardoublepage
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\begin{frontmatter}
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\title{Burst boost second-order susceptibility in a three-fish setting}
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\author[1]{Alexandra Barayeu\corref{fnd1}}
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%\ead{alexandra.rudnaya@uni-tuebingen.de}
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\author[2,3]{Maria Schlungbaum}
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\author[2,3]{Benjamin Lindner}
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\author[1,4,5]{Jan Benda}
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\author[1]{Jan Grewe\corref{cor1}}
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\ead{jan.greqe@uni-tuebingen.de}
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% \ead[url]{home page}
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\cortext[cor1]{Corresponding author}
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\affiliation[1]{organization={Neuroethology, Institute for Neurobiology, Eberhard Karls University},
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city={T\"ubingen}, postcode={72076}, country={Germany}}
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\affiliation[4]{organization={Bernstein Center for Computational Neuroscience T\"ubingen},
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city={T\"ubingen}, postcode={72076}, country={Germany}}
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\affiliation[5]{organization={Werner Reichardt Centre for Integrative Neuroscience},
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city={T\"ubingen}, postcode={72076}, country={Germany}}
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\affiliation[2]{organization={Bernstein Center for Computational Neuroscience Berlin}, city={Berlin}, country={Germany}}
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\affiliation[3]{organization={Physics Department of Humboldt University Berlin},city={Berlin}, country={Germany}}
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\begin{abstract}
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In this work, the influence of nonlinearities on stimulus encoding in the primary sensory afferents of weakly electric fish of the species \lepto{} was investigated. These fish produce an electric organ discharge (EOD) with a fish-specific frequency. When the EOD of one fish interferes with the EOD of another fish, it results in a signal with a periodic amplitude modulation, called beat. The beat provides information about the sex and size of the encountered conspecific and is the basis for communication. The beat frequency is predicted as the difference between the EOD frequencies and the beat amplitude corresponds to the size of the smaller EOD field. Primary sensory afferents, the P-units, phase-lock to the EOD and encode beats with changes in their firing rate. In this work, the nonlinearities of primary electrosensory afferents, the P-units of weakly electric fish of the species \lepto{} and \eigen{} were addressed. Nonlinearities were characterized as the second-order susceptibility of P-units, in a setting where at least three fish were present. The nonlinear responses of P-units were especially strong in regular firing P-units and bursty P-units. Bursting was identified as a mechanism to boost nonlinearity.% with bursting being identified as a factor enhancing nonlinear interactions.
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\end{abstract}
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\end{frontmatter}
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\section{Introduction}%\label{chapter2}
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% of spiking responses influences the encoding in P-units Nonlinearities can arise not only between cells but also inside single neurons,
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%, as has been shown in modeling studies % In the previous chapter, it was elaborated that the encoding of high beat frequencies requires a nonlinearity at the synapse between the electroreceptors and the afferent P-unit.
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In this work, nonlinearities are studied in the gymnotiform weakly electric fish of the species \lepto. These fish are equipped with an electric organ \citep{Salazar2013}, that is constantly active and produces a quasi sinusoidal electric organ discharge (EOD), with a long-term stable and fish-specific frequency (\feod, \citealp{Knudsen1975, Henninger2020}). These fish use their EOD for electrolocation \citep{Fotowat2013, Nelson1999} and communication \citep{Fotowat2013, Walz2014, Henninger2018}. When two fish are nearby the EOD of the receiver and the EOD of the encountered fish interfere and result in a new signal with a characteristic periodic amplitude modulation (AM), which is also called beat. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citep{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing the context for electrocommunication in weakly electric fish \citep{Engler2001, Hupe2008, Henninger2018, Benda2020}. The amplitude of a beat is defined by the amplitude of the smaller EOD, here \fstimintro{}. To account for the varying field size of different fish the beat amplitude is expressed as a percentage of the receiver EOD and termed contrast. If \fstimintro{} is 20\,$\%$ of the receiver EOD the beat contrast will also be 20\,$\%$. The beat frequency is calculated as the difference between the two present EOD frequencies $\Delta f=\ffstimintro -\ffeodintro{}$ (\citealp{Joris2004, Walz2014}). \lepto{} is sexually dimorphic with females having lower EOD frequencies (500--750\,Hz) and males having higher EOD frequencies (750--1000\,Hz). When both encountered fish have the same sex, their frequencies are drawn from the same frequency distribution, resulting in a low beat frequency $\Delta f$ and a slowly oscillating beat. When both fish have different sexes, their frequencies are drawn from two different frequency distributions and a high beat frequency $\Delta f$ and a fast oscillating beat occur.
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%\lepto{} is sexually dimorphic with females having lower EOD frequencies (500--750\,Hz) and males having higher EOD frequencies (750--1000\,Hz).
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Cutaneous tuberous organs, that are distributed all over the body of these fish
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(\citealp{Carr1982}), sense the actively generated electric field and its modulations.
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Within a single tuberous electroreceptor organ about 30 primary electroreceptors form ribbon synapses onto the dendrites of the same afferent fiber
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\citep{Szabo1965, Wachtel1966} that projects via the lateral line nerve to the hindbrain. There, the afferent fiber synapses onto pyramidal neurons in the
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electrosensory lateral line lobe (ELL, \citealp{Maler2009a}). The EOD is encoded by the p-type electroreceptor afferents, the P-units. These fire phase-locked to the EOD of the fish with a spike at a similar phase in the EOD cycle, with the phase depending on the receptor location \citep{Hladnik2023}. P-units fire not once in every EOD cycle but in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at multiples of the EOD period. P-units have heterogeneous baseline firing properties with the mean baseline firing rate \fbase{} varying between 50 and 450\,Hz \citep{Grewe2017, Hladnik2023} and the coefficient of variation (CV) of their ISIs varying between 0.2 and 1.7 \citep{Grewe2017, Hladnik2023}.
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In previous works, P-units have been mainly considered to be linear encoders \citep{Xu1996, Benda2005, Gussin2007, Grewe2017}. Still, P-units exhibit nonlinear effects, especially for strong beat amplitudes \citep{Nelson1997}. Although some P-units can be described by a linear decoder, other P-units require a nonlinear decoder \citep{Chacron2004}. In the following, the focus will be on nonlinear effects, as the second-order susceptibility in the time-resolved firing rate, that has been analytically derived based on leaky integrate-and-fire (LIF) models by \citealt{Voronenko2017}. In that framework, second-order susceptibility was predicted to appear in the spiking response at specific frequencies in the one and two sine-wave settings. As described by \citet{Voronenko2017} nonlinearities, at the sum and difference frequencies, are expected only for certain frequencies, that can be predicted based on the mean baseline firing rate \fbase{} of the cell. In their work, the second-order susceptibility was analytically retrieved based on LIF models, where the input were two pure sine waves. A triangular nonlinear shape was predicted, with nonlinearities appearing at the sum of the two input frequencies \fsum{} in the response, if one of the beat frequencies \fone{}, \ftwo{} or the sum of the beat frequencies \fsum{} was equal to \fbase{} (see diagonal, vertical and horizontal lines in the upper right quadrant in\citealp{Voronenko2017}). In addition, a triangular nonlinear shape was predicted, with nonlinearities appearing at the difference of the two input frequencies \fdiff{} in the response, if one of the input frequencies \fone{}, \ftwo{} or the difference of the input frequencies \fdiff{} was equal to \fbase{} (see diagonal, vertical and horizontal lines in the lower right quadrant in \citealp{Voronenko2017}). Whereas these predictions of the nonlinear response theory presented in \citet{Voronenko2017} apply to P-units, where the main driving force are beats and not the whole signal, will be addressed in the following.
