969 lines
95 KiB
TeX
969 lines
95 KiB
TeX
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%% Copyright 2007-2020 Elsevier Ltd
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%% $Id: elsarticle-template-num.tex 190 2020-11-23 11:12:32Z rishi $
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%%%%% hyperef %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\usepackage[breaklinks=true,colorlinks=true,citecolor=blue!30!black,urlcolor=blue!30!black,linkcolor=blue!30!black]{hyperref}
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%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\figitem}[2][]{\ifthenelse{\equal{#1}{}}{\textsf{\bfseries #2}}{\textsf{\bfseries #2}}$_{\sf #1}$}
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\newcommand{\panel}[2][]{\ifthenelse{\equal{#1}{}}{\textsf{#2}}{\textsf{#2}$_{\sf #1}$}}%\ifthenelse{\equal{#1}{}}{\textsf{#2}}{\textsf{#2}$_{\sf #1}$}
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\newcommand{\figsrefb}[1]{\figsb~\fref{#1}}
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\newcommand{\subfigrefb}[2]{\figb~\subfref{#1}{#2}}
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\newcommand{\subfigsrefb}[2]{\figsb~\subfref{#1}{#2}}
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%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\fstim}{\ensuremath{f_{stim}}}
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\newcommand{\feod}{\ensuremath{f_{EOD}}}
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\newcommand{\feodsolid}{\ensuremath{f_{\rm{EOD}}}}
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\newcommand{\feodhalf}{\ensuremath{f_{EOD}/2}}
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\newcommand{\fbase}{\ensuremath{f_{Base}}}
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\newcommand{\fbasesolid}{\ensuremath{f_{\rm{Base}}}}
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\newcommand{\fstimintro}{\ensuremath{\rm{EOD}_{2}}}
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\newcommand{\feodintro}{\ensuremath{\rm{EOD}_{1}}}
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\newcommand{\ffstimintro}{\ensuremath{f_{2}}}
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\newcommand{\ffeodintro}{\ensuremath{f_{1}}}
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%intro
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%\newcommand{\signalcomp}{$s$}
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\newcommand{\baseval}{134}
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\newcommand{\bone}{$\Delta f_{1}$}
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\newcommand{\btwo}{$\Delta f_{2}$}
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\newcommand{\fone}{$f_{1}$}
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\newcommand{\ftwo}{$f_{2}$}
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\newcommand{\ff}{$f_{1}$--$f_{2}$}
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\newcommand{\auc}{\rm{AUC}}
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\newcommand{\n}{\ensuremath{N}}
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\newcommand{\bvary}{\ensuremath{\Delta f_{1}}}
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\newcommand{\avary}{\ensuremath{A(\Delta f_{1})}}
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\newcommand{\afvary}{\ensuremath{A(f_{1})}}
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\newcommand{\avaryc}{\ensuremath{A(c_{1})}}
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\newcommand{\fvary}{\ensuremath{f_{1}}}
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\newcommand{\bstable}{\ensuremath{\Delta f_{2}}}
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\newcommand{\astable}{\ensuremath{A(\Delta f_{2})}}
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\newcommand{\astablec}{\ensuremath{A(c_{2})}}
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\newcommand{\cstable}{\ensuremath{c_{2}}}
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\newcommand{\fstable}{\ensuremath{f_{2}}}
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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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%\newcommand{\colstabletwo}{blue}%sum
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%\newcommand{\colvaryone}{dark red}%sum
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%\newcommand{\colvarytwo}{red}%sum
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%\newcommand{\colstableoneb}{Dark blue}%sum
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%\newcommand{\colstabletwob}{Blue}%sum
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%\newcommand{\colvaryoneb}{Dark red}%sum
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%\newcommand{\colvarytwob}{Red}%sum
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\newcommand{\colstableone}{blue}%sum
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\newcommand{\colstabletwo}{cyan}%sum
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\newcommand{\colvaryone}{brown}%sum
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\newcommand{\colvarytwo}{red}%sum
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\newcommand{\colstableoneb}{Blue}%sum
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\newcommand{\colstabletwob}{Cyan}%sum
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\newcommand{\colvaryoneb}{Brown}%sum Brown
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\newcommand{\colvarytwob}{Red}%sum
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\newcommand{\resp}{$A(f)$}%sum %\Delta
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\newcommand{\respb}{$A(\Delta f)$}%sum %\Delta
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\newcommand{\rocf}{98}%sum %\Delta
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\newcommand{\roci}{36}%sum %\Delta
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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{\bsumb}{$\bsum{}=\fbase{}$}%su\right m
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\newcommand{\btwob}{$\Delta f_{2}=\fbase{}$}%sum
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\newcommand{\boneb}{$\Delta f_{1}=\fbase{}$}%sum
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\newcommand{\bsumbtwo}{$\bsum{}=2 \fbase{}$}%sum
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\newcommand{\bsumbc}{$\bsum{}=\fbasecorr{}$}%sum
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\newcommand{\bsume}{$\bsum{}=f_{EOD}$}%sum
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\newcommand{\bsumehalf}{$\bsum{}=f_{EOD}/2$}%sum
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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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\newcommand{\fsumb}{$\fsum=\fbase{}$}%su\right m
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\newcommand{\ftwob}{$f_{2}=\fbase{}$}%sum
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\newcommand{\foneb}{$f_{1}=\fbase{}$}%sum
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\newcommand{\ftwobc}{$f_{2}=\fbasecorr{}$}%sum
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\newcommand{\fonebc}{$f_{1}=\fbasecorr{}$}%sum
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\newcommand{\fsumbtwo}{$\fsum{}=2 \fbase{}$}%sum
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\newcommand{\fsumbc}{$\fsum{}=\fbasecorr{}$}%sum
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\newcommand{\fsume}{$\fsum{}=f_{EOD}$}%sum
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\newcommand{\fsumehalf}{$\fsum{}=f_{EOD}/2$}%sum
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\newcommand{\fdiffb}{$\fdiff{}=\fbase{}$}%diff of both beat frequencies
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\newcommand{\fdiffbc}{$\fdiff{}=\fbasecorr{}$}%diff of both beat frequencies
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\newcommand{\fdiffe}{$\fdiff{}=f_{EOD}$}%diff of both
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\newcommand{\fdiffehalf}{$\fdiff{}=f_{EOD}/2$}%diff of both
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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{\bcone}{$\Delta f_{Intruder}$}%sum
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%\newcommand{\bcsum}{$|\Delta f_{Female} + \Delta f_{Intruder}|$}%sum
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%\newcommand{\abcsum}{$A(|\Delta f_{Female} + \Delta f_{Intruder}|)$}%sum
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%\newcommand{\bcdiff}{$|\Delta f_{Female} - \Delta f_{Intruder}|$}%diff of both beat frequencies
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%\newcommand{\bcsumb}{$|\Delta f_{Female} + \Delta f_{Intruder}| =f_{Base}$}%su\right m
