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\journal{iScience}

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\begin{document}

%\maketitle
\begin{frontmatter}

\title{Second-order susceptibility in electrosensory primary afferents in a three-fish setting}


\author[1]{Alexandra Barayeu} %\corref{fnd1}}
% \ead{alexandra.rudnaya@uni-tuebingen.de}
\author[4,5]{Maria Schlungbaum}
\author[4,5]{Benjamin Lindner}
\author[1,2,3]{Jan Benda}
\author[1]{Jan Grewe\corref{cor1}}
\ead{jan.grewe@uni-tuebingen.de}
% \ead[url]{home page}

\cortext[cor1]{Corresponding author}
\affiliation[1]{organization={Neuroethology, Institute for Neurobiology, Eberhard Karls University},
  city={T\"ubingen}, postcode={72076}, country={Germany}}
\affiliation[4]{organization={Bernstein Center for Computational Neuroscience T\"ubingen},
  city={T\"ubingen}, postcode={72076},  country={Germany}}
\affiliation[5]{organization={Werner Reichardt Centre for Integrative Neuroscience},
  city={T\"ubingen}, postcode={72076},  country={Germany}}
\affiliation[2]{organization={Bernstein Center for Computational Neuroscience Berlin}, city={Berlin},  country={Germany}}
\affiliation[3]{organization={Physics Department of Humboldt University Berlin},city={Berlin}, country={Germany}}

%Nonlinearities contribute to the encoding of the full behaviorally relevant signal range in primary electrosensory afferents.

%Nonlinear effects identified as mechanisms that contribute 
%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

%Nonlinearities in primary electrosensory afferents contribute to the representation of a wide range of beat frequencies and amplitudes

%Nonlinearities contribute to the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents

%The role of nonlinearities in the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents

%Nonlinearities facilitate the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents


\begin{abstract}
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. 
\end{abstract}

\end{frontmatter}

\section{Introduction}

% of spiking responses influences the encoding in P-units Nonlinearities can arise not only between cells but also inside single neurons, 
%, as has been shown in modeling studies % In the previous chapter, it was elaborated that the encoding of high beat frequencies requires a nonlinearity at the synapse between the electroreceptors and the afferent P-unit. 

In this work, nonlinearities are studied in the gymnotiform weakly electric fish of the species \lepto. These fish are equipped with an electric organ \citep{Salazar2013}, that is constantly active and produces a quasi sinusoidal electric organ discharge (EOD), with a long-term stable and fish-specific frequency (\feod, \citealp{Knudsen1975, Henninger2020}). \lepto{} is sexually dimorphic with females having lower EOD frequencies (500--750\,Hz) and males having higher EOD frequencies (750--1000\,Hz). These fish use their EOD for electrolocation \citep{Fotowat2013, Nelson1999} and communication \citep{Fotowat2013, Walz2014, Henninger2018}. Cutaneous tuberous organs, that are distributed all over the body of these fish
[\citealp{Carr1982}], sense the actively generated electric field and its modulations.
Within a single tuberous electroreceptor organ about 30 primary electroreceptors form ribbon synapses onto the dendrites of the same afferent fiber
\citep{Szabo1965, Wachtel1966} that projects via the lateral line nerve to the hindbrain. There, the afferent fiber synapses onto pyramidal neurons in the
electrosensory lateral line lobe (ELL, \citealp{Maler2009a}). The EOD is encoded by the p-type electroreceptor afferents, the P-units. These fire phase-locked to the EOD of the fish with a spike at a similar phase in the EOD cycle, with the phase depending on the receptor location \citep{Hladnik2023}. P-units fire not once in every EOD cycle but in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at multiples of the EOD period. P-units have heterogeneous baseline firing properties with the mean baseline firing rate \fbase{} varying between 50 and 450\,Hz \citep{Grewe2017, Hladnik2023} and the coefficient of variation (CV) of their ISIs varying between 0.2 and 1.7 \citep{Grewe2017, Hladnik2023}.


In previous works, P-units have been mainly considered to be linear encoders \citep{Xu1996, Benda2005, Gussin2007, Grewe2017}. Still, P-units exhibit nonlinear effects, especially for strong beat amplitudes \citep{Nelson1997}. Although some P-units can be described by a linear decoder, other P-units require a nonlinear decoder \citep{Chacron2004}. In the following, the focus will be on nonlinear effects in the time-resolved firing rate. 


In the following, a three-fish setting and its encoding in P-units in \lepto{} will be introduced. When the receiver fish with EOD frequency \feod{} is alone, a peak at the mean baseline firing rate \fbase{} is present in the power spectrum of the firing rate of a recorded P-unit (\subfigrefb{motivation}{A}, bottom, see section \ref{baselinemethods}). P-units also represent \feod{} with a peak in the power spectrum of their firing rate, but this peak is beyond the range of frequencies addressed in this figure. If two fish with the EOD frequencies \feod{}  and $f_{1}$ meet, their EODs interfere and result in a new signal with a characteristic periodic amplitude modulation, that will be called beat. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \citep{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing the context for electrocommunication in weakly electric fish \citep{Engler2001, Hupe2008, Henninger2018, Benda2020}.
The beat amplitude is defined by the smaller EOD of the encountered fish, is expressed in relation to the receiver EOD and is termed contrast (contrast of 20\,$\%$ in \subfigrefb{motivation}{B}). The beat frequency is defined as the difference between the two EOD frequencies ($\Delta f_{1} = f_{1}-\feod{}$). \lepto{} is sexually dimorphic with females having lower EOD frequencies (500--750\,Hz) and males having higher EOD frequencies (750--1000\,Hz). If two fish with the same sex and with similar EOD frequencies meet, this results in a low beat frequency $\Delta f_{1}$ and a slowly oscillating beat (\subfigrefb{motivation}{B}, top). A P-unit represents this beat frequency in its spike trains and firing rate (\subfigrefb{motivation}{B}, middle). When two fish from opposite sex with different frequencies meet a high difference frequency $\Delta f_{2}$ and a fast beating signal occurs (\subfigrefb{motivation}{C}, top). This beat is represented in the spike trains and firing rate of the P-unit (\subfigrefb{motivation}{C}, middle). In this example, $\Delta f_{2}$ is similar to \fbase{} of the cell, with a strong beat/baseline peak in the power spectrum of the firing rate when both fish are present (green circle in  \subfigrefb{motivation}{C}, bottom). When three fish encounter, as e.g. during an electrosensory cocktail party observed the field (\citealp{Henninger2018}), all their waveforms interfere with both beats with frequencies $\Delta f_{1}$ and $\Delta f_{2}$ being present in the superimposed signal (\subfigrefb{motivation}{D}, top). The power spectrum of the firing rate contains both beat frequencies $\Delta f_{1}$ and $\Delta f_{2}$, but also nonlinear peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} of the two beat frequencies (\subfigrefb{motivation}{D}, bottom).