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%The response of neurons can be retrieved by probing the system with pure sine waves or with a band-pass filtered white noise stimulus, that simultaneously includes all frequencies of interest, each with randomly drawn amplitudes and phases (\citealp{Chacron2005, Grewe2017}). Noise stimuli are commonly used protocols in electrophysiological recordings and once recorded they can always be reused to retrieve the tuning curve of the neuron \citep{Grewe2017, Neiman2011fish}.
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Nonlinearity can be influenced not only by the stimulus properties, as the input frequencies, but also by the cell properties during baseline, as the coefficient of variation (CV) of the interspike intervals (ISI) or the mean baseline firing rate \fbase{} \citep{Voronenko2017, Savard2011}. P-units are very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. Bursting, the repeated firing of groups of spikes interleaved with quiescence, is also an important factor influencing nonlinearity \citep{Chacron2004, Oswald2004}. Some P-units are non-bursty, always firing a single spike interleaved with quiescence. Others are bursty, firing a burst package with several spikes at subsequent EOD periods followed by quiescence. Bursting has been investigated in P-unit models, where the detailed time course of the stimulus was better encoded by non-bursty neurons, while feature extraction of the rising stimulus phase was better performed by bursty cells \citep{Chacron2004}. In addition, it was demonstrated that a linear decoder is sufficient for non-bursty firing but a nonlinear decoder is necessary to describe bursty neurons \citep{Chacron2004}. Slice recordings revealed that the ISI statistics of P-units are transported to higher-order neurons, the pyramidal cells in the electrosensory line lobe (ELL, \citealp{Khanbabaie2010}), making burst information potentially available to pyramidal cells.
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%Khanbbaie 2010: ok das waren slices also wenn P-units bursten sollte man das in pyramiden zellen sehen können, Berman1999
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Bursting has been found not only in primary sensory afferents of the \lepto{} but has also been observed in pyramidal cells of the ELL, where bursts better encode low-frequency events whereas single spikes encode the whole frequency range \citep{Oswald2004}. There it was also demonstrated that the first spike in a burst package carries linear information about the stimulus, that is not increased by additional spikes in a burst package. Instead, bursts were shown to improve feature detection in pyramidal cells, by improving the signal-to-noise ratio \citep{Oswald2004}. An improved signal-to-noise ratio and sharpened tuning curves by bursts have also been found in the auditory system \citep{Eggermont1996}. This improved signal-to-noise ratio might be related to bursts supporting reliable synaptic transmission \citep{Csicsvari1998}.
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Bursts in higher order neurons are often generated via feedback loops on a network level \citep{Zeldenrust2018, Krahe2004} that can be influenced by environmental variables \citep{Krahe2004}. P-units do not receive feedback from other cells and are considered to be intrinsic bursters. Bursting has behavioral implications e.g. for avoidance behavior in crickets, where bursts provide information about the stimulus \citep{Marsat2010burst, Marsat2012bursting}. Bursts are ubiquitous in sensory modalities as in chattering cells in the visual system \citep{Nowak2003}, fast rhythmic bursting cells in the auditory system \citep{Cunningham2004}, Purkinje cells in the cerebellum \citep{Womack2004} or thalamic relay neurons \citep{Destexhe1993}.
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%neuron, the burst structure provides temporal information about the stimulus \citep{Marsat2010burst}.
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%with neuromodulation of the spinal cord
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%Doiron2002, low and high-frequency events
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%Stimulus encoding of bursts is different in different cell types and species,
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% considered as intrinsic bursters \citep{Zeldenrust2018review} since there are no feedback known in these cells
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%\citep{Metzen2016}
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%\subsection{Outlook}when \fsum{} and \fdiff{} are equal to \fbase
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In this work, the second-order susceptibility in the spiking responses of P-units will be accessed with white noise stimulation. The influence of the baseline firing properties, such as the CV, on nonlinear interactions will be investigated. It will be demonstrated that some P-units exhibit nonlinearities in relation to \fbase{}, as predicted by \citet{Voronenko2017}, but some cells diverge from this theoretical prediction, with bursting influencing the occurrence of the nonlinearities. P-unit models will be used to highlight bursting as a mechanism to boost nonlinearity.
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%will be addressed. How many P-units exhibit such nonlinearities
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%The nonlinearity in P-units in \lepto{} will be compared to other cell types as ampullary cells and with other species as \eigen{}.
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% and show that the number of spikes in a burst package, and not the first burst spike timing, are critical for this increase in nonlinearity
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%influence the occurrence of nonlinearity by comparing the nonlinearity in P-units with other cell populations as the ampullary cells (lower CVs) and consider not only \lepto{} but also the P-units of \eigen{} (lower CV)
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% if the nonlinear frequency combinations occur where they are predicted based on simple LIF models without a carrier.
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\section{Results}
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\begin{figure*}[h]%hp!
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\includegraphics{cells_suscept}
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\caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. \figitem{A} Regular firing low-CV P-unit. \figitem[i]{A} Left: Interspike intervals (ISI) distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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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} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. \figitem{B} Noisy high-CV P-Unit. Panels as in \panel{A}.
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}
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\end{figure*}
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\subsection{Bursting boosts second-order susceptibility}
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%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.
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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}).
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\begin{figure*}[h!]%p!
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\includegraphics[width=\columnwidth]{burst_cells_suscept}%
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\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.}
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\end{figure*}
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\begin{figure*}[ht!]
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\includegraphics[width=\columnwidth]{burst_add}
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\caption{\label{burst_add} 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}.
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}
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\end{figure*}
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%To investigate the influence of the spike number in a burst package on the nonlinearity would require a larger experimentally recorded bursty P-unit population to represent all heterogeneous P-unit parameters as \feod, \fbase{} and CV. , but also in \feod, \fbase{} and CV, \subfigrefb{burst_add}{C}
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%\newpage
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\newpage
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\subsection{More spikes in burst package increase the nonlinearity}
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The two shown P-units differ in their average number of spikes per burst package (\subfigrefb{burst_cells_suscept}\,\panel[ii]{A},\,\panel[ii]{B}). In the following the influence of the spike number in a burst package on nonlinearity was investigated in a low-CV P-unit model (spikes in the first row in \subfigrefb{burst_add}{B}). In this model, the nonlinear structures occur in relation to \fbasecorr{} and its harmonic (yellow lines in pink edges, \subfigrefb{burst_add}{C}, top). Spikes were added after exactly one, two or three EOD periods, creating artificial burst packages (\subfigrefb{burst_add}{B}, see section \ref{modelburstadd_method}). Adding one spike adds a wide band in the second-order susceptibility (\subfigrefb{burst_add}{D}, top) and elevations in the projected diagonal (\subfigrefb{burst_add}{D}, bottom). Increasing the number of spikes in a burst to three (\subfigrefb{burst_add}{E}) or four spikes (\subfigrefb{burst_add}{F}) adds more bands in the second-order susceptibility (\subfigrefb{burst_add}{E, F}, top) and introduces several elevations in the projected diagonals (bottom).