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%\newcommand{\bctwob}{$\Delta f_{Female} =f_{Base}$}%sum
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%\newcommand{\bconeb}{$\Delta f_{Intruder}=f_{Base}$}%sum
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%\newcommand{\bcsumbtwo}{$|\Delta f_{Female} + \Delta f_{Intruder}|=2 f_{Base}$}%sum
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%\newcommand{\bcsumbc}{$|\Delta f_{Female} + \Delta f_{Intruder}|=\fbasecorr{}$}%sum
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%\newcommand{\bcsume}{$|\Delta f_{Female} + \Delta f_{Intruder}|=f_{EOD}$}%sum
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%\newcommand{\bcsumehalf}{$|\Delta f_{Female} + \Delta f_{Intruder}|=f_{EOD}/2$}%sum
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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{\bcdiffehalf}{$|\Delta f_{Female} - \Delta f_{Intruder}|=f_{EOD}/2$}%diff of both
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\newcommand{\fctwo}{\ensuremath{f_{\rm{Female}}}}%sum
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\newcommand{\fcone}{\ensuremath{f_{\rm{Intruder}}}}%sum
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\newcommand{\bctwo}{\ensuremath{\Delta \fctwo{}}}%sum
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\newcommand{\bcone}{\ensuremath{\Delta \fcone{}}}%sum
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\newcommand{\bcsum}{\ensuremath{|\bctwo{}| + |\bcone{}|}}%sum
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\newcommand{\abcsumabbr}{\ensuremath{A(\Delta f_{\rm{Sum}})}}%sum
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\newcommand{\abcsum}{\ensuremath{A(\bcsum{})}}%sum
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\newcommand{\abcone}{\ensuremath{A(\bcone{})}}%sum
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\newcommand{\bctwoshort}{\ensuremath{\Delta f_{F}}}%sum
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\newcommand{\bconeshort}{\ensuremath{\Delta f_{I}}}%sum
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\newcommand{\bcsumshort}{\ensuremath{|\bctwoshort{} + \bconeshort{}|}}%sum
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\newcommand{\abcsumshort}{\ensuremath{A(\bcsumshort{})}}%sum
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\newcommand{\abconeshort}{\ensuremath{A(|\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{\rif}{\ensuremath{\rm{RIF}}}
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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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% references to tables:
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\newcommand{\tref}[1]{\textup{\ref{#1}}}
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% references to tables in normal text:
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\newcommand{\tab}{Tab.}
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\newcommand{\Tab}{Table}
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\newcommand{\tabs}{Tabs.}
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\newcommand{\Tabs}{Tables}
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\newcommand{\tabref}[1]{\tab~\tref{#1}}
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\newcommand{\Tabref}[1]{\Tab~\tref{#1}}
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\usepackage{xr}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% notes
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%% The line no packages add line numbers. Start line numbering with
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%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
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%% for the whole article with \linenumbers.
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\usepackage{lineno}
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%\linenumbers
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%\setlength{\linenumbersep}{3pt}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% caption packages
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%\usepackage{graphicx}
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\usepackage[format=plain,labelfont=bf]{caption}
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%\usepackage{capt-of}
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\captionsetup[figure2]{labelformat=empty}
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%\usepackage[onehalfspacing]{setspace}
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\journal{iScience}
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\setcounter{secnumdepth}{0}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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%\maketitle
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\begin{frontmatter}
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\title{Second-order susceptibility in electrosensory primary afferents 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[4,5]{Maria Schlungbaum}
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\author[4,5]{Benjamin Lindner}
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\author[1,2,3]{Jan Benda}
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\author[1]{Jan Grewe\corref{cor1}}
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\ead{jan.grewe@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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%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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\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. White noise stimulation was confirmed as a method to retrieve the socond-order suscepitbility in P-units.% 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}
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%with nonlinearities being observed in all sensory modalities
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Neuronal systems are inherently nonlinear \citealp{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. A prominent example of a nonlinearity is rectification that is assumed to occur through the transduction machinery of inner hair cells \citealp{Peterson2019}, signal rectification in receptor cells \citealp{Chacron2000, Chacron2001} or the rheobase of action-potential generation \citealp{Middleton2007, Longtin2008}. Nonlinearity can be necessary to explain the behavior of complex cells in the visual system \citealp{Adelson1985}, to extract information about the stimulus \citealp{Barayeu2023} and to encode stimulus features as up- and down-strokes \citealp{Gabbiani1996}.
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The time-resolved firing rate of a neuron can be described by the Volterra series where the first-order term describes the linear contribution and all higher-order terms the nonlinear contributions \citealp{Voronenko2017}. In previous work, the second-order response of the Volterra series, the second-order susceptibility, was analytically retrieved based on leaky integrate-and-fire (LIF) models, where the input were two sine waves \citealp{Voronenko2017}. There second-order susceptibility was demonstrated to be very pronounced at specific input frequencies. A triangular nonlinear shape, with nonlinearities appearing if one of the input frequencies, the sum or the difference of the frequencies was equal to the baseline firing rate \fbase{} of the cell, was predicted. Since these nonlinear effects were especially pronounced if one signal was weak, they might influence the faint signal detection as it was observed in the field in the framework of the electrosensory cocktail party \citealp{Henninger2018}.
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%. The EOD
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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 \citealp{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}), that is used for electrolocation \citealp{Fotowat2013, Nelson1999} and communication \citealp{Fotowat2013, Walz2014, Henninger2018}. If two fish meet, their EODs interfere and result in a new signal with a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD of the encountered fish and is expressed in relation to the receiver EOD as a contrast. The beat frequency is defined as the difference between the two EOD frequencies ($\Delta f_{1} = f_{1}-\feod{}$). Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citealp{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing the context for electrocommunication in weakly electric fish \citealp{Engler2001, Hupe2008, Henninger2018, Benda2020}. Whereas the predictions of the nonlinear response theory \citealp{Voronenko2017} are applicable if the main driving force for a neuron are beats and not the whole signal, will be addressed in this work.