\begin{figure*}[h!]
  \includegraphics[width=\columnwidth]{motivation}
  \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 mean 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.
  }
\end{figure*}

As described by \citet{Voronenko2017} such nonlinearities, at the sum and difference frequencies, are expected only for certain frequencies, that can be predicted based on the mean baseline firing rate \fbase{} of the cell. In their work, the second-order susceptibility was analytically retrieved based on LIF models, where the input were two pure sine waves. A triangular nonlinear shape was predicted, with nonlinearities appearing at the sum of the two input frequencies \fsum{} in the response, if one of the beat frequencies \fone{}, \ftwo{} or the sum of the beat frequencies \fsum{} was equal to \fbase{} (see diagonal, vertical and horizontal lines in the upper right quadrant in\citealp{Voronenko2017}). In addition, a triangular nonlinear shape was predicted, with nonlinearities appearing at the difference of the two input frequencies \fdiff{} in the response, if one of the input frequencies \fone{}, \ftwo{} or the difference of the input frequencies \fdiff{} was equal to \fbase{} (see diagonal, vertical and horizontal lines in the lower right quadrant in \citealp{Voronenko2017}). Whereas these predictions of the nonlinear response theory presented in \citet{Voronenko2017} apply to P-units, where the main driving force are beats and not the whole signal, will be addressed in the following.

%\bsumb{}, \bdiffb{},and to which extend nonlinearities appear at the beat frequencies \bone{} and \btwo{}, since it belongs to the fish in the experimental setup

The EOD with frequency \feod{} is fixed, thus the varied parameters in a three-fish setting are the EOD frequencies \fone{} and \ftwo{}, resulting in a two-dimensional stimulus space spanned by the two corresponding beat frequencies \bone{} and \btwo{}. It is challenging to sample this whole stimulus space in an electrophysiological experiment. A different solution has to be implemented, where nonlinear frequency candidates can be quickly identified and then only these be probed in an electrophysiological recording. For this white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been proposed \citep{Egerland2020}. During this procedure, the cell has to be presented with several white noise stimulus realizations, each time with randomly drawn amplitudes and phases. Setting the stimulus in relation to the firing rate response in the frequency domain, as in \eqnref{susceptibility}, allows to quantify the second-order susceptibility of the system and highlight the frequency combinations prone to nonlinearity. This method was utilized with bandpass limited white noise being the direct input to a LIF model in previous literature \citep{Egerland2020}. In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD (see methods section \ref{rammethods} and \eqnref{ram_equation}). Whether it is possible to access the second-order susceptibility of P-units with RAM stimuli will be addressed in the following chapter. %the presented method is still

%Tuning curves can also be retrieved by probing the system with a band-pass filtered white noise stimulus, that simultaneously includes all frequencies of interest, each with randomly drawn amplitudes and phases (\figrefb{whitenoise_didactic}, \citealp{Chacron2005, Grewe2017}). Noise stimuli are commonly used protocols in electrophysiological recordings and once recorded they can always be reused to retrieve the tuning curve of the neuron \citep{Grewe2017, Neiman2011fish}.


%where the frequencies have randomly drawn phases and random amplitudes.
% a neuron after a single stimulus presentation








%Nonlinearity can be influenced not only by the stimulus properties, as the input frequencies, but also by the cell properties during baseline, as the coefficient of variation (CV) of the interspike intervals (ISI) or the mean baseline firing rate \fbase{} \citep{Voronenko2017, Savard2011}. P-units are very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. 





%\subsection{Outlook}when \fsum{} and \fdiff{} are equal to \fbase
In this work, the second-order susceptibility in the spiking responses of P-units will be accessed with white noise stimulation. The influence of the baseline firing properties, such as the CV, on nonlinear interactions will be investigated. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by \citet{Voronenko2017}. White noise stimulation will be confirmed as a method to access the second-order susceptibility in P-units. 

%will be addressed. How many P-units exhibit such nonlinearities
%The nonlinearity in P-units in \lepto{} will be compared to other cell types as ampullary cells and with other species as \eigen{}.
% and show that the number of spikes in a burst package, and not the first burst spike timing, are critical for this increase in nonlinearity
%influence the occurrence of nonlinearity by comparing the nonlinearity in P-units with other cell populations as the ampullary cells (lower CVs) and consider not only \lepto{} but also the P-units of \eigen{} (lower CV)
% if the nonlinear frequency combinations occur where they are predicted based on simple LIF models without a carrier.




\section{Results}


\subsection{Low-CV P-units exhibit nonlinear interactions} %frequency combinations withappearing when the input frequencies are related to \fbase{} are

P-units are heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023} and differ in their noisiness, which is represented by the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a regular firing pattern and are less noisy, whereas high-CV P-units have a less regular firing pattern.
Second-order susceptibility is expected to be especially pronounced for low-CV cells \citep{Voronenko2017}. In the following first low-CV P-units will be addressed in \subfigrefb{cells_suscept}{A}.

P-units probabilistically phase-lock to the EOD of the fish, firing at the same phase but not in every EOD cycle, resulting in a multimodal ISI histogram with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}\,\panel[i]{A}, left). The strongest peak in the baseline power spectrum of the firing rate of a P-unit is the \feod{} peak, and the second strongest peak is the mean baseline firing rate \fbase{} peak (\subfigrefb{cells_suscept}\,\panel[i]{A}, right). The power spectrum of P-units is symmetric around half \feod, with baseline peaks appearing at $\feod \pm \fbase{}$. 

Noise stimuli, as random amplitude modulations (RAM) of the EOD, are common stimuli during P-unit recordings. In the following, the amplitude of the noise stimulus will be quantified as the standard deviation and will be expressed as a contrast (unit \%) in relation to the receiver EOD. The spikes of P-units slightly align with the RAM stimulus with a low contrast (light purple) and are stronger driven in response to a higher RAM contrast (dark purple,  \subfigrefb{cells_suscept}\,\panel[ii]{A}). The linear encoding (see \eqnref{linearencoding_methods}) is comparable between the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}\,\panel[iii]{A}).%visualized by the gain of the transfer function,\suscept{} 


To quantify the second-order susceptibility in a three-fish setting the noise stimulus was set in relation to the corresponding P-unit response in the Fourier domain, resulting in a matrix where the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{susceptibility}, \subfigrefb{cells_suscept}\,\panel[iv]{A}--\panel[v]{A}). Note that the RAM stimulus can be decomposed in frequencies $f$, that 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}\,\panel[iv]{A}) is comparable to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). Based on the theory \citep{Voronenko2017} nonlinearities should arise when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (upper right quadrant in \figrefb{plt_RAM_didactic2}), which would imply a triangular nonlinearity shape highlighted by the pink triangle corners in \subfigrefb{cells_suscept}\,\panel[iv]{A}--\panel[v]{A}. A slight diagonal nonlinearity band appears for the low RAM contrast when \fsumb{} is satisfied (yellow diagonal between pink edges, \subfigrefb{cells_suscept}\,\panel[iv]{A}). Since the matrix contains only anti-diagonal elements, the structural changes were quantified by the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}\,\panel[vi]{A}). For a low RAM contrast the  \fbase{} peak in the projected diagonal is slightly enhanced (\subfigrefb{cells_suscept}\,\panel[vi]{A}, gray dot on light purple line). For the higher RAM contrast, the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}\,\panel[v]{A}), with no pronounced \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}\,\panel[vi]{A}, dark purple line). In addition, there is an offset between the projected diagonals, demonstrating that the second-order susceptibility is reduced for RAM stimuli with a higher contrast (\subfigrefb{cells_suscept}\,\panel[vi]{A}).