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Although with more spikes per burst package the overall nonlinearity increases (more yellow in \subfigrefb{burst_add}{F} than in \subfigrefb{burst_add}{C}), some frequencies are enhanced but others are dampened. The peak at \fbasecorr{} in the projected diagonal is increased with more spikes in a burst package (compare \subfigrefb{burst_add}{C} and \subfigrefb{burst_add}{D--F}). The frequencies on the vertical and horizontal lines at \fonebc{} and \ftwobc{} in the second-order susceptibility are enhanced or dampened depending on their position in the antidiagonal bands induced by bursts. Adding bursts in the model revealed that the burst-induced increase of the second-order susceptibility is mediated not by the timing of the first spikes but by the number of spikes in a burst package.
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%The frequencies on the vertical and horizontal lines at \fonebc{} and \ftwobc{} in the second-order susceptibility matrix are altered differentially.
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%\newpage
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\subsection{Bursts influence nonlinearity on a population level}
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%The burst fraction is highly correlation with the CV$_{Base}$ ( \subfigrefb{data_overview}\,\panel[v]{A}).
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So far, bursting was identified to increase nonlinearity on a single cell level (\figrefb{burst_cells_suscept}) and in the following its influence on a population level will be addressed. Burst correction leads to a reduction of the CV during baseline from \cv{} to \cvbasecorr{} and of the firing rate during baseline from \fbase{} to \fbasecorr{} in a population of P-units in \lepto{} (compare purple in \subfigrefb{data_overview}\,\panel[i, ii]{A}). \cv{}, that previously has been identified to increase nonlinearity (\subfigrefb{data_overview_mod}{A, B}), is highly correlated with the burst fraction, the number of spikes in a burst package divided by the number of all spikes (see methods section \ref{burstfraction}, $r=0.81$, $p<0.001$, \subfigrefb{data_overview}\,\panel[iii]{A}). Here, the nonlinearity was characterized as the peakedness of \fbasecorr{} in the projected diagonal with the score \nlicorr{} (\eqnref{nli_equation}). Bursty cells have higher nonlinearity values \nlicorr{} than non-bursty cells (\subfigrefb{data_overview}\,\panel[iv]{A}). When correlating \nli{} with \nlicorr{} bursty cells strongly deviate from the equality line but non-bursty cells do not change in their NLI values (\subfigrefb{data_overview}\,\panel[v]{A}). These non-bursty cells have high \nlicorr{} values in the cluster close to the origin in \subfigrefb{data_overview}\,\panel[iv]{A} and simultaneously are the low-CV cells that contributed to the high \nli{} values in \subfigrefb{data_overview_mod}{B}.
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Most cells increase bursting with higher RAM contrasts (red circles are above blue circles and above the equality line, \subfigrefb{data_overview}\,\panel[vi]{A}). These findings highlight that some P-units use bursting as a mechanism to encode a wide dynamic range of contrasts. Cells that already possess high burst fractions during baseline cannot increase their firing even further (high burst fractions values close to the equality line, \subfigrefb{data_overview}\,\panel[vi]{A}) and thus cannot use bursting as a mechanism for contrast encoding.
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% The example cells in \subfigrefb{cells_suscept}{A--B} slightly increase in their burst fraction for higher RAM contrasts (grey circles in \subfigrefb{data_overview}\,\panel[vi]{A}).
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\subsection{Nonlinearity in P-units of \eigen{}}% is implemented not with bursting but present in lower CV cells
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So far it was demonstrated in \lepto{} that P-units implement intermediate nonlinearity values with non-bursty low-CV cells and high nonlinearity values with bursty high-CV P-units (\subfigrefb{data_overview}\,\panel[iv]{A}). P-units of \eigen{} do not burst but have lower CVs than the P-units of \lepto{} (\subfigrefb{data_overview}\,\panel[i]{B}). Similar nonlinearity values are implemented in these low-CV P-units of \eigen{} (\subfigrefb{data_overview}\,\panel[ii]{B}) as they were observed in bursty P-units of \lepto{} (\subfigrefb{data_overview}\,\panel[iv]{A}).
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The baseline firing rate power spectrum of P-units of \eigen{} is also mirrored around half \feod{} (\subfigrefb{cells_eigen}\,\panel[i]{A}, right). Pronounced nonlinearity bands can be observed at \fsumb{} in the second-order susceptibility of low-CV P-units (\subfigrefb{cells_eigen}\,\panel[iv]{A}). \eigen{} have lower EOD frequencies than \lepto{} and the \fbase{} of P-units can be as high as the \feod{} of \eigen{}. In some P-units \fbase{} is close to \feodhalf{}, with very pronounced nonlinearity bands at \fsumb{} and \fsumehalf{} (\subfigrefb{cells_eigen}\,\panel[iv]{B}). Nonlinearities at \fsumehalf{} can also be observed in \lepto{} (\subfigrefb{model_and_data}\,\panel[ii]{A}), still the mean baseline firing rates \fbase{} of these fish are usually below half \feod{}, with no overlap of the according nonlinearities.
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%High-CV P-units of \eigen{} do neither burst nor have pronounced nonlinearity bands in the second-order susceptibility matrix (not shown).
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% (as in,\subfigrefb{burst_cells_suscept})
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%keep the same burst fraction during baseline and stimulation
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%The cells have another common baseline parameter the burst fraction, the number of burst spikes below 1.5 EOD multiples divided by all spikes. The correlation with the burst fraction is also negative and significant (r=-0.24 p$<$0.001***, n=220).
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\begin{figure*}[hp!]
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\includegraphics[width=\columnwidth]{data_overview}
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\caption{\label{data_overview} Population statistics of ampullary cells (green) and P-units (purple) in \lepto{} (\panel{A}) and \eigen{} (\panel{B}). If only P-units are presented in the subplots (\panel[iii-vi]{A}), the marginal distributions are purple and the scatter is color-coded, as indicated in the color bar below the subplot. \fbase{} -- mean baseline firing rate before burst correction. \fbasecorr{} -- mean baseline firing rate after burst correction. \cv{} -- baseline CV before burst correction. \cvbasecorr{} -- baseline CV after burst correction. \nli{} \nlicorr{} -- see methods section \ref{projected_method}. Burst fraction -- see methods section \ref{burstfraction}. Response modulation -- see methods section \ref{response_modulation}. \figitem[i]{A},\textbf{\,\panel[ii]{A},\,\panel[i]{B}} Baseline statistics. Burst correction leads to a reduction of baseline CVs and mean firing rates in \lepto{} (\panel[i, ii]{A}). \eigen{} has lower CVs (\panel[i]{B}). \figitem[iv]{A} Gray circles -- the lower marker corresponds to \subfigrefb{burst_cells_suscept}{B} and the higher marker to \subfigrefb{burst_cells_suscept}{A}. \figitem[v]{A} Cells with a high burst fraction deviate from the equality line. \figitem[vi]{A} High bust fraction cells do not increase their bursting during stimulation ($_{Stim}$) and are close to the gray equality line. The example cells in \figrefb{cells_suscept} are marked with gray circles.