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%apply to P-units,
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Cutaneous tuberous organs, that are distributed all over the body of these fish\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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\citealp{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 \citealp{Hladnik2023}. P-units fire in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at integer multiples of the EOD period. In previous works, P-units have mainly been considered to be linear encoders \citealp{Xu1996, Benda2005, Gussin2007, Grewe2017}. Still, P-units exhibit nonlinear effects, especially for strong beat amplitudes \citealp{Nelson1997}. Although some P-units can be described by a linear decoder, other P-units require a nonlinear decoder \citealp{Chacron2004}.
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When the receiver fish with EOD frequency \feod{} is alone, a peak at \fbase{} is present in the power spectrum of the firing rate of a recorded P-unit (\subfigrefb{motivation}{A}, bottom). P-units also represent \feod{} in their firing rate, but this peak is beyond the range of frequencies addressed in this figure. The beat frequency in a two fish setting is represented in the spike trains, firing rate and the corresponding power spectrum of P-units (\subfigrefb{motivation}{B, C}). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{motivation}{D}, top). The power spectrum of the P-unit's firing rate contains both beat frequencies, but also nonlinear peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). This difference of the two beat frequencies is known as the social envelope \citealp{Stamper2012Envelope, Savard2011} that often emerges as the modulation of two beats in the superimposed signal. The question whereas P-units encode envelopes has been the subject of controversy, with some works not considering P-units as envelope encoders \citealp{Middleton2006}, while others identify some P-unit populations as successful envelope encoders \citealp{Savard2011}. In this work the second-order susceptibility at the sum and difference frequency will be characterized for the three fish setting, addressing whereas P-units encode envelopes and how this encoding is influenced by their baseline firing properties.
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%encoding of envelopes in
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%$\Delta f_{1}$ and $\Delta f_{2}$ \bone{} and \btwo{}
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If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}. In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD.
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Whether it is possible to access the second-order susceptibility of P-units with white noise stimulation will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
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%is not the direct input to the neurons. with RAM stimuli
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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.
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%White noise stimulation will be confirmed as a method to access the second-order susceptibility in P-units.
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%The influence of the baseline firing properties, such as the CV, on nonlinear interactions will be investigated.
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\begin{figure*}[h!]
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\includegraphics[width=\columnwidth]{motivation}
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\caption{\label{motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting. Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both encountered fish lead to a beat contrast of 10\,\%. Top: Interference of the receiver EOD with the EODs of other fish. Second row: Spike trains of the P-unit. Third row: Firing rate, retrieved as the convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: Only the receiver is present. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate. \figitem{B} The receiver and the fish with EOD frequency $f_{1}=631$\,Hz are present. \figitem{C} The receiver and the fish with EOD frequency $f_{2}=797$\,Hz are present. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate.
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}
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\end{figure*}
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\section{Results}
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Theoretical work shows that leaky-integrate-and-fire (LIF) model neurons show a distinct pattern of nonlinear stimulus encoding when the model is driven by two cosine signals. In the context of the weakly electric fish, such a setting is part of the animal's everyday life as the sinusoidal electric-organ discharges (EODs) of neighboring animals interfere with the own field and each lead to sinusoidal amplitude modulations (AMs) that are called beats and envelopes \citealp{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units, respond to such AMs of the underlying EOD carrier and their time-dependent firing rate carries information about the stimulus' time-course. P-units are heterogeneous in their baseline firing properties \citealp{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy, more regular, firing pattern whereas high-CV P-units show a less regular firing pattern in their baseline activity.
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\subsection{Low-CV P-units exhibit nonlinear interactions} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
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Second-order susceptibility is expected to be especially pronounced for low-CV cells \citealp{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\citealp{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{cells_suscept}{B}).
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\begin{figure*}[!h]
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\includegraphics{cells_suscept}
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\caption{\label{cells_suscept} \notejg{dashed lines still a little faint} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. \textbf{A} Interspike-interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \textbf{B} Power-spectrum of the baseline response. \textbf{C} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) of the same P-unit. The stimulus contrast reflects the strength of the AM. \textbf{D} Transfer function (first-order susceptibility, \eqnref{linearencoding_methods}) of the responses to 10\% (light purple) and 20\% contrast (dark purple) RAM stimulation. \textbf{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \textbf{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.}
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\end{figure*}
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Noise stimuli, the random amplitude modulations (RAM, \subfigref{cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{cells_suscept}{C}). The linear encoding (see \eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
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Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{eq:susceptibility}, \subfigrefb{cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}{G}, compare light and dark purple lines).
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\subsection{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
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CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023}. \Figref{cells_suscept_high_CV} shows the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{cells_suscept_high_CV}{F}).
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\subsection{Ampullary afferents exhibit strong nonlinear interactions}
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Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\citealp{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{ampullary}{G}, dark green).
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\begin{figure*}[ht]%hp!
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\includegraphics{ampullary}
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\caption{\label{ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. \textbf{A} Interspike-interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \textbf{B} Power-spectrum of the baseline response. \textbf{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \textbf{D} Transfer function (first-order susceptibility, \eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \textbf{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \textbf{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
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}
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\end{figure*}
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% \subsection{Full nonlinear structure visible only in P-unit models}
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\subsection{Internal noise hides parts of the nonlinearity structure}
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Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\citealp{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}).\notejg{a problem, that we use a new noise for each trial?}\notesr{Using a new noise for each trial, is the way this method is defined. When using the same noise for one million repetiotions we will not see the triangular shape at any time. I tried this and Benjamin confirmed that this would be not possible.}
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%Note that this line doesn't appear in the susceptibility matrix of the model (\subfigrefb{model_and_data}{C})Each cell has an intrinsic noise level (\subfigrefb{model_and_data}{A}, bottom).
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%The signal component (purple) compensates for this total noise reduction, is not a weak signal anymore and
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Based on the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the intrinsic noise of a LIF model can be split up into several independent noise processes with the same correlation function. We make use of this and split the intrinsic noise $\xi$ into two parts: 90\% are then treated as signal ($\xi_{signal}$) while the remaining 10\% are treated as noise ($\xi_{noise}$, see methods for details). With this, the signal-to-noise ratio in the simulation can be arbitrarily varied and the combination of many repetitions and noise-split indeed reveals the triangular shape shown theoretically and for LIF models without carrier\citealp{Voronenko2017}(\subfigrefb{model_and_data}\,\panel[iii]{C}).