%There a triangle is plotted not only if the frequency combinations are equal to the \fbase{} fundamental but also to the \fbase{} harmonics (two triangles further away from the origin). 
%In this figure a part of \fsumehalf{} is marked with the orange diagonal line. 

\begin{figure*}[h]%hp!
\includegraphics{cells_suscept}%cells_suscept  
  \caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. \figitem{A} Regular firing low-CV P-unit. \figitem[i]{A} Left: Interspike intervals (ISI) distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast. 
 Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. \figitem{B} Noisy high-CV P-Unit. Panels as in \panel{A}.
  }
\end{figure*}

\subsection{High-CV P-units do not exhibit any nonlinear interactions}%frequency combinations
Based on the theory strong nonlinearities in spiking responses are not predicted for cells with irregular firing properties and high CVs \citep{Voronenko2017}. CVs in P-units can range up to 1.5 \citep{Grewe2017, Hladnik2023} and as a next step the second-order susceptibility of high-CV P-units will be presented. As low-CV P-units, high-CV P-units fire at multiples of the EOD period (\subfigrefb{cells_suscept_high_CV}\,\panel[i]{A}, left). In contrast to low-CV P-units high-CV P-units are noisier in their firing pattern and have a less pronounced mean baseline firing rate peak \fbase{} in the power spectrum of their firing rate during baseline (\subfigrefb{cells_suscept_high_CV}\,\panel[i]{A}, right). High-CV P-units do not exhibit any nonlinear structures related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{cells_suscept_high_CV}\,\panel[iv]{A}--\panel[v]{A}), nor in the projected diagonals (\subfigrefb{cells_suscept_high_CV}\,\panel[vi]{A}). As in low-CV P-units (\subfigrefb{cells_suscept}\,\panel[v]{A}), the mean second-order susceptibility decreases with higher RAM contrasts in high-CV P-units (\subfigrefb{cells_suscept_high_CV}\,\panel[v]{A}).

\begin{figure*}[ht]%hp!
\includegraphics{ampullary}  
  \caption{\label{ampullary} Response of an experimentally measured ampullary cell. Light green -- low noise stimulus contrast. Dark green -- high noise stimulus contrast. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) plus a band-pass limited white noise (red, see methods section \ref{rammethods}). Middle: Spike trains in response to a low noise stimulus contrast. Bottom: Spike trains in response to a high noise stimulus contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility (\eqnref{susceptibility}) for the low noise stimulus contrast. 
 Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Second-order susceptibility matrix for the higher noise stimulus contrast. Colored lines as in \panel[iv]{A}. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dot: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. 
  }
\end{figure*}

\subsection{Ampullary cells exhibit strong nonlinear interactions}%with lower CVs as P-units
\lepto{} posses another primary sensory afferent population, the ampullary cells, with overall low \fbase{} (80--200\,Hz) and low CV values (0.08--0.22, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the EOD, with no maxima at multiples of the EOD period and smoothly unimodal distributed ISIs (\subfigrefb{ampullary}\,\panel[i]{A}, left). Ampullary cells do not have a peak at \feod{} in the baseline power spectrum of the firing rate with no symmetry around it (\subfigrefb{ampullary}\,\panel[i]{A}, right). Instead, the \fbase{} peak is very pronounced with clear harmonics. When being exposed to a noise stimulus with a low contrast, 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}\,\panel[iv]{A}). With higher noise stimuli contrasts these bands disappear (\subfigrefb{ampullary}\,\panel[v]{A}) and the projected diagonal is lowered (\subfigrefb{ampullary}\,\panel[vi]{A}, dark green). 

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 \citep{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citep{Voronenko2017}. 
%and here this could be confirmed experimentally.


\subsection{Full nonlinear structure visible only in P-unit models}
In the following nonlinear interactions were systematically compared between an electrophysiologically recorded low-CV P-unit and the according P-unit LIF models with a RAM contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{A}). For a homogeneous population with size $\n{}=11$ one could observe a diagonal band in the absolute value of the second-order susceptibility at \fsumb{} of the recorded P-unit (yellow diagonal in pink edges, \subfigrefb{model_and_data}\,\panel[ii]{A}) and in the according model (\subfigrefb{model_and_data}\,\panel[iii]{A}). A nonlinear band appeared at \fsumehalf{}, but only in the recorded P-unit (orange line, \subfigrefb{model_and_data}\,\panel[ii]{A}). The signal-to-noise ratio and estimation of the nonlinearity structures can be improved if the number of RAM stimulus realizations is increased. Models have the advantage that they allow for data amounts that cannot be acquired experimentally. Still, even if a RAM stimulus is generated 1 million times, no changes are observable in the nonlinearity structures in the model second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iv]{A}).

%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). 
%The signal component (purple) compensates for this total noise reduction, is not a weak signal anymore and 
Based on the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} the intrinsic noise of a LIF model can be split up into several independent noise processes with the same correlation function. 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 \citep{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}). 

If a high-CV P-unit is investigated (not shown), there would be no nonlinear structures, neither in the electrophysiologically recorded data nor in the according model, corresponding to the theoretical predictions \citep{Voronenko2017}.