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}
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\end{figure*}
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% \figitem{H} Mutual information RR-SR Score ($\frac{MI_{RR}-MI_{SR}}{MI_{SR}}$) for different burst fractions with no significant correlation. \figitem{I} \nli{} to response modulation for ampullary cells and P-units in \eigen, with negative correlations.
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%\figitem{D} Positive correlation between burst fraction, CV$_{Base}$ and response modulation.
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%Here we calculated the susceptibility for 42 model cells and sorted them by the CV during the stimulation (left to right, top to bottom).
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%of 42 cells was fitted to 42 recorded cells based on the parameters described in the methods section (\ref{cell_characteristics}).
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%\begin{figure*}[hp!]
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% \includegraphics[width=\columnwidth]{p_units}
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% \caption{\label{p_units}.
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% }
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%\end{figure*}
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%\begin{figure*}[h]
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% \includegraphics[width=\columnwidth]{burst_occluded}
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% \caption{\label{burst_occluded} Burst correction affects the suscepstibility %considering the very bride diagonal, that vanishes. In addition, it lowers the CV and %firing rates.
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% }
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%\end{figure*}
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\begin{figure*}[hp!]
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\includegraphics[width=\columnwidth]{cells_eigen} % {ampullary}{burst_cells_suscept}, {cells_eigen} , calculated as the gain of the transfer function
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\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}).}
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\end{figure*}
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%%Intro einleitung, rekapitulation, was wurde in dem Kapitel angeguckt , dann sagen wir warum könnte die nichtlienaritäten wichtig sein da gibts den socialen mechanismen, bursts, low CV und ampullären, high cv zellen zeigens nicht es sei den sie sind bursty, dann nochmal eine andere Art und die haben alle low cvs und strong nonlinearities, und dann sagen wir die könnte eine wichtige rolle spielen im threefish scenario und wir können sie auf unterschiedlichen wegen herbei führen, und dann dreh in die große weite welt sondern auch in den anderen
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% encode the social context,.
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%Intro in discussion wird besser an resultate gemacht und ein erster satz
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%\subsection{Summary}These nonlinear effects might have an influence on the encoding of conspecifics in P-units. when \fone{}, \ftwo{}, \fsum{} or \fdiff{}
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\section{Discussion}
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In this chapter, the second-order susceptibility in spiking responses of P-units
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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.
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\subsection{Bursting as a mechanism for amplitude encoding}% which already burst during baseline or do no
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In this chapter, bursty and non-bursty cells during baseline were identified as two subpopulations of P-units with different stimulus amplitude encoding mechanisms. One subpopulation were P-units that already bursted during baseline (\figrefb{burst_cells_suscept}) and could not increase bursting even further for stronger stimulus amplitudes (\subfigrefb{data_overview}\,\panel[vi]{A}, equality line), with bursting not contributing to the encoding of amplitude changes. The other identified subpopulation were P-units that did not burst during baseline, but gradually increased bursting with stronger stimulus amplitudes (\subfigrefb{cells_suscept}, \subfigrefb{data_overview}\,\panel[vi]{A}). In these cells bursting contributed to the encoding of a wide dynamic range of stimulus amplitudes. An increasing burst fraction is known to encode different stimulus strengths as e.g. temperature changes \citep{Longtin1996}, muscle stretch intensity \citep{Birmingham1999} or the strength of electrical white noise signals in ampullary cells of paddlefish \citep{Neiman2011fish}.
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The two subpopulations identified here are in line with previous literature, where it has been demonstrated that different features are in parallel encoded in bursty and non-bursty P-units \citep{Chacron2004}. In slice recordings it was shown that the ISI statistics of P-units are transported to higher-order neurons, the pyramidal cells in the electrosensory line lobe (ELL, \citealp{Khanbabaie2010}), where bursting might have an impact on higher-order cells.
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%\subsection{Bursting in P-units}%that bursting influences nonlinearity
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\subsection{Bursting increases nonlinearity and linearity in P-units}%that bursting influences nonlinearity
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%In the following, the influence of bursts on the nonlinearity and linearity in P-units will be discussed.
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In this chapter, bursty high-CV P-units in \lepto{} were identified to have strong nonlinearity, exceeding the one of non-bursty low-CV P-units (\subfigrefb{data_overview}\,\panel[iv]{A}). These findings are in line with previous literature, where it was derived in P-unit models that non-bursty cells can be described by a linear decoder, whereas bursty cells require a nonlinear decoder \citep{Chacron2004}. Bursting has previously been associated with an increased second-order susceptibility in ampullary cells of paddlefish \citep{Neiman2011fish}.
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%These findings are similar to the results in this chapter, where more bursts in a spike package lead to an improved signal-to-noise ratio (\figrefb{burst_add}).
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%. It could be demonstrated that bursts enhanced the nonlinearity at low frequencies, as \fsumbc{}
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%, but could increase or decrease it at higher frequencies ore fluctuations in the second-order susceptibility
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In bursty cells two nonlinearity structures were present in the second-order susceptibility: the wide burst-induced bands and the sharp bands at \fbasecorr{} (\subfigrefb{burst_cells_suscept}\,\panel[iv]{A},\,\panel[iv]{B}), that were enhanced by bursts (\subfigrefb{burst_add}{D--F}). It was derived in the model that the enhanced nonlinearity is not driven by the timing of the first spike in the burst package but is defined by the number of spikes in a burst package, with more burst spikes inducing higher nonlinearity and increasing the signal-to-noise ratio. In pyramidal cells of the ELL, it was demonstrated that bursting improves feature detection by improving the signal-to-noise ratio \citep{Oswald2004, Gabbiani1996}. An improved signal-to-noise ratio and sharpened tuning curves have been associated with bursts in the auditory system \citep{Eggermont1996}. The reason behind an improved signal-to-noise ratio might be a decrease of the synaptic failure probability \citep{Lisman1997} and a more reliable synaptic transmission \citep{Csicsvari1998} induced by bursts.
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Besides the influence on the nonlinearity, bursting was demonstrated to contribute to the linear encoding of the stimulus, which is reduced after burst correction (\figrefb{burst_cells_suscept}). Usually, bursts in P-units have been discussed considering their contribution to nonlinearity \citep{Chacron2004} with their impact on linear encoding being mainly neglected. The increase of linearity via bursts is not universal since in other cells, as in pyramidal cells of the ELL, the linear encoding of the stimulus is attributed to isolated spikes and not to bursts \citep{Oswald2004}.
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The investigation of bursts is important since bursts are ubiquitous in sensory modalities as in chattering cells in the visual system \citep{Nowak2003}, in fast rhythmic bursting cells in the auditory system \citep{Cunningham2004}, Purkinje cells in the cerebellum \citep{Womack2004} or thalamic relay neurons \citep{Destexhe1993}. Bursts have an important application and are successfully used for chronic pain treatment with neuromodulation of the spinal cord \citep{Ridder2010Cord, Chakravarthy2018, Karri2020}, but also for tinnitus suppression \citep{Ridder2010Auditory}. The dynamics leading to bursting have been addressed with bifurcation analysis and are well understood \citep{Izhikevich2000}.