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%Adding an additional, independent, RAM stimulus to the simulation does not heavily influence the qualitative observations but the nonlinearity becomes weaker (compare \subfigrefb{model_and_data}\,\panel[iii]{C} and \panel[iv]{C})
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% Based on this a weak RAM signal as the input to the P-unit model (red in \subfigrefb{model_and_data}\,\panel[i]{A}) can be approximated by a model where no RAM stimulus is present (red) but instead the total noise of the model is split up into a reduced intrinsic noise component (gray) and a signal component (purple), maintaining the CV and \fbase{} as during baseline (\subfigrefb{model_and_data}\,\panel[i]{B}, see methods section \ref{intrinsicsplit_methods} for more details). This signal component (purple) can be used for the calculation of the second-order susceptibility. With the reduced noise component the signal-to-noise ratio increases and the number of stimulus realizations can be reduced. This noise split cannot be applied in experimentally measured cells. If this noise split is applied in the model with $\n{}=11$ stimulus realizations the nonlinearity at \fsumb{} is still present (\subfigrefb{model_and_data}\,\panel[iii]{B}). If instead, the RAM stimulus is drawn 1 million times the diagonal nonlinearity band is complemented by vertical and horizontal lines appearing at \foneb{} and \ftwob{} (\subfigrefb{model_and_data}\,\panel[iv]{B}). These nonlinear structures correspond to the ones observed in previous works \citealp{Voronenko2017}. If now a weak RAM stimulus is added (\subfigrefb{model_and_data}\,\panel[i]{C}, red), simultaneously the noise is split up into a noise and signal component (gray and purple) and the calculation is performed on the sum of the signal component and the RAM (red plus purple) only the diagonal band is present with $\n{}=11$ (\subfigrefb{model_and_data}\,\panel[iii]{C}) but also with 1\,million stimulus realizations (\subfigrefb{model_and_data}\,\panel[iv]{C}).
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In a high-CV P-unit we could not find such nonlinear structures, neither in the electrophysiologically recorded data nor in the respective model (not shown), corresponding to the theoretical predictions \citealp{Voronenko2017}.\notejg{How can we understand this? If we take a larger chunk of the intrinsic noise should it not help? Is there more than just noise that changes the structure?}\notejg{No this would not help. Splitting up the noise doesn't change the total noise. Nonlinear effects are expected only until a certain amount of noise. Once this noise is surpassed a better signal-to-noise ratio will not help to see that nonlinear effects that are not there. In the low CV cells the noise split changes the signal-to-noise ratio, not the total noise.}
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\begin{figure*}[hb!]
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\includegraphics[width=\columnwidth]{model_and_data}
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\caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details).
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}
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\end{figure*}
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\subsection{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
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We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral component of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{motivation}{D}) and at the difference between the two beat frequencies (rosa marker, \subfigrefb{motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (rosa circle, \subfigrefb{motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed (\eqnref{eq:crosshigh}).
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\begin{figure*}[h]
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\includegraphics[width=\columnwidth]{model_full}
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\caption{\label{model_full} Second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods). The colored markers highlight the nonlinear effects found in \subfigrefb{motivation}{D}.}
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\end{figure*}
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%section \ref{intrinsicsplit_methods}).\figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B} The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.
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The second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \citealp{Voronenko2017} (\figref{model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \figref{model_full} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in \figref{motivation}.
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The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{motivation}{D}) can be explained by the fading out horizontal line in the upper-left quadrant (\figrefb{model_full}, \citealp{Schlungbaum2023}). Even though the second-order susceptibilities here were estimated form data and models with an modulated (EOD) carrier (\figrefb{model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\citealp{Voronenko2017, Schlungbaum2023}.
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\begin{figure*}[ht!]
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\includegraphics[width=\columnwidth]{data_overview_mod}
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\caption{\label{data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity index \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
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}
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\end{figure*}
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%\eqnref{response_modulation}
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\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
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The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three that depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's $r=-0.16$, $p<0.01$). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{A, C}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
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%(Pearson's $r=-0.35$, $p<0.001$)
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%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
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The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{data_overview_mod}{B}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{data_overview_mod}{D}).
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%(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
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\section{Discussion}
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In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory \citealp{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
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%\fsum{} or \fdiff{} were equal to \fbase{} \bsum{} or \bdiff{},
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%\subsection{Methodological implications}%implying that the
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\subsection{Full nonlinear structure present in LIF models with and without carrier}%implying that the
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%occurring nonlinearities could only be explained based on a (\foneb{}, \ftwob{}, \fsumb{}, \fdiffb{},
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In this chapter, it was demonstrated that in small homogeneous populations of electrophysiological recorded cells, a diagonal structure appeared when the sum of the input frequencies was equal to \fbase{} in the second-order susceptibility (\figrefb{cells_suscept}, \figrefb{ampullary}). The size of a population is limited by the possible recording duration in an experiment, but there are no such limitations in P-unit models, where an extended nonlinear structure with diagonals, vertical and horizontal lines at \fsumb{}, \fdiffb{}, \foneb{} or \ftwob{} appeared with increased stimulus realizations (\figrefb{model_full}). This nonlinear structure is in line with the one predicted in the framework of the nonlinear response theory, which was developed based on LIF models without carrier \citealp{Voronenko2017} and was here confirmed to be valid in models with carrier (\figrefb{model_and_data}).
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\subsection{Noise stimulation approximates the nonlinearity in a three-fish setting}%implying that the
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In this chapter, the nonlinearity of P-units and ampullary cells was retrieved based on white noise stimulation, where all behaviorally relevant frequencies were simultaneously present in the stimulus. The nonlinearity of ampullary cells in paddlefish \citealp{Neiman2011fish} has been previously accessed with bandpass limited white noise.
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Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells.
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%the power spectrum of the firing rate
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%The existence of the extended nonlinearity structure (\figrefb{model_full}) was
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%\bsum{} and \bdiff{}
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%, where nonlinearities in the second-order susceptibility matrix are predicted to appear at frequency combinations at the diagonals \fsumb{}, \fdiffb{}, at the vertical line \foneb{} and the horizontal line at \ftwob.