% (see methods, \eqnref{Noise_split_intrinsic}, \citealp{Novikov1965, Furutsu1963}) or its harmonics
\begin{figure*}[hb!]
  \includegraphics[width=\columnwidth]{model_and_data}% is equal to \fbase  is equal to half the \feod
  \caption{\label{model_and_data} The influence of the RAM stimulus realization number $\n$, the RAM contrast $c$, and the split of the total intrinsic noise in a signal and noise component on the nonlinearity structures of the second-order susceptibility of an electrophysiologically recorded low-CV P-unit and its LIF model (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). Pink lines in the matrices mark the edges of the structure when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{}. The orange line in the matrices marks a part of the line at \fsumehalf. 
\figitem[i]{A},\,\panel[i]{\textbf{B}},\,\panel[i]{\textbf{C}} Red -- RAM stimulus. The total intrinsic noise can be split into a noise component (gray) and a signal component (purple), maintaining the same CV and \fbase{} as before the split (see methods section \ref{intrinsicsplit_methods}). The calculation is performed on the sum of the signal component (purple) and the RAM (red) in \eqnref{susceptibility}. 
\figitem[ii]{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit, with $\n{}=11$ RAM stimulus realizations. 
\figitem[iii]{A},\,\panel[iii]{\textbf{B}},\,\panel[iii]{\textbf{C}} Absolute value of the model second-order susceptibility with $\n{}=11$ RAM stimulus realizations. 
\figitem[iv]{A},\,\panel[iv]{\textbf{B}},\,\panel[iv]{\textbf{C}} Absolute value of the model second-order susceptibility with 1 million RAM stimulus realizations. 
\figitem{A} RAM contrast of 1\,$\%$. The band at \fsumb{} is visible in the matrices. 
\figitem{B} No RAM stimulus, but a total noise split into a signal component (purple) and a noise component (gray). The band at \fsumb{} is visible in \panel[iii]{B} and \panel[iv]{B}. Besides that horizontal and vertical nonlinearities appear at \foneb{} and \ftwob{} in \panel[iv]{B}.
\figitem{C} A RAM stimulus (red) and a total noise split into a signal component (purple) and a noise component (gray). Only the band at \fsumb{} is visible in the matrices.
}
\end{figure*}



\begin{figure*}[h]
  \includegraphics[width=\columnwidth]{model_full}
  \caption{\label{model_full} Second-order susceptibility of an electrophysiologically recorded P-unit and of the corresponding model (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{susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Mean baseline firing rate $\fbase{}=120$\,Hz. \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} Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods section \ref{intrinsicsplit_methods}). The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}.}
\end{figure*}
\subsection{Similar nonlinear effects with RAM and sine-wave stimulation}
In the previous paragraphs, the nonlinearity at \fsum{} in the P-unit response was identified for the RAM frequencies \fone{} and \ftwo{}. This RAM-based second-order susceptibility can be used to approximate the nonlinearity in the three-fish setting, where two beats with frequencies \bone{} and \btwo{} are the driving forces for the P-unit response. In the previously shown three-fish setting a nonlinear peak occurred at the sum of the two beat frequencies (orange circle, \subfigrefb{motivation}{D}). In that example $\Delta f_{1}$ was similar to \fbase{}, corresponding to the vertical line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iv]{B}. In the three-fish example, there was a second less prominent nonlinearity at the difference of the two beat frequencies (purple circle, \subfigrefb{motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iv]{B}, where only the nonlinearity at \fsum{} in the response is addressed. 



Instead, the full second-order susceptibility matrix in \figrefb{model_full}, which depicts nonlinearities in the P-unit response at \fsum{} in the upper right and lower left quadrants and nonlinearities at \fdiff{} in the lower right and upper left quadrants (\eqnref{susceptibility}, \citealp{Voronenko2017}), has to be considered. Once calculating this full second-order susceptibility matrix based on the experimentally recorded data (\subfigrefb{model_full}{A}) and the corresponding model (\subfigrefb{model_full}{B}), one can observe that the diagonal structures are present in the upper right quadrant and for the lower right quadrants. The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper right quadrant for the nonlinearity at \fsum{} and are prolonged to the lower right quadrant with lower nonlinearity values at \fdiff{} in the P-unit response. 

%, that quantifies the nonlinearity at \fdiff{} in the response , that quantifies the nonlinearity at \fsum{} in the response,


The small \fdiff{} peak in the power spectrum of the firing rate appearing during pure sine-wave stimulation (\subfigrefb{motivation}{D}) can be explained by the vertical line in the lower right quadrant (\subfigrefb{model_full}{B}, \citealp{Schlungbaum2023}). The here presented full second-order susceptibilities matrix was retrieved based on data and models with EOD carrier (\figrefb{model_full}) and is in accordance with the second-order susceptibilities calculated based on models without a carrier (\citealp{Voronenko2017, Schlungbaum2023}).
% When pure sine wave stimulation is happening it is expected that both nonlinear effects observed at \fsum{} and \fdiff{} (upper right and lower right quadrant, \subfigrefb{model_full}{B}) for a stimulation with positive frequencies \citep{Schlungbaum2023}.

\begin{figure*}[ht!]
  \includegraphics[width=\columnwidth]{data_overview_mod}
  \caption{\label{data_overview_mod} Nonlinearity for a population of ampullary cells (\panel{A, C}) and P-units (\panel{B, D}). \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{nli_equation}). There are maximally two noise contrasts per cell in a population. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} Response modulation, \eqnref{response_modulation}, is an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. 
  }
\end{figure*}
%\label{response_modulation}

%Second-order susceptibility for all frequencies
\subsection{Low CVs are associated with strong nonlinearity on a population level}%when considering
So far second-order susceptibility was illustrated only with single-cell examples (\figrefb{cells_suscept}, \figrefb{ampullary}). For a P-unit comparison on a population level, the second-order susceptibility of P-units was expressed in a nonlinearity index \nli{}, see \eqnref{nli_equation}, that characterized the peakedness of the \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}\,\panel[vi]{A}). \nli{} has high values when the \fbase{} peak in the projected diagonal is especially pronounced, as in the low-CV ampullary cell (\subfigrefb{ampullary}\,\panel[vi]{A}, light green). The two noise stimulus contrasts of this ampullary cell are highlighted in the population statics of ampullary cells with dark circles (\subfigrefb{data_overview_mod}{A}). The higher noise stimulus contrast is associated with a less pronounced peak in the projected diagonal (\subfigrefb{ampullary}\,\panel[vi]{A}, dark green) and is represented with a lower \nli{} value (\subfigrefb{data_overview_mod}{A}, dark circle close to the origin). In an ampullary cell population, there is a negative correlation between the CV during baseline and \nli{}, meaning that the diagonals are pronounced for low-CV cells and disappear towards high-CV cells (\subfigrefb{data_overview_mod}{A}). Since the same stimulus can be strong for some cells and faint for others, the noise stimulus contrast is not directly comparable between cells. A better estimation of the subjective stimulus strength is the response modulation of the cell (see methods section \ref{response_modulation}). Ampullary cells with stronger response modulations have lower \nli{} scores (red in \subfigrefb{data_overview_mod}{A}, \subfigrefb{data_overview_mod}{C}). The so far shown population statistics comprised several RAM contrasts per cell and if instead each ampullary cell is represented with the lowest recorded contrast, then \nli{} significantly correlates with the CV during baseline ($r=-0.46$, $p<0.001$), the response modulation ($r=-0.6$, $p<0.001$) but not with \fbase{} ($r=0.2$, $p=0.16$).%, $\n{}=51$, $\n{}=51$, $\n{}=51${*}{*}{*}^*^*^*each cell can contribute several RAM contrasts in 


The P-unit population has higher baseline CVs and lower \nli{} values (\subfigrefb{data_overview_mod}{B}) that are weaker correlated than in the population of ampullary cells. The negative correlation (\subfigrefb{data_overview_mod}{B}) is increased when \nli{} is plotted against the response modulation of P-units (\subfigrefb{data_overview_mod}{D}). The two example P-units shown before (\figrefb{cells_suscept}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. High-CV P-units and strongly driven P-units have lower \nli{} values (\subfigrefb{data_overview_mod}{B, D}). 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{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$






\section{Discussion}
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 mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. 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.