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%\subsubsection{Bursting increases nonlinearity}%that bursting influences nonlinearity
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%In crickets in the auditory command neuron, the burst structure was demonstrated to provide temporal information about the stimulus \citep{Marsat2010burst}. I
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%\subsubsection{More spikes in a burst package increase the signal-to-noise ratio}%that bursting influences nonlinearity
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%\subsubsection{Bursting increases linearity}%that bursting influences nonlinearity
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%These low-frequency modulations of the amplitude modulation are
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\subsection{Encoding of secondary envelopes}%($\n{}=49$) Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study \citet{Savard2011}.
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The RAM stimulus used in this chapter is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citep{Stamper2012Envelope}. Low-frequency secondary envelopes are extracted downstream of P-units in the ELL \citep{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citep{Middleton2007}. The encoding of social envelopes can also be attributed to P-units with stronger nonlinearities, lower firing rates and higher CVs \citep{Savard2011}. In this chapter high CVs were associated with increased bursting (\subfigrefb{data_overview}\,\panel[iii]{A}). Thus it is likely that the high-CV nonlinear envelope encoding P-units found by \citet{Savard2011} are bursty. Then these findings \citep{Savard2011} would be in line with the in this chapter demonstrated increased nonlinearity in bursty cells (\figrefb{burst_cells_suscept}, \figrefb{burst_add}).
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%e.g. at \bdiff{}
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\subsection{Nonlinearity necessary to sustain in different species?}
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In this chapter, it was demonstrated that similar nonlinearity values can be implemented with different mechanisms, namely with low-CV P-units in \eigen{} or bursting in \lepto{} (\subfigrefb{data_overview}\,\panel[iv]{A},\,\panel[ii]{B}). This suggests that nonlinearity might be an important feature, necessary to sustain in different species of weakly electric fish. If traits develop independently, with the trait not being present in a common ancestor, the process is termed convergent evolution. Examples of convergent evolution are blue eyes and light skin in humans \citep{Edwards2010}, the camera eye present in mammals, jellyfish and squids \citep{Kozmik2008} or the evolution of wave-type and pulse-type weakly electric fish \citep{Lavoue2012}. Slightly different traits in closely related species imply divergent evolution. An example of divergent evolution are dogs and wolfs that descend from the same ancestor \citep{Vila1999} or Darwin's finches that developed beaks with different shapes \citep{Ford1973}. \lepto{} and \eigen{} are closely related species belonging to the same superfamily Apteronotoidea, but to different families of Apteronotidae and Sternopygidae \citep{Albert2001}. The nonlinearity in the primary sensory afferents of more weakly electric fish species should be accessed in further studies, to increase the certainty if such nonlinearity might have been present in the common ancestor of \eigen{} and \lepto{} and test whereas this sustained nonlinearity might be an example of divergent evolution.% to address such questions.
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\subsection{Conclusion} In this chapter, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. Nonlinear effects were identified in experimentally recorded bursty high-CV P-units. P-units could be subdivided into two populations, where some P-units did not burst during baseline and increased bursting to encode stimulus amplitudes, and others bursted during baseline and did not use bursting as an amplitude encoding mechanism. Bursting was identified as a factor increasing linear and nonlinear encoding in P-units. Such nonlinearity was identified as a feature potentially necessary to sustain in primary sensory afferents of different weakly electric fish species. Nonlinearity was found to be decreased the more fish were present, thus keeping the signal representation in the firing rate simple.
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\section{Methods}
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\subsection{Experimental subject details}
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Experiments were performed on male and female weakly electric fish
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of the species \lepto{} obtained from a
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commercial tropical fish supplier (Aquarium Glaser GmbH, Rodgau,
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Germany). The fish were kept in tanks with a water temperature
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of $25\,^\circ$C and a conductivity of around
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$270\,\micro\siemens\per\centi\meter$ under a 12\,h:12\,h light-dark
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cycle. All experimental
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protocols complied with national and European law and were approved by
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the Ethics Committee of the Regierungspräsidium T\"ubingen (permit no: ZP1-16).
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No experiments were performed. Instead, cells recorded in \lepto{} and \eigen{} at the Ludwig Maximilian University (LMU) M\"unchen and at the Eberhard-Karls University T\"ubingen between 2010 and 2023 were used. The final sample consisted of 222 P-units and 45 ampullary cells from 71 weakly electric fish of the species \lepto{} and 60 P-units and 18 ampullary cells from 17 weakly
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electric fish of the species \eigen.
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%\subsection{Experimental model and subject details In chapter \ref{chapter4} }
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% glued to the skull
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\subsection{Surgery} Before surgery, anesthesia was provided via bath application with a solution of MS222 (120\,mg/l, PharmaQ, Fordingbridge, UK) buffered with Sodium Bicarbonate (120\,mg/l). The posterior anterior lateral line nerve (pALLN) above the gills before its descent towards the anterior lateral line ganglion (ALLNG) was disclosed for subsequent P-unit recordings. During the surgery water supply was ensured by a mouthpiece, sustaining
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anesthesia with a solution of MS222 (100\,mg/l) buffered with Sodium Bicarbonate (100\,mg/l).
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%For the surgery the fish was fixed on a stage via a metallic rod.
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\subsection{Experimental setup} During the experiments fish were immobilized by a single
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intramuscular injection of Tubocurarine (Sigma-Aldrich, Steinheim,
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Germany; 25--50\,\micro\litre{} of 5\,mg/ml solution). For the
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recordings fish were positioned on a stage in a tank, with a major part
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of their body in the water. Analgesia was refreshed in intervals of two hours by cutaneous Lidocaine application (2\,\%; bela-pharm, Vechta, Germany) around the nerve. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P;
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Science Products, Hofheim, Germany) were pulled to a resistance of 50--100\,\mega\ohm{} (model P-97; Sutter Instrument, Novato, CA) and filled with 1\,M KCl solution. Electrodes were fixed in a microdrive (Luigs-Neumann, Ratingen, Germany) and lowered into the nerve (\figrefb{Settup}, blue triangle). Recordings of electroreceptor afferents were amplified (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, P-unit recording, recorded EOD and the generated
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stimulus, were digitized with sampling rates of 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{www.relacs.net}) running on a Linux computer was used for online spike and EOD detection, stimulus generation, and calibration.
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%For parts of the data %0, 20 or
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\subsection{Identification of P-units and ampullary cells}
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The neurons were classified into cell types during the recording by the experimenter. P-units were classified based on mean baseline firing rates of 50--450\,Hz \citep{Grewe2017, Hladnik2023} and phase-locking to the EOD, leading to a multimodal interspike interval (ISI) distribution (as in \subfigrefb{heterogeneity}{B}). Ampullary cells were classified based on mean firing rates of 80--200\,Hz and no phase-locking to the EOD with an unimodal ISI distribution. Only cells with a baseline recording were included in the final sample. Cells that ceased spiking during the baseline recording were excluded.
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\subsection{Field recordings} The EOD of fish without the stimulus was termed global EOD and measured with two vertical carbon rods ($11\,\centi\meter$
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long, 8\,mm diameter) in a head-tail configuration (\figrefb{Settup}, green bars). This signal was
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amplified 200--500 times and band-pass filtered (3 to 1\,500\,Hz
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passband, DPA2-FX; npi electronics, Tamm, Germany).