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%llowing for data amounts that could not be acquired experimentally
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%. It could be shown that if the nonlinearity at \fsumb{} could be found with a low stimulus repeat number in electrophysiologically recorded P-units,
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%\subsection{Nonlinearity and CV}%
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\subsection{Nonlinearity and CV}%
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In this chapter, the CV has been identified as an important factor influencing nonlinearity in the spiking response, with strong nonlinearity in low-CV cells (\subfigrefb{data_overview_mod}{A--B}). These findings of strong nonlinearity in low-CV cells are in line with previous literature, where it has been proposed that noise linearizes the system \citealp{Roddey2000, Chialvo1997, Voronenko2017}. More noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citealp{Schneider2011}. Reduced noise of low-CV cells has been associated with stronger nonlinearity in pyramidal cells of the ELL \citealp{Chacron2006}.
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\subsection{Ampullary cells}%
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In this chapter strong nonlinear interactions were found in a subpopulation of low-CV P-units and low-CV ampullary cells (\subfigrefb{data_overview_mod}{A, B}). Ampullary cells have lower CVs than P-units, do not phase-lock to the EOD, and have unimodal ISI distributions. With this, ampullary cells have very similar properties as the LIF models, where the theory of weakly nonlinear interactions was developed on \citealp{Voronenko2017}. Almost all here investigated ampullary cells had pronounced nonlinearity bands in the second-order susceptibility for small stimulus amplitudes (\figrefb{ampullary}). With this ampullary cells are an experimentally recorded cell population close to the LIF models that fully meets the theoretical predictions of low-CV cells having very strong nonlinearities \citealp{Voronenko2017}. Ampullary cells encode low-frequency changes in the environment e.g. prey movement \citealp{Engelmann2010, Neiman2011fish}. The activity of a prey swarm of \textit{Daphnia} strongly resembles Gaussian white noise \citealp{Neiman2011fish}, as it has been used in this chapter. How such nonlinear effects in ampullary cells might influence prey detection could be addressed in further studies.
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%\notejg{dumped here since not strictly result...}
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%These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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%Nonlinear effects only for specific frequency combinations
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\subsection{Heterogeneity of P-units might influence nonlinearity}
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%is required to cover all female-intruder combinations} %gain_via_mean das ist der Frequenz plane faktor
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%associated with nonlinear effects as \fdiffb{} and \fsumb{} were with the same firing rate low-CV P-units
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%and \figrefb{ROC_with_nonlin}
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P-units are very heterogeneous in their baseline firing properties considering their baseline firing rates (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}), their CV and also their bursting, the repeating firing of spikes after an EOD cycle \citealp{Chacron2004}. How this heterogeneity influences nonlinear effects when an population of P-units is integrated by the next connected pyramidal cells in the electrosensory lateral line lobe (ELL) should be addressed in further studies.
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The nonlinearity in this work were found in low-CV P-units. A selective readout from a homogeneous population of low-CV cells might be required for this nonlinear effects to sustain on a population level. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citealp{Maler2009a} and P-units have heterogeneous baseline properties that do not depend on the location of the receptor on the fish body \citealp{Hladnik2023} as a result each pyramidal cell in the ELL will integrate over a heterogeneous and not homogeneous P-unit population. The center-surround receptive field organization of superficial and intermediate layer pyramidal cells in addition increases this heterogeneity of the integrated population. Heterogeneous populations with different baseline properties can be more advantageous for the encoding compared to homogeneous populations, as has been shown in P-units and models \citealp{Beiran2018, Hladnik2023}.
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%\subsubsection{A heterogeneous readout is required to cover all female-intruder combinations}
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Nonlinear effects might facilitate the encoding of faint signals during a three fish setting, the electrosensory cocktail party. The EOD frequencies of the encountered three fish would be drawn from the EOD frequency distribution and a stable faint signal detection would require a response irrespective of the individual EOD frequencies. Here nonlinear effects, that might influence the detection of faint signals, was found only at specific frequencies in relation to the baseline firing rate \fbase{}. Weather integrating from a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) would cover the behaviorally relevant range in the electrosensory cocktail party could be addressed in further studies.
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%Thus, a heterogeneous readout might be not only physiologically plausible but also required since nonlinear effects depending on cell property, as the baseline firing rate \fbase{}, and not on stimulus properties, might be not behaviorally relevant.
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%In this work, nonlinear effects were always found only for specific frequencies in relation to \fbase{}, corresponding to findings from previous literature \citealp{Voronenko2017}.
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%Only a heterogeneous population could cover the whole stimulus space required during the electrosensory cocktail party.
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%If pyramidal cells would integrate only from P-units with the same baseline firing rate \fbasesolid{} not all fish encounters, relevant for the context of the electrosensory cocktail party, could be covered.
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%
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%Second-order susceptibility was strongest the lower the CV of the use LIF model \citealp{Voronenko2017}.
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% These nonlinear effects characterized In this chapter, (\bone{}, \btwo{}, \bsum{} or \bdiff{}) are specific for a three-fish setting. could be identified at \fsumb{} in small ow-CV ampullary cells and low-CV P-units
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%\subsection{Full nonlinear structure corresponds to theoretical predictions}
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%where second-order susceptibility could be explored systematically,
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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
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The RAM stimulus used in this work 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 \citealp{Stamper2012Envelope}.
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In previous works it was demonstrated that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citealp{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citealp{Middleton2007}. Based on our work we would predict that only a small class of cells, with very low CVs, should encode the social envelope at the difference frequency. If the sample in that previous work \citealp{Middleton2007} did not contain low CVs cells, this could explain the conclusion that P-units were not identified as envelope encoders.
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On the other hand in previous literature the encoding of social envelopes was attributed to a subpopulation of P-units with strong nonlinearities, low firing rates and high CVs \citealp{Savard2011}. These findings are in contrast to the findings in the previously mentioned work \citealp{Middleton2007} and on first glance also to our findings. The missing link, that has not been considered in this work, might be bursting of P-units, the repeated firing of spikes after one EOD period interleaved with quiescence (unpublished work). Bursting was not explicitly addressed in the previous work, still the high CVs of the envelope encoding P-units indicate a higher rate of bursting \citealp{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
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%This small percentage of the low-CV cells would be in line with no P-units found in the work.