%\fsum{} or \fdiff{} were equal to \fbase{} \bsum{} or \bdiff{}, 

\subsection{Methodological implications}%implying that the
\subsubsection{Full nonlinear structure present in LIF models with and without carrier}%implying that the
 %occurring nonlinearities could only be explained based on a  (\foneb{},  \ftwob{}, \fsumb{}, \fdiffb{},
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 (\figrefb{plt_RAM_didactic2}, \citealp{Voronenko2017}) and was here confirmed to be valid in models with carrier (\figrefb{model_and_data}).

\subsubsection{Noise stimulation approximates the nonlinearity in a three-fish setting}%implying that the
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 LIF models (\citealp{Egerland2020}) and of ampullary cells in paddlefish \citep{Neiman2011fish} has been previously accessed with bandpass limited white noise. 

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.

%the power spectrum of the firing rate 
%The existence of the extended nonlinearity structure (\figrefb{model_full}) was
%\bsum{} and \bdiff{} 
%, 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.
%llowing for data amounts that could not be acquired experimentally
%. It could be shown that if the nonlinearity at \fsumb{} could be found with a low stimulus repeat number in electrophysiologically recorded P-units, 


\subsection{Nonlinearity and CV}%
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 \citep{Roddey2000, Chialvo1997, Voronenko2017}. More noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citep{Schneider2011}. Reduced noise of low-CV cells has been associated with stronger nonlinearity in pyramidal cells of the ELL \citep{Chacron2006}. 

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 \citep{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 \citep{Voronenko2017}. Ampullary cells encode low-frequency changes in the environment e.g. prey movement \citep{Engelmann2010, Neiman2011fish}. The activity of a prey swarm of \textit{Daphnia} strongly resembles Gaussian white noise \citep{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. 



%
%Second-order susceptibility was strongest the lower the CV of the use LIF model \citep{Voronenko2017}.  
% 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
%\subsection{Full nonlinear structure corresponds to theoretical predictions}
%where second-order susceptibility could be explored systematically, 




\subsubsection{Neuronal delays might deteriorate nonlinear effects}
A potential restriction of the analysis in this chapter is that all stimulus repeats in a population started with the same phase. In previous works \citep{Hladnik2023} it was demonstrated in P-units that depending on the receptor position, the same signal arrives with different delays at the target neurons, thus deteriorating the stimulus encoding, with higher frequencies being affected stronger than lower frequencies. The high mean baseline frequencies \fbasesolid{} of P-units (up to 400\,Hz) might be especially affected by such neuronal delays. How neuronal delays influence nonlinear effects and intruder detection should be tested in further studies. 




%These low-frequency modulations of the amplitude modulation are
\subsection{Encoding of secondary envelopes}%($\n{}=49$)	Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study \citet{Savard2011}.
The RAM stimulus used in this chapter is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citep{Stamper2012Envelope}. Low-frequency secondary envelopes are extracted downstream of P-units in the ELL \citep{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citep{Middleton2007}. The encoding of social envelopes can also be attributed to P-units with stronger nonlinearities, lower firing rates and higher CVs \citep{Savard2011}.  In this chapter high CVs were associated with increased bursting (\subfigrefb{data_overview}\,\panel[iii]{A}). 


\subsection{More fish would decrease second-order susceptibility}%
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 \citep{Tan2005} and \lepto{} in groups of two \citep{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 (see section \ref{cocktail party}, \citealp{Henninger2018}) will be addressed in the next chapter.


\subsection{Conclusion} In this chapter, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. Nonlinear effects were identified in experimentally recorded non-bursty low-CV cells and bursty high-CV P-units. It was demonstrated that the theory of weakly nonlinear interactions \citep{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinearity was found to be decreased the more fish were present, thus keeping the signal representation in the firing rate simple. 


\section{Methods}

\subsection{Experimental subject details}
Experiments were performed on male and female weakly electric fish
of the species \lepto{} obtained from a
commercial tropical fish supplier (Aquarium Glaser GmbH, Rodgau,
Germany). The fish were kept in tanks with a water temperature
of $25\,^\circ$C and a conductivity of around
$270\,\micro\siemens\per\centi\meter$ under a 12\,h:12\,h light-dark
cycle. All experimental
protocols complied with national and European law and were approved by
the Ethics Committee of the Regierungspräsidium T\"ubingen (permit no: ZP1-16). 



No experiments were performed. Instead, cells recorded in \lepto{} and \eigen{} at the Ludwig Maximilian University (LMU) M\"unchen and at the Eberhard-Karls University T\"ubingen between 2010 and 2023 were used. The final sample consisted of 222 P-units and 45 ampullary cells from 71 weakly electric fish of the species \lepto{} and 60 P-units and 18 ampullary cells from 17 weakly
electric fish of the species \eigen.




%\subsection{Experimental model and subject details In  chapter \ref{chapter4} }



% glued to the skull
\subsection{Surgery} Before surgery, anesthesia was provided via bath application with a solution of MS222 (120\,mg/l, PharmaQ, Fordingbridge, UK) buffered with Sodium Bicarbonate (120\,mg/l). The posterior anterior lateral line nerve (pALLN) above the gills before its descent towards the anterior lateral line ganglion (ALLNG) was disclosed for subsequent P-unit recordings. During the surgery water supply was ensured by a mouthpiece, sustaining
anesthesia with a solution of MS222 (100\,mg/l) buffered with Sodium Bicarbonate (100\,mg/l).

%For the surgery the fish was fixed on a stage via a metallic rod. 