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The EOD of the fish with the stimulus was termed local EOD and was measured between two 1\,cm-spaced silver
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wires located next to the left gill of the fish and orthogonal to its longitudinal
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body axis (amplification 200--500 times, band-pass filtered with 3 to
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1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm,
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Germany, \figrefb{Settup}, red markers).
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\subsection{Stimulation}
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The stimulus was isolated from the ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located
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$15\,\centi\meter$ laterally to the fish (\figrefb{Settup}, gray bars).
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\begin{figure*}[h!]%(\subfigrefb{beat_amplitudes}{B}).
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\includegraphics[width=\columnwidth]{Settup}
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\caption{\label{Settup} Electrophysiolocical recording set-up. The fish, depicted as a black scheme and surrounded by isopotential lines, was positioned in the middle of the tank. Blue triangle -- electrophysiological recordings were conducted at the posterior anterior lateral line nerve (pALLN) above the
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gills. Gray horizontal bars -- electrodes for the stimulation. Green vertical bars -- electrodes to measure the global EOD (the EOD of the fish without the stimulus). Red dots -- electrodes to measure the local EOD (the EOD of the fish with the stimulus). The local EOD was measured with a distance of 1 \,cm between the electrodes. All measured signals were amplified, filtered and stored on a local computer.}
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\end{figure*}
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%Fish sketch adapted base on \citep{Hagedorn1985} next to the gills,
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%Blue circles - P-units.
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\subsection{White noise stimuli}\label{rammethods}
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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).
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%A stimulus had a duration of 10\,s and was subdivided into $\n{}=20$ windows with no overlap, each with the duration of $T=0.5$\,s. S
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%The size of the sinewaves was set to a contrast of 10\,$\%$ of the EOD of the receiver.
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%If one sine wave was present this resulted in the one-beat condition, if two sine waves were present in the two-beat condition.
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\subsection{Data analysis} Data analysis was performed with Python~3
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using the packages matplotlib, numpy, scipy, sklearn, pandas, nixio
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\citep{Stoewer2014}, and thunderfish
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(\url{https://github.com/bendalab/thunderfish}).
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\paragraph{Baseline calculation}\label{baselinemethods}%
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The mean baseline firing rate \fbase{} was calculated as the number
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of spikes divided by the duration of the baseline recording (on
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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.
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%The serial correlation was calculated with lag 1 ($SC_1$, \eqnref{serial}).
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\paragraph{Bursting}\label{burstfraction}
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Bursty cells are characterized by a bimodal ISI distribution, with the burst spikes having ISI values smaller than the burst threshold, which is usually set to 1.5\,EOD periods in P-units \citep{Chacron2004, Metzen2016}. Here the bimodal ISI distribution of bursty P-units was not always properly separated by a threshold of 1.5\,EOD periods. Therefore, a threshold best separating the bimodal distribution was set by visual inspection. In the final P-units sample, 149 cells had a burst threshold of 1.5\,EOD periods, 59 cells had a burst threshold of 2.5\,EOD periods, 7 cells had a burst threshold of 3.5\,EOD periods, 4 cells had a burst threshold of 4.5\,EOD periods and 1 cell had a burst threshold of 5.5\,EOD periods. The burst fraction was calculated as the number of spikes below the burst threshold divided by all spikes.
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The baseline characteristics (see methods section \ref{baselinemethods}), as the mean baseline firing rate after burst correction \fbasecorr{} and the baseline CV after burst correction \cvbasecorr{}, were calculated based on the burst-corrected spike train, where all spikes in a burst package after the first spike were removed.
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%To get a burst-corrected spike train all spikes in a burst package after the first spike were removed.
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%as the mean number of spikes in the burst-corrected spike train. % CV \cvbasecorr{} after burst correction
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%\subsubsection{burstfraction} %1.5: 147,2.5: 59,3.0: 1,3.5: 6,4.5: 3,5.5: 1,
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%\subsection{Data analysis for chapter 2}
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\paragraph{Response modulation} \label{response_modulation}
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The firing rate of a cell is modulated around an average firing rate similar to \fbase{} in response to a stimulus. The response modulation was calculated based on the mean firing rate $f(t)$. For $f(t)$ the binary spike trains, with zero everywhere no spike occurred and the sampling rate everywhere a spike occurred, were convolved with a Gaussian with a standard deviation of 2.5\,ms and the mean over the stimulus repeats was calculated. The response modulation $\sigma_{M} = \sqrt{\langle (f(t)-\langle f(t) \rangle)^2\rangle}$ was calculated as the standard deviation of the mean firing rate $f(t)$, where the averages are taken over time. The response modulation was calculated as an estimate of the effective stimulus strength.
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% _{t}_{n}
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% smoothed = pad_for_mean(smoothed, max_len)
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% response_modulation = np.nanstd(np.mean(smoothed, axis=0)) Spectral analysis and linear encoding
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%the transfer function
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%Applying the Fourier transform to the stimulus $s(t)$ resulted in $\tilde{s}(f)$ and applying the Fourier transform to the response of the neuron, the firing rate of the spike trains $r(t)$ resulted in $R(f)$.
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%, therefore the stimulusSecond-order susceptibility
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\paragraph{Spectral analysis}\label{susceptibility_methods}%chapter 2In this score the stimulus $s(t)$ and the response $r(t)$ were set into relation.
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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.
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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
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\begin{equation}
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\label{powereq}
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\begin{split}
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S_{ss}(\omega) = \frac{\langle \tilde s(\omega) \tilde s^* (\omega)\rangle}{T}
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\end{split}
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\end{equation}
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with $\tilde s^* $ being the complex conjugate and $\langle ... \rangle$ the averaging over the segments.
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The cross-spectrum was calculated with \eqnref{cross}, linear encoding was estimated with \eqnref{linearencoding_methods} and the higher-order cross-spectrum was calculated with \eqnref{crosshigh}.
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\begin{equation}
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\label{cross}
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\begin{split}
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S_{rs}(\omega) = \frac{\langle \tilde r(\omega) \tilde s^* (\omega)\rangle}{T}
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\end{split}
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\end{equation}
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\begin{equation}
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\label{linearencoding_methods}
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\begin{split}
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\chi_{1}(\omega) = \frac{S_{rs}(\omega) }{S_{ss}(\omega) }
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\end{split}
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\end{equation}
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\begin{equation}
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\label{crosshigh}
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S_{rss} (\omega_{1},\omega_{2}) = \frac{\langle \tilde r (\omega_{1}+\omega_{2}) \tilde s^* (\omega_{1})\tilde s^* (\omega_{1}) \rangle}{T}
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\end{equation}
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The second-order susceptibility was calculated as
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\begin{equation}
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\label{susceptibility0}
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\begin{split}
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\chi_{2}(\omega_{1}, \omega_{2}) = \frac{S_{rss} (\omega_{1},\omega_{2})}{2S_{ss} (\omega_{1}) S_{ss} (\omega_{2})}
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\end{split}
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\end{equation}
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by dividing the higher-order cross-spectrum by the power spectra. Applying the Fourier transform this can be rewritten resulting in:
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\begin{equation}
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\label{susceptibility}
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\begin{split}
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\chi_{2}(\omega_{1}, \omega_{2}) = \frac{TN \sum_{n=1}^N \int_{0}^{T} dt\,r_{n}(t) e^{-i(\omega_{1}+\omega_{2})t} \int_{0}^{T}dt'\,s_{n}(t')e^{i \omega_{1}t'} \int_{0}^{T} dt''\,s_{n}(t'')e^{i \omega_{2}t''}}{2 \sum_{n=1}^N \int_{0}^{T} dt\, s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{T} dt\,s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{2}t'}}
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\end{split}
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\end{equation}
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The absolute value of a second-order susceptibility matrix is visualized in \figrefb{model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $r(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $r(t)$ at the difference of the input frequencies.