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\subsection{More fish would decrease second-order susceptibility}%
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When using noise stimulation strong nonlinearity was demonstrated to appear for small noise stimuli but to decrease for stronger noise stimuli (\figrefb{cells_suscept}). A white noise stimulus is a proxy of many fish being present simultaneously. When the noise amplitude is small, those fish are distant and the nonlinearity is maintained. When the stimulus amplitude increases, many fish close to the receiver are mimicked and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \eigen{} can usually be found in groups of three to four fish \citealp{Tan2005} and \lepto{} in groups of two \citealp{Stamper2010}. Thus the here described second-order susceptibility might still be behaviorally relevant for both species. The decline of nonlinear effects when several fish are present might be adaptive for the receiver, reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons. How nonlinear effects might influence the three-fish setting known as the electrosensory cocktail party (\citealp{Henninger2018}) will be addressed in the next chapter.
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\notejg{Why do we see peaks at the vertical lines in the three fish setting but not in the RAM situation? SNR? Discussion?}
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\subsection{Conclusion} In this work, 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. It was demonstrated that the theory of weakly nonlinear interactions \citealp{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinear effects were identified in experimentally recorded low-CV cells primary sensory afferents, the P-units and the ampullary cells. P-units share several features with mammalian
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auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citealp{Joris2004}.
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\section{Methods}
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\subsection{Experimental subjects and procedures}
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Within this project we re-evaluated datasets that were recorded between 2010 and 2023 at the Ludwig Maximilian University (LMU) M\"unchen and the Eberhard-Karls University T\"ubingen. All experimental protocols complied with national and European law and were approved by the respective Ethics Committees of the Ludwig-Maximilians Universität München (permit no. 55.2-1-54-2531-135-09) and the Eberhard-Karls Unversität Tübingen (permit no. ZP 1/13 and ZP 1/16).
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The final sample consisted of 221 P-units and 45 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
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Germany). The fish were kept in tanks with a water temperature of $25\pm1\,^\circ$C and a conductivity of around $270\,\micro\siemens\per\centi\meter$ under a 12\,h:12\,h light-dark cycle.
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Before surgery, the animals were deeply anesthetized 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) was exposed by making a small cut into the skin covering the nerve. The cut was placed dorsal of the operculum just before the nerve descends towards the anterior lateral line ganglion (ALLNG). Those parts of the skin that were to be cut were locally anesthetized by cutaneous application of liquid lidocaine hydrochloride (20\,mg/ml, bela-pharm GmbH). During the surgery water supply was ensured by a mouthpiece to maintain anesthesia with a solution of MS222 (100\,mg/l) buffered with Sodium Bicarbonate (100\,mg/l). After surgery fish were immobilized by intramuscular injection of from 25\,$\micro$l to 50\,$\micro$l of tubocurarine (5\,mg/ml dissolved in fish saline; Sigma-Aldrich).
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Respiration was then switched to normal tank water and the fish was transferred to the experimental tank.
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\subsection{Experimental setup}
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For the recordings fish were positioned centrally in the experimental tank, with the major parts of their body submerged into the water. Those body parts that were above the water surface were covered with paper tissue to avoid drying of the skin. Local analgesia was refreshed in intervals of two hours by cutaneous reapplication of Lidocaine (2\,\%; bela-pharm, Vechta, Germany) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB1These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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50F-8P; 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{Setup}, blue triangle). Recordings of electroreceptor afferents were amplified and lowpass filtered at 10\,kHz (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, neuronal recordings, recorded EOD and the generated stimulus, were digitized with sampling rates of 20 or 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. Recorded data was then stored on the hard drive for offline analysis.
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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 baseline firing rates of 50--450\,Hz and a clear phase-locking to the EOD and their responses to amplitude modulations of their own EOD\citealp{Grewe2017, Hladnik2023}. Ampullary cells were classified based on mean firing rates of 80--200\,Hz absent phase-locking to the EOD and responses to low-frequency sinusoidal stimuli\citealp{Grewe2017}. We here selected only those cells of which the neuron's baseline activity as well as the responses to frozen noise stimuli were recorded.
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\subsection{Electric field recordings}
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The electric field of the fish was recorded in two ways: 1. we measured the so-called \textit{global EOD} with two vertical carbon rods ($11\,\centi\meter$ long, 8\,mm diameter) in a head-tail configuration (\figrefb{Setup}, green bars). The electrodes were placed isopotential to the stimulus. This signal was differentially amplified with a factor between 100 and 500 (depending on the recorded animal) and band-pass filtered (3 to 1500\,Hz pass-band, DPA2-FX; npi electronics, Tamm, Germany). 2. The so-called \textit{local EOD} was measured with 1\,cm-spaced silver wires located next to the left gill of the fish and orthogonal to the fish's longitudinal body axis (amplification 100 to 500 times, band-pass filtered with 3 to 1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm, Germany, \figrefb{Setup}, red markers). This local measurement recorded the combination of the fish's own field and the applied stimulus and thus serves as a proxy of the transdermal potential that drives the electroreceptors.
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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 $15\,\centi\meter$ laterally to the fish (\figrefb{Setup}, gray bars). The stimulus was calibrated with respect t>>>>>>> Stashed changes
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o the local EOD.
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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{Setup} Electrophysiolocical recording setup. The fish, depicted as a black scheme and surrounded by isopotential lines, was positioned in the center of the tank. Blue triangle -- electrophysiological recordings were conducted in the posterior anterior lateral line nerve (pALLN). Gray horizontal bars -- electrodes for the stimulation. Green vertical bars -- electrodes to measure the \textit{global EOD} placed isopotential to the stimulus, i.e. recording fish's unperturbed EOD. Red dots -- electrodes to measure the \textit{local EOD} picking up the combination of fish's EOD and the stimulus. The local EOD was measured with a distance of 1 \,cm between the electrodes. All measured signals were amplified, filtered, and stored for offline analysis.}
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\end{figure*}
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\subsection{White noise stimulation}\label{rammethods}
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The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The stimulus intensity is given as the contrast, i.e. the standard deviation of the white noise stimulus in relation to the fish's EOD amplitude. 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. To create random amplitude modulations (RAM) for P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation profile (MXS-01M; npi electronics).
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\subsection{Data analysis} Data analysis was performed with Python 3 using the packages matplotlib\cite{Hunter2007}, numpy\cite{Walt2011}, scipy\cite{scipy2020}, sklearn\cite{scikitlearn2011}, pandas\cite{Mckinney2010}, nixio\cite{Stoewer2014}, and thunderfish (\url{https://github.com/bendalab/thunderfish}).