\subsection{Experimental setup} During the experiments fish were immobilized by a single
intramuscular injection of Tubocurarine (Sigma-Aldrich, Steinheim,
Germany; 25--50\,\micro\litre{} of 5\,mg/ml solution). For the
recordings fish were positioned on a stage in a tank, with a major part
of their body in the water. Analgesia was refreshed in intervals of two hours by cutaneous Lidocaine application (2\,\%; bela-pharm, Vechta, Germany) around the nerve. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P;
Science Products, Hofheim, Germany) were pulled to a resistance of 50--100\,\mega\ohm{} (model P-97; Sutter Instrument, Novato, CA) and filled with 1\,M KCl solution. Electrodes were fixed in a microdrive (Luigs-Neumann, Ratingen, Germany) and lowered into the nerve (\figrefb{Settup}, blue triangle). Recordings of electroreceptor afferents were amplified (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, P-unit recording, recorded EOD and the generated
stimulus, were digitized with sampling rates of 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{www.relacs.net}) running on a Linux computer was used for online spike and EOD detection, stimulus generation, and calibration.

%For parts of the data  %0, 20 or 

\subsection{Identification of P-units and ampullary cells}
The neurons were classified into cell types during the recording by the experimenter. P-units were classified based on mean baseline firing rates of 50--450\,Hz \citep{Grewe2017, Hladnik2023} and phase-locking to the EOD, leading to a multimodal interspike interval (ISI) distribution (as in \subfigrefb{heterogeneity}{B}). Ampullary cells were classified based on mean firing rates of 80--200\,Hz and no phase-locking to the EOD with an unimodal ISI distribution. Only cells with a baseline recording were included in the final sample. Cells that ceased spiking during the baseline recording were excluded. 


\subsection{Field recordings} The EOD of fish without the stimulus was termed global EOD and measured with two vertical carbon rods ($11\,\centi\meter$
long, 8\,mm diameter) in a head-tail configuration (\figrefb{Settup}, green bars). This signal was
amplified 200--500 times and band-pass filtered (3 to 1\,500\,Hz
passband, DPA2-FX; npi electronics, Tamm, Germany). 

The EOD of the fish with the stimulus was termed local EOD and was measured between two 1\,cm-spaced silver
wires located next to the left gill of the fish and orthogonal to its longitudinal
body axis (amplification 200--500 times, band-pass filtered with 3 to
1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm,
Germany, \figrefb{Settup}, red markers). 





\subsection{Stimulation} 
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{Settup}, gray bars).




\begin{figure*}[h!]%(\subfigrefb{beat_amplitudes}{B}).
  \includegraphics[width=\columnwidth]{Settup}
  \caption{\label{Settup} Electrophysiolocical recording set-up. The fish, depicted as a black scheme and surrounded by isopotential lines, was positioned in the middle of the tank. Blue triangle -- electrophysiological recordings were conducted at the posterior anterior lateral line nerve (pALLN) above the
gills. Gray horizontal bars -- electrodes for the stimulation. Green vertical bars -- electrodes to measure the global EOD (the EOD of the fish without the stimulus). Red dots -- electrodes to measure the local EOD (the EOD of the fish with the stimulus). The local EOD was measured with a distance of 1 \,cm between the electrodes. All measured signals were amplified, filtered and stored on a local computer.}
\end{figure*}
%Fish sketch adapted base on \citep{Hagedorn1985} next to the gills, 
%Blue circles - P-units. 


\subsection{White noise stimuli}\label{rammethods}
The fish were stimulated with band-pass limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The standard deviation of the white noise was expressed in relation to the EOD size of the fish in the experimental set-up and termed contrast. The contrast varied between 1 and 20\,$\%$ for \lepto{} and between 2.5 and 40\,$\%$ for \eigen. Only cell recordings with at least 10\,s of white noise stimulation were included for the analysis. When ampullary cells were recorded the white noise was directly applied as the stimulus. For P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation (RAM, MXS-01M; npi electronics).



%A stimulus had a duration of 10\,s and was subdivided into $\n{}=20$ windows with no overlap, each with the duration of $T=0.5$\,s. S












%The size of the sinewaves was set to a contrast of 10\,$\%$ of the EOD of the receiver. 
%If one sine wave was present this resulted in the one-beat condition, if two sine waves were present in the two-beat condition. 







\subsection{Data analysis} Data analysis was performed with Python~3
using the packages matplotlib, numpy, scipy, sklearn, pandas, nixio
\citep{Stoewer2014}, and thunderfish
(\url{https://github.com/bendalab/thunderfish}).


\paragraph{Baseline calculation}\label{baselinemethods}
The mean 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 mean ISI: $\rm{CV} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle}\langle ISI \rangle$. If the baseline was recorded several times in a cell, the mean \fbase{} and mean CV were calculated.


%The serial correlation was calculated with lag 1 ($SC_1$,  \eqnref{serial}).









%To get a burst-corrected spike train all spikes in a burst package after the first spike were removed.
%as the mean number of spikes in the burst-corrected spike train. % CV \cvbasecorr{} after burst correction 
%\subsubsection{burstfraction} %1.5: 147,2.5: 59,3.0: 1,3.5: 6,4.5: 3,5.5: 1,

%\subsection{Data analysis for  chapter 2}
\paragraph{Response modulation}  \label{response_modulation}
The firing rate of a cell is modulated around an average firing rate similar to \fbase{} in response to a stimulus. The response modulation was calculated based on the mean firing rate $f(t)$. For $f(t)$ the binary spike trains, with zero everywhere no spike occurred and the sampling rate everywhere a spike occurred, were convolved with a Gaussian with a standard deviation of 2.5\,ms and the mean over the stimulus repeats was calculated. The response modulation $\sigma_{M} = \sqrt{\langle (f(t)-\langle f(t) \rangle)^2\rangle}$ was calculated as the standard deviation of the mean firing rate $f(t)$,  where the averages are taken over time. The response modulation was calculated as an estimate of the effective stimulus strength.

% _{t}_{n}
% smoothed = pad_for_mean(smoothed, max_len)
% response_modulation = np.nanstd(np.mean(smoothed, axis=0)) Spectral analysis and linear encoding


  
  %the transfer function
  %Applying the Fourier transform to the stimulus $s(t)$ resulted in  $\tilde{s}(f)$ and applying the Fourier transform to the response of the neuron, the firing rate of the spike trains $r(t)$ resulted in $R(f)$. 
  







%, therefore the stimulusSecond-order susceptibility
\paragraph{Spectral analysis}\label{susceptibility_methods}%chapter 2In this score the stimulus $s(t)$ and the response $r(t)$ were set into relation. 
The first-order and second-order susceptibility between the stimulus $s(t)$ and the response $r(t)$ were calculated (\eqnref{linearencoding_methods}, \eqnref{susceptibility}). The response $r(t)$ was the firing rate of the neuron, calculated as the binary spike train representation, with the sampling rate value when a spike occurred, zero everywhere else and the unit Hz. The noise stimulus $s(t)$ (see section \ref{rammethods}), was expressed in relation to the EOD and had no unit. A stimulus had a duration of 10\,s that were subdivided into $\n{}=20$ segments with no overlap, each with the duration of $T=0.5$\,s. Since the sampling rate $S_{r}$ varied between cell recordings (20\,kHz, 40\,kHz or 100\,kHz) $n_{\rm fft}$ was set to $S_{r}\cdot0.5$\,s, resulting in a frequency resolution of 2\,Hz for each cell. 