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%\subsection{First-order susceptibility} \label{linearencoding_methods}
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% The lower right and upper left quadrants in the susceptibility matrix in \figrefb{model_full} were calculated as
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%\begin{equation}
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% \label{susceptibility2}
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% \chi_{2} = \frac{TN |\sum_{n=1}^N \int_{0}^{T} dt r_{n}(t) e^{i( \omega_{1}- \omega_{2})t} \int_{0}^{T}dt' s_{n}(t')e^{-i \omega_{1}t'} \int_{0}^{T} dt'' s_{n}(t'')e^{i \omega_{2}t''}|}{2 \sum_{n=1}^N \int_{0}^{T} dt s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt' s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{N} dt s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{N} dt' s_{n}(t')e^{i \omega_{2}t'}}.
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%\end{equation}
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\paragraph{Nonlinearity index}\label{projected_method}%chapter 2
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The structural changes in the absolute value of the second-order susceptibility (as the matrix in \figrefb{model_full}) were quantified in a nonlinearity index:
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\begin{equation}
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\label{nli_equation}
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NLI(f_{0}) = \frac{\max_{f_{0}-5\,\rm{Hz} \leq f \leq f_{0}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
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\end{equation}
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For this index, the second-order susceptibility matrix was projected on the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness of a frequency $f_{0}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $f_{0} \pm 5$\,Hz (\subfigrefb{cells_suscept}\,\panel[vi]{A}, gray area) and dividing it by the median of $D(f)$. This was calculated for the frequencies \fbase{} and \fbasecorr{} resulting in \nli{} and \nlicorr{}.
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If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} and \nlicorr{} values was used for the population statistics in \figref{data_overview_mod} and \figrefb{data_overview}. The second-order susceptibility matrices depicted in \figrefb{cells_suscept}, \figrefb{ampullary}, \figrefb{burst_cells_suscept} and \figrefb{cells_eigen} were calculated based on the first frozen noise repeat.
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% \includegraphics[width=\columnwidth]{cells_eigen} % {ampullary}{burst_cells_suscept}, {cells_eigen}
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%\nli{} and \nlicorr{} scores shown in were calculated as the mean of the \nli{} and \nlicorr{} values for all frozen noise RAM repetitions in a cel
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\subsection{Leaky integrate-and-fire models}\label{lifmethods}
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Leaky integrate-and-fire (LIF) models with a carrier were constructed to reproduce the specific firing
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properties of P-units \citep{Chacron2001,Sinz2020}. The input into the P-unit model during baseline was the fish's own EOD
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\begin{equation}
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\label{eod}
|
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x(t) = x_{EOD}(t) = \cos(2\pi f_{EOD} t)
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\end{equation}
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with EOD frequency $f_{EOD}$ and amplitude normalized to one.
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In the model, the input $x(t)$ was first thresholded.
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\begin{equation}
|
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\label{threshold2}
|
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\lfloor x(t) \rfloor_0 = \left\{ \begin{array}{rcl} x(t) & ; & x(t) \ge 0 \\ 0 & ; & x(t) < 0 \end{array} \right.
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\end{equation}
|
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$\lfloor x(t) \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (box left to \subfigrefb{flowchart}\,\panel[i]{A}). Thresholding potentially happens at the synapse between the receptor cells and the afferent.
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The resulting signal was then low-pass filtered with a time constant $\tau_{d}$ by the afferent's dendrite (box left to \subfigrefb{flowchart}\,\panel[ii]{A}).
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\begin{equation}
|
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\label{dendrite}
|
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\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}^{p}
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\end{equation}
|
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Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high
|
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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 $x(t)$.
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The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
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\begin{equation}
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\label{LIF}
|
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\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D}\xi(t)
|
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\end{equation}
|
|
where $\tau_{m}$ is the membrane time constant, $\mu$ is a fixed bias current, $\alpha$ is a scaling factor for $V_{d}$, and $\sqrt{2D}\xi(t)$ is a white noise with strength $D$. All state variables in the LIF are unitless. The output of the model is in seconds.
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The adaptation current $A$ followed
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\begin{equation}
|
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\label{adaptation}
|
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\tau_{A} \frac{d A}{d t} = - A
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\end{equation}
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with adaptation time constant $\tau_A$.
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Whenever the membrane voltage $V_m(t)$ crossed the threshold $\theta=1$ a spike was generated, $V_{m}(t)$ was reset to $0$, the adaptation current was incremented by $\Delta A$, and integration of $V_m(t)$ was paused for the duration of a refractory period $t_{ref}$ (\subfigrefb{flowchart}\,\panel[iv]{A}).
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\begin{equation}
|
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\label{spikethresh}
|
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V_m(t) \ge \theta \; : \left\{ \begin{array}{rcl} V_m & \mapsto & 0 \\ A & \mapsto & A + \Delta A/\tau_A \end{array} \right.
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\end{equation}
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The static nonlinearity $f(V_m)$ was equal to zero for the LIF. In the case of an exponential integrate-and-fire model (EIF), this function was set to
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\begin{equation}
|
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\label{eifnl}
|
|
f(V_m)= \Delta_V \text{e}^{\frac{V_m-1}{\Delta_V}}
|
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\end{equation}
|
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\citep{Fourcaud-Trocme2003}, where $\Delta_V$ was varied from 0.001 to 0.1.
|
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%, \figrefb{eif}
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\begin{figure*}[hb!]
|
|
\includegraphics[width=\columnwidth]{flowchart}
|
|
\caption{\label{flowchart} Flowchart of a LIF P-unit model with EOD carrier. Model cell identifier 2012-07-03-ak (see table~\ref{modelparams} for model parameters). \figitem[i]{A}--\,\panel[i]{\textbf{D}} Rectification of the input. Positive values are maintained and negative discarded (see box on the left). \figitem[ii]{A}--\,\panel[ii]{\textbf{D}} Dendritic low-pass filtering. \figitem[iii]{A}--\,\panel[iii]{\textbf{D}} The noise component in $\sqrt{2D}\,\xi(t)$ \eqnref{LIF} or $\sqrt{2D \, c_{noise}}\,\xi(t)$ in \eqnref{Noise_split_intrinsic}. \figitem[iv]{A}--\,\panel[iv]{\textbf{D}} Spikes generation in the LIF model. Spikes are generated when the voltage of 1 is crossed (markers). Then the voltage is again reset to 0. \figitem[v]{A}--\,\panel[v]{\textbf{D}} Power spectrum of the spikes above. The first peak in panel \panel[v]{A} is the \fbase{} peak. The peak at 1 is the \feod{} peak. The other two peaks are at $\feod{} \pm \fbase{}$. \figitem{A} Baseline condition: The input to the model is a sinus with frequency \feod{}. \figitem{B} The EOD carrier is multiplied with a band-pass limited random amplitude modulation (RAM) with a contrast of 2\,$\%$, as in \eqnref{ram_equation}. \figitem{C} The EOD carrier is multiplied with a band-pass limited RAM signal with a contrast of 20\,$\%$. \figitem{D} The total noise of the model is split into a signal component regulated by $c_{signal}$ in \eqnref{ram_split}, and a noise competent regulated by $c_{noise}$ in \eqnref{Noise_split_intrinsic}. The intrinsic noise in \panel[iii]{D} is reduced compared to \panel[iii]{A}--\panel[iii]{C}. To maintain the CV during the noise split in \panel{D} comparable to the CV during the baseline in \panel{A} the RAM contrast is increased in \panel[i]{D}.}
|
|
\end{figure*}
|
|
%\figitem[i]{C}$RAM(t)$
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\subsection{Numerical implementation}
|
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|
|
The ODEs were integrated by the Euler forward method with a time-step of $\Delta t = 0.00005$\,s.