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\paragraph{Baseline analysis}\label{baselinemethods}
|
|
The baseline firing rate \fbase{} was calculated as the number of spikes divided by the duration of the baseline recording (on average 18\,s). The coefficient of variation (CV) was calculated as the standard deviation of the interspike intervals (ISI) divided by the average ISI: $\rm{CV} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle} / \langle ISI \rangle$. If the baseline was recorded several times in a recording, the average \fbase{} and CV were calculated.
|
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|
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\paragraph{White noise analysis} \label{response_modulation}
|
|
In the stimulus driven case, the neuronal activity of the recorded cell is modulated around the average firing rate that is similar to \fbase{} and in that way encodes the time-course of the stimulus.
|
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The time-dependent response of the neuron was estimated from the spiking activity $x_k(t) = \sum_i\delta(t-t_{k,i})$ recorded for each stimulus presentation, $k$, by kernel convolution with a Gaussian kernel
|
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\begin{equation}
|
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K(t) = \scriptstyle \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{t^2}{2\sigma^2}}
|
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\end{equation}
|
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with $\sigma$ the standard deviation of the Gaussian which was set to 2.5\,ms if not stated otherwise. For each trial $k$ the $x_k(t)$ is convolved with the kernel $K(t)$
|
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\begin{equation}
|
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r_k(t) = x_k(t) * K(t) = \int_{-\infty}^{+\infty} x_k(t') K(t-t') \, \mathrm{d}t' \;,
|
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\end{equation}
|
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where $*$ denotes the convolution. $r(t)$ is then calculated as the across-trial average
|
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\begin{equation}
|
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r(t) = \left\langle r_k(t) \right\rangle _k.
|
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\end{equation}
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To quantify how strongly the neuron is driven by the stimulus we quantified the response modulation as the standard deviation $\sigma_{M} = \sqrt{\langle (r(t)-\langle r(t) \rangle)^2\rangle}$.
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\paragraph{Spectral analysis}\label{susceptibility_methods}
|
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The neuron is driven by the stimulus and thus the neuronal response $r(t)$ depends on the stimulus $s(t)$. To investigate the relation between stimulus and response we calculated the first- and second-order susceptibility of the neuron to the stimulus in the frequency domain. The Fourier transforms of $s(t)$ and $r(t)$ are denoted as $\tilde s(\omega)$ and $\tilde r(\omega)$ and were calculated according to $\tilde x(\omega) = \int_{0}^{T} \, x(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. Stimuli had a duration of 10\,s and spectra of stimulus and response were calculated in separate segments of 0.5\,s with no overlap resulting in a spectral resolution of 2\,Hz. $r(t)$ was estimated by kernel convolution with a box kernel that had a width matching the sampling interval to preserve temporal accuracy as far as possible.
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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 $\langle ... \rangle$ denotes averaging over the segments. The cross-spectrum $S_{rs}(\omega)$ was calculated according to
|
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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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From $S_{rs}(\omega)$ and $ S_{ss}(\omega)$ we calculated the linear susceptibility (transfer function) as
|
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\begin{equation}
|
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\label{linearencoding_methods}
|
|
\begin{split}
|
|
\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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The second-order cross-spectrum that depends on the two frequencies $\omega_1$ and $\omega_2$ was calculated according to
|
|
\begin{equation}
|
|
\label{eq:crosshigh}
|
|
S_{rss} (\omega_{1},\omega_{2}) = \frac{\langle \tilde r (\omega_{1}+\omega_{2}) \tilde s^* (\omega_{1})\tilde s^* (\omega_{2}) \rangle}{T}
|
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\end{equation}
|
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The second-order susceptibility was calculated by dividing the higher-order cross-spectrum by the spectral power at the respective frequencies.
|
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\begin{equation}
|
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\label{eq:susceptibility}
|
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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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% Applying the Fourier transform this can be rewritten resulting in:
|
|
% \begin{equation}
|
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% \label{susceptibility}
|
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% \begin{split}
|
|
% \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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% \notejg{Wofuer genau brauchen wir equation 9?}
|
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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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\paragraph{Nonlinearity index}\label{projected_method}
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\notejg{use of $f_{Base}$ or $f_{base}$ or $f_0$ should be consistent throughout the manuscript.}
|
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We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = f_{Base}$. To characterize this we calculated the nonlinearity index (NLI) as
|
|
\begin{equation}
|
|
\label{eq:nli_equation}
|
|
NLI(f_{Base}) = \frac{\max_{f_{Base}-5\,\rm{Hz} \leq f \leq f_{Base}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
|
|
\end{equation}
|
|
\notejg{sollte es $D(\omega)$ sein?}
|
|
For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $f_{Base}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $f_{Base} \pm 5$\,Hz (\subfigrefb{cells_suscept}{G}, gray area) and dividing it by the median of $D(f)$.
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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{} values was used for the population statistics in \figref{data_overview_mod}.
|
|
\notejg{should go to the legend: calculated based on the first frozen noise repeat.}
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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 properties of P-units \citealp{Chacron2001,Sinz2020}. The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD modeled as a cosine wave
|
|
\begin{equation}
|
|
\label{eq:eod}
|
|
x(t) = x_{EOD}(t) = \cos(2\pi f_{EOD} t)
|
|
\end{equation}
|
|
with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
|
|
|
|
In the model, the input $x(t)$ was then first thresholded to model the synapse between the primary receptor cells and the afferent.
|
|
\begin{equation}
|
|
\label{eq:threshold2}
|
|
\lfloor x(t) \rfloor_0 = \left\{ \begin{array}{rcl} x(t) & ; & x(t) \ge 0 \\ 0 & ; & x(t) < 0 \end{array} \right.
|
|
\end{equation}
|
|
$\lfloor x(t) \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (box left to \subfigrefb{flowchart}\,\panel[i]{A}).
|
|
|
|
The resulting receptor signal was then low-pass filtered to approximate passive signal conduction in the afferent's dendrite (box left to \subfigrefb{flowchart}\,\panel[ii]{A}).
|
|
\begin{equation}
|
|
\label{eq:dendrite}
|
|
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}
|
|
\end{equation}
|
|
with $\tau_{d}$ as the dendritic time constant. Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. Because the input was dimensionless, the dendritic voltage was dimensionless, too. The combination of threshold and low-pass filtering extracts the amplitude modulation of the input $x(t)$.
|
|
|
|
The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
|
|
\begin{equation}
|
|
\label{eq:LIF}
|
|
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D}\xi(t)
|
|
\end{equation}
|
|
where $\tau_{m}$ is the membrane time-constant, $\mu$ is a fixed bias current, $\alpha$ is a scaling factor for $V_{d}$, $A$ is an inhibiting adaptation current, and $\sqrt{2D}\xi(t)$ is a white noise with strength $D$. All state variables except $\tau_m$ are dimensionless.