The Fourier transform of a time signal was calculated as $\tilde s(\omega) = \int_{0}^{T} \, s(t) \cdot e^{- i \omega t}\,dt$, with $T$ being the signal duration. The power spectrum was calculated as 
\begin{equation}
  \label{powereq}
	\begin{split}  
  S_{ss}(\omega) = \frac{\langle \tilde s(\omega) \tilde s^* (\omega)\rangle}{T}
	\end{split}
\end{equation}
with $\tilde s^* $ being the complex conjugate and $\langle ... \rangle$ the averaging over the segments.

The cross-spectrum was calculated with \eqnref{cross}, linear encoding was estimated with \eqnref{linearencoding_methods} and the higher-order cross-spectrum was calculated with \eqnref{crosshigh}.
\begin{equation}
  \label{cross}
	\begin{split}  
  S_{rs}(\omega) = \frac{\langle \tilde r(\omega) \tilde s^* (\omega)\rangle}{T}
	\end{split}
\end{equation}
\begin{equation}
  \label{linearencoding_methods}
	\begin{split}  
  \chi_{1}(\omega)  = \frac{S_{rs}(\omega) }{S_{ss}(\omega) }
	\end{split}
\end{equation}
\begin{equation}
  \label{crosshigh}
S_{rss} (\omega_{1},\omega_{2}) = \frac{\langle \tilde r (\omega_{1}+\omega_{2}) \tilde s^*  (\omega_{1})\tilde s^*  (\omega_{1}) \rangle}{T}
\end{equation}
The second-order susceptibility was calculated as
\begin{equation}
  \label{susceptibility0}
	\begin{split}  
  \chi_{2}(\omega_{1}, \omega_{2}) = \frac{S_{rss} (\omega_{1},\omega_{2})}{2S_{ss} (\omega_{1}) S_{ss} (\omega_{2})}
	\end{split}
\end{equation}
by dividing the higher-order cross-spectrum by the power spectra. Applying the Fourier transform this can be rewritten resulting in:
\begin{equation}
  \label{susceptibility}
	\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'}}
	\end{split}
\end{equation}
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.

%\subsection{First-order susceptibility} \label{linearencoding_methods} 


% The lower right and upper left quadrants in the susceptibility matrix in \figrefb{model_full} were calculated as
%\begin{equation}
%  \label{susceptibility2}
%  \chi_{2} = \frac{TN |\sum_{n=1}^N \int_{0}^{T} dt r_{n}(t) e^{i( \omega_{1}- \omega_{2})t} \int_{0}^{T}dt' s_{n}(t')e^{-i \omega_{1}t'} \int_{0}^{T} dt'' s_{n}(t'')e^{i \omega_{2}t''}|}{2 \sum_{n=1}^N \int_{0}^{T} dt s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt' s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{N} dt s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{N} dt' s_{n}(t')e^{i \omega_{2}t'}}.
%\end{equation}



\paragraph{Nonlinearity index}\label{projected_method}%chapter 2
The structural changes in the absolute value of the second-order susceptibility (as the matrix in \figrefb{model_full}) were quantified in a nonlinearity index:
  \begin{equation}
    \label{nli_equation}
    NLI(f_{0}) = \frac{\max_{f_{0}-5\,\rm{Hz} \leq f \leq f_{0}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
  \end{equation}
For this index, the second-order susceptibility matrix was projected on the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness of a frequency $f_{0}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $f_{0} \pm 5$\,Hz (\subfigrefb{cells_suscept}\,\panel[vi]{A}, gray area) and dividing it by the median of $D(f)$. This was calculated for the frequencies \fbase{} and \fbasecorr{} resulting in \nli{} and \nlicorr{}.


If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} and \nlicorr{} values was used for the population statistics in \figref{data_overview_mod} and \figrefb{data_overview}. The second-order susceptibility matrices depicted in \figrefb{cells_suscept}, \figrefb{ampullary}, \figrefb{burst_cells_suscept} and \figrefb{cells_eigen} were calculated based on the first frozen noise repeat. 
% \includegraphics[width=\columnwidth]{cells_eigen} %  {ampullary}{burst_cells_suscept}, {cells_eigen} 
%\nli{} and \nlicorr{} scores shown in were calculated as the mean of the \nli{} and \nlicorr{} values for all frozen noise RAM repetitions in a cel







\subsection{Leaky integrate-and-fire models}\label{lifmethods}

Leaky integrate-and-fire (LIF) models with a carrier were constructed to reproduce the specific firing
properties of P-units \citep{Chacron2001,Sinz2020}. The input into the P-unit model during baseline was the fish's own EOD
  \begin{equation}
    \label{eod}
    x(t) = x_{EOD}(t) = \cos(2\pi f_{EOD} t)
  \end{equation}
  with EOD frequency $f_{EOD}$ and amplitude normalized to one.

In the model, the input $x(t)$ was first thresholded. 
\begin{equation}
  \label{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}). Thresholding potentially happens at the synapse between the receptor cells and the afferent.



The resulting signal was then low-pass filtered with a time constant $\tau_{d}$ by the afferent's dendrite (box left to \subfigrefb{flowchart}\,\panel[ii]{A}).
\begin{equation}
  \label{dendrite}
  \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor x(t) \rfloor_{0}^{p}
\end{equation}
Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high
sensitivity to small amplitude modulations. Because the input was unitless, the dendritic voltage was unitless, too. $p$ was set to one resulting in a pure threshold. This thresholding and low-pass filtering extracted the amplitude modulation of the input $x(t)$. 


The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
\begin{equation}
  \label{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 $V_{d}$, and $\sqrt{2D}\xi(t)$ is a white noise with strength $D$. All state variables in the LIF are unitless. The output of the model is in seconds.