|
|
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|
|
|
For the intrinsic noise $\xi(t)$ (\eqnref{LIF}, \subfigrefb{flowchart}\,\panel[iii]{A}) in each time step $i$ a random number was drawn from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$:
|
|
\begin{equation}
|
|
\label{LIFintegration}
|
|
V_{m_{i+1}} = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
|
|
\end{equation}
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\subsection{Model parameters}\label{paramtext}
|
|
The 8 free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$,
|
|
$D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to
|
|
both baseline activity (mean baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector
|
|
strength of spike coupling to EOD) and responses to step increases and
|
|
decreases in EOD amplitude (onset-state and steady-state responses,
|
|
effective adaptation time constant) of 42 specific P-units for a fixed power of $p=1$ (table~\ref{modelparams}, \citealp{Ott2020}). When modifying
|
|
the model (e.g. varying the threshold nonlinearity or the power $p$ in \eqnref{dendrite}) the bias current $\mu$ was adapted to restore the original mean baseline firing rate. For each stimulus repetition the start parameters $A$, $V_{d}$ and $V_{m}$ were drawn randomly from a starting value distribution, retrieved from a 100\,s baseline, recorded in the model after a 100\,s transient.
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\subsection{Stimuli for the model}%analysis of the
|
|
For the beat protocols, the stimulus was the EOD of the receiving fish, normalized to an
|
|
amplitude of one plus the EOD of a second or third fish. If not stated
|
|
otherwise, a superposition of cosine waves was used to
|
|
mimic the EODs (\eqnref{beat}).
|
|
|
|
The input for the model during RAM stimulation was the EOD multiplied with a random amplitude modulation $RAM(t)$:
|
|
\begin{equation}
|
|
\label{ram_equation}
|
|
x(t) = (1+ RAM(t)) \cdot \cos(2\pi f_{EOD} t)
|
|
\end{equation}
|
|
The $RAM(t)$ was generated by drawing random numbers for each frequency up to 300\,Hz in Fourier space. After back transformation, the resulting signal was scaled to the desired standard deviation relative to the EOD carrier, which was termed contrast. %Rectified $x(t)$ with RAM contrasts of $2$ and $20\,\%$ are depicted in \subfigrefb{flowchart}\,\panel[i]{B},\,\panel[i]{C}.%$f_{EOD}/2$
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|
\subsection{Second-order susceptibility analysis of the model}
|
|
%\subsubsection{Model second-order nonlinearity}
|
|
|
|
The second-order susceptibility in the model was calculated with \eqnref{susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz.
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\subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
|
|
Based on the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} the total noise of a LIF model can be split up into several independent noise processes with the same correlation function. Based on this theorem a weak input signal to the LIF model can be approximated with no input signal but instead, a split of the total noise into a noise component and a signal component (\citealp{Egerland2020}). This signal component can be used for the cross-spectrum calculation in \eqnref{susceptibility}, where it is not a weak signal anymore. This approach has the advantage that the signal-to-noise ratio is increased and the number of noise stimulus realizations \n, which would be required in case of a weak input signal for the same signal-to-noise ratio, is reduced. The signal component is regulated by $c_{signal}$ in \eqnref{ram_split} and the noise component is regulated by $c_{noise}$ in \eqnref{Noise_split_intrinsic}.
|
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|
|
\begin{equation}
|
|
\label{ram_split}
|
|
x(t) = (1+ \sqrt{\rho \, 2D \,c_{signal}} \cdot RAM(t)) \cdot \cos(2\pi f_{EOD} t)
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
\label{Noise_split_intrinsic_dendrite}
|
|
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}^{p}
|
|
\end{equation}
|
|
|
|
|
|
%\begin{equation}
|
|
% \label{Noise_split_intrinsic}
|
|
% V_{m_{i+1}} = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{noise}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
|
|
%\end{equation}
|
|
|
|
\begin{equation}
|
|
\label{Noise_split_intrinsic}
|
|
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{noise}}\,\xi(t)
|
|
\end{equation}
|
|
% das stimmt so, das c kommt unter die Wurzel!
|
|
|
|
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\,\%.
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|
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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\subsection{Artificial bursts in the model}\label{modelburstadd_method}
|
|
For the analysis in \figrefb{burst_add} the spikes in the non-busty model with identifier 2013-01-08-aa (see table~\ref{modelparams} for model parameters) were supplemented by burst spikes after exactly one, two or three EOD periods. A spike was not added if the refractory time to the next spike could not be maintained.
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% to the spikes generated in the model
|
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%bursts were artificially supplemented to the spikes created with the
|
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%\section{Second-order susceptibility analysis of the model}
|
|
%the changes of the field potential with distance can be described as an ideal dipole with the field
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% the 5\,\% maximal values of the centered EOD
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\begin{table*}[hp!]
|
|
\caption{\label{modelparams} Model parameters of LIF models, fitted to 42 electrophysiologically recorded P-units (\citealp{Ott2020}). See section \ref{lifmethods} for model and parameter description.}
|
|
\begin{center}
|
|
\begin{tabular}{lrrrrrrrr}
|
|
\hline
|
|
\bfseries $cell$ & \bfseries $\beta$ & \bfseries $\tau_{m}$/ms & \bfseries $\mu$ & \bfseries $D$/$\mathbf{ms}$ & \bfseries $\tau_{A}$/ms & \bfseries $\Delta_A$ & \bfseries $\tau_{d}$/ms & \bfseries $t_{ref}$/ms \\\hline
|
|
2013-01-08-aa& $4.5$& $1.20$& $0.59$& $0.001$& $37.52$& $0.01$& $1.18$& $0.38$ \\
|
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\hline
|
|
\end{tabular}
|
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\end{center}
|
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\end{table*}% 2013-01-08-aa
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\appendix
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\setcounter{secnumdepth}{2}
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\section{Appendix}
|
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\begin{figure*}[h!]%p!
|
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\includegraphics[width=\columnwidth]{burst_cells_suscept_appendix}%
|
|
\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.}
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\end{figure*}
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\newpage
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%Recording at the frequency combinations \bcsum{} and \bcdiff{} \fbasecorr{}at the burst-corrected firing rate
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\appendix
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