|
|
|
|
The adaptation current $A$ followed
|
|
\begin{equation}
|
|
\label{eq:adaptation}
|
|
\tau_{A} \frac{d A}{d t} = - A
|
|
\end{equation}
|
|
with adaptation time constant $\tau_A$.
|
|
|
|
Whenever the membrane voltage $V_m(t)$ crossed the spiking 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}).
|
|
\begin{equation}
|
|
\label{spikethresh}
|
|
V_m(t) \ge \theta \; : \left\{ \begin{array}{rcl} V_m & \mapsto & 0 \\ A & \mapsto & A + \Delta A/\tau_A \end{array} \right.
|
|
\end{equation}
|
|
|
|
% 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
|
|
% \begin{equation}
|
|
% \label{eifnl}
|
|
% f(V_m)= \Delta_V \text{e}^{\frac{V_m-1}{\Delta_V}}
|
|
% \end{equation}
|
|
% \citealp{Fourcaud-Trocme2003}, where $\Delta_V$ was varied from 0.001 to 0.1.
|
|
%, \figrefb{eif}
|
|
|
|
\begin{figure*}[hb!]
|
|
\includegraphics[width=\columnwidth]{flowchart}
|
|
\caption{\label{flowchart}
|
|
Flowchart of a LIF P-unit model with EOD carrier. The main steps of the model are depicted in the left model column. The three columns show the relevant signals in three different settings. (i) the baseline situation, no external stimulus, only the animal's self-generated EOD (i.e. the carrier) is present (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\% contrast) stimulus, (iii) Noise split condition in which 90\% of the internal noise is used as a driving stimulus scaled with the correction factor $\rho$ (see text). Note: that the firing rate and the CV of the ISI distribution is the same in this and the baseline condition. (Model cell identifier 2012-07-03-ak, see table~\ref{modelparams} for model parameters. \textbf{A} Thresholding: a simple linear threshold was applied to the EOD carrier (eq,\,\ref{eq:eod}) The red line on top depicts the amplitude modulation (AM). \textbf{B} Dendritic low-pass filtering attenuates the carrier. \textbf{C} An Gaussian noise is added to the signal in B. Note the reduced internal noise amplitude in the split (iii) condition. \textbf{D} Spiking output of the LIF model in response to the addition of B and C. \textbf{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (D$_i$) there are several peaks at, from left to right, the baseline firing rate $f_{base}$, $f_{EOD} - f_{base}$ $f_{EOD}$, and $f_{EOD} + f_{base}$, in the stimulus driven regime, there is only a peak at $f_{eod}$, while under the noise split condition (D$_iii$) again all peaks are present.}
|
|
\end{figure*}
|
|
|
|
\subsection{Numerical implementation}
|
|
The model's ODEs were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. The intrinsic noise $\xi(t)$ (\eqnref{eq:LIF}, \subfigrefb{flowchart}\,\panel[iii]{A}) was added by drawing a random number from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one in each time step $i$. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$:
|
|
\begin{equation}
|
|
\label{eq: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}
|
|
|
|
\subsection{Model parameters}\label{paramtext}
|
|
The eight 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 the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step-like increases and decreases in EOD amplitude (onset-state and steady-state responses, effective adaptation time constant). For each simulation, the start parameters $A$, $V_{d}$ and $V_{m}$ were drawn from a random starting value distribution, estimated from a 100\,s baseline simulation after an initial 100\,s of simulation that was discarded as a transient.
|
|
|
|
\subsection{Stimuli for the model}
|
|
The model neurons were driven with similar stimuli as the real neurons in the original experiments. To mimic the interaction with one or two foreign animals the receiving fish's EOD (eq.\,\ref{eq:eod}) was normalized to an amplitude of one and the respective EODs of a second or third fish were added.
|
|
|
|
The random amplitude modulation (RAM) input to the model was created by drawing random amplitude and phases from Gaussian distributions for each frequency component in the range $0-300$ Hz. An inverse Fourier transform was applied to get the final amplitude RAM time-course. The input to the model was then
|
|
\begin{equation}
|
|
\label{eq:ram_equation}
|
|
x(t) = (1+ RAM(t)) \cdot \cos(2\pi f_{EOD} t)
|
|
\end{equation}
|
|
|
|
From each simulation run, the first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
|
|
% \subsection{Second-order susceptibility analysis of the model}
|
|
% %\subsubsection{Model second-order nonlinearity}
|
|
|
|
% The second-order susceptibility in the model was calculated with \eqnref{eq: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.
|
|
|
|
\subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
|
|
According to the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the total noise of a LIF ($\xi$) model can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal ($\xi_{signal} = c_{signal} \cdot \xi$) and used to calculate the cross-spectra \eqnref{eq:crosshigh} and (ii) the remaining noise ($\xi_{noise} = (1-c_{signal})\cdot\xi$) that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus.
|
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|
|
\begin{equation}
|
|
\label{eq:ram_split}
|
|
x(t) = (1+ \sqrt{\rho \, 2D \,c_{signal}} \cdot \xi(t)) \cdot \cos(2\pi f_{EOD} t)
|
|
\end{equation}
|
|
with $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass.
|
|
\begin{equation}
|
|
\label{eq:Noise_split_intrinsic_dendrite}
|
|
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}
|
|
\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{eq: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}). For the application of the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
|
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\notejg{Etwas kurz, die Tabelle. Oder benutzen wir hier tatsaechlich nur die zwei Modelle?}
|
|
\begin{table*}[hp!]
|
|
\caption{\label{modelparams} Model parameters of LIF models, fitted to 2 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
|
|
2012-07-03-ak& $10.6$& $1.38$& $-1.32$& $0.001$& $96.05$& $0.01$& $1.18$& $0.12$ \\
|
|
2018-05-08-ae& $139.6$& $1.49$& $-21.09$& $0.214$& $123.69$& $0.16$& $3.93$& $1.31$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{table*}% 2013-01-08-aa % 2012-07-03-ak
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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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\setcounter{secnumdepth}{2}
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\section{Appendix}
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\begin{figure*}[hp]%hp!
|
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\includegraphics{cells_suscept_high_CV}
|
|
\caption{\label{cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} 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{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{eq: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{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
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}
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\end{figure*}
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