The adaptation current $A$ followed
\begin{equation}
  \label{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 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}
\citep{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. Model cell identifier 2012-07-03-ak (see table~\ref{modelparams} for model parameters). \figitem[i]{A}--\,\panel[i]{\textbf{D}} Rectification of the input. Positive values are maintained and negative discarded (see box on the left). \figitem[ii]{A}--\,\panel[ii]{\textbf{D}} Dendritic low-pass filtering. \figitem[iii]{A}--\,\panel[iii]{\textbf{D}} The noise component in $\sqrt{2D}\,\xi(t)$ \eqnref{LIF} or $\sqrt{2D \, c_{noise}}\,\xi(t)$ in \eqnref{Noise_split_intrinsic}. \figitem[iv]{A}--\,\panel[iv]{\textbf{D}} Spikes generation in the LIF model. Spikes are generated when the voltage of 1 is crossed (markers). Then the voltage is again reset to 0. \figitem[v]{A}--\,\panel[v]{\textbf{D}} Power spectrum of the spikes above. The first peak in panel \panel[v]{A} is the \fbase{} peak. The peak at 1 is the \feod{} peak. The other two peaks are at $\feod{} \pm \fbase{}$. \figitem{A} Baseline condition: The input to the model is a sinus with frequency \feod{}. \figitem{B} The EOD carrier is multiplied with a band-pass limited random amplitude modulation (RAM) with a contrast of 2\,$\%$, as in \eqnref{ram_equation}. \figitem{C} The EOD carrier is multiplied with a band-pass limited RAM signal with a contrast of 20\,$\%$. \figitem{D} The total noise of the model is split into a signal component regulated by $c_{signal}$ in \eqnref{ram_split}, and a noise competent regulated by $c_{noise}$ in \eqnref{Noise_split_intrinsic}. The intrinsic noise in \panel[iii]{D} is reduced compared to \panel[iii]{A}--\panel[iii]{C}. To maintain the CV during the noise split in \panel{D} comparable to the CV during the baseline in \panel{A} the RAM contrast is increased in \panel[i]{D}.}
\end{figure*}
%\figitem[i]{C}$RAM(t)$


\subsection{Numerical implementation}

The ODEs were integrated by the Euler forward method with a time-step of $\Delta t = 0.00005$\,s.


For the intrinsic noise $\xi(t)$ (\eqnref{LIF}, \subfigrefb{flowchart}\,\panel[iii]{A}) in each time step $i$ a random number was drawn from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$:
\begin{equation}
  \label{LIFintegration}
  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
\end{equation}








\subsection{Model parameters}\label{paramtext}
The 8 free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, 
$D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to
both baseline activity (mean baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector
strength of spike coupling to EOD) and responses to step increases and
decreases in EOD amplitude (onset-state and steady-state responses,
effective adaptation time constant) of 42 specific P-units for a fixed power of $p=1$ (table~\ref{modelparams}, \citealp{Ott2020}). When modifying
the model (e.g. varying the threshold nonlinearity or the power $p$ in \eqnref{dendrite}) the bias current $\mu$ was adapted to restore the original mean baseline firing rate. For each stimulus repetition the start parameters $A$, $V_{d}$ and $V_{m}$ were drawn randomly from a starting value distribution, retrieved from a 100\,s baseline, recorded in the model after a 100\,s transient.





\subsection{Stimuli for the model}%analysis of the 
For the beat protocols, the stimulus was the EOD of the receiving fish, normalized to an
amplitude of one plus the EOD of a second or third fish. If not stated
otherwise, a superposition of cosine waves was used to
mimic the EODs (\eqnref{beat}).

The input for the model during RAM stimulation was the EOD multiplied with a random amplitude modulation $RAM(t)$:
  \begin{equation}
    \label{ram_equation}
    x(t) = (1+ RAM(t)) \cdot \cos(2\pi f_{EOD} t)
  \end{equation}
The $RAM(t)$ was generated by drawing random numbers for each frequency up to 300\,Hz in Fourier space. After back transformation, the resulting signal was scaled to the desired standard deviation relative to the EOD carrier, which was termed contrast. %Rectified $x(t)$ with RAM contrasts of $2$ and $20\,\%$ are depicted in \subfigrefb{flowchart}\,\panel[i]{B},\,\panel[i]{C}.%$f_{EOD}/2$


\subsection{Second-order susceptibility analysis of the model}
%\subsubsection{Model second-order nonlinearity}

The second-order susceptibility in the model was calculated with \eqnref{susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz. 





\subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
Based on the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} the total noise of a LIF model can be split up into several independent noise processes with the same correlation function. Based on this theorem a weak input signal to the LIF model can be approximated with no input signal but instead, a split of the total noise into a noise component and a signal component (\citealp{Egerland2020}). This signal component can be used for the cross-spectrum calculation in \eqnref{susceptibility}, where it is not a weak signal anymore. This approach has the advantage that the signal-to-noise ratio is increased and the number of noise stimulus realizations \n, which would be required in case of a weak input signal for the same signal-to-noise ratio, is reduced. The signal component is regulated by $c_{signal}$ in \eqnref{ram_split} and the noise component is regulated by $c_{noise}$ in \eqnref{Noise_split_intrinsic}. 

  \begin{equation}
    \label{ram_split}
    x(t) = (1+ \sqrt{\rho \, 2D \,c_{signal}} \cdot RAM(t)) \cdot \cos(2\pi f_{EOD} t)
  \end{equation}

\begin{equation}
  \label{Noise_split_intrinsic_dendrite}
  \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+  \lfloor x(t) \rfloor_{0}^{p}
\end{equation}


%\begin{equation}
%  \label{Noise_split_intrinsic}
%  V_{m_{i+1}}  = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{noise}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
%\end{equation}

\begin{equation}
  \label{Noise_split_intrinsic}
  \tau_{m} \frac{d V_{m}}{d t}  = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{noise}}\,\xi(t)
\end{equation}
% das stimmt so, das c kommt unter die Wurzel!

A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{D}). Both components have to add up to the initial 100\,$\%$ of the total noise, otherwise the Novikov-Furutsu Theorem \citep{Novikov1965, Furutsu1963} would not be applicable. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied \citep{Egerland2020}. In the here used LIF model with EOD carrier, this is more complicated since the noise stimulus $RAM(t)$ is first multiplied with the carrier (\eqnref{ram_split}), the signal is then subjected to rectification and subsequent dendritic low-pass filtering and becomes colored (\eqnref{Noise_split_intrinsic_dendrite}). This is the component that is added to the noise component in \eqnref{Noise_split_intrinsic} and should in sum lead to a total noise of 100\,\%.

 To compensate for these transformations, the generated noise $RAM(t)$ was scaled up by a factor $\rho$ (\eqnref{ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). The $\rho$ scaling factor was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier present) and the CV during stimulation (total noise split with $c_{signal}$ and $c_{noise}$). The assumption behind this approach was that as long the CV stays the same between baseline and stimulation both components have added up to 100\,$\%$ of the total noise and the noise split is valid.










\begin{table*}[hp!]
  \caption{\label{modelparams} Model parameters of LIF models, fitted to 42 electrophysiologically recorded P-units (\citealp{Ott2020}). See section \ref{lifmethods} for model and parameter description.}
  \begin{center}
  \begin{tabular}{lrrrrrrrr}
    \hline
    \bfseries $cell$ & \bfseries $\beta$  & \bfseries $\tau_{m}$/ms & \bfseries $\mu$ & \bfseries $D$/$\mathbf{ms}$ & \bfseries $\tau_{A}$/ms & \bfseries $\Delta_A$ & \bfseries $\tau_{d}$/ms & \bfseries $t_{ref}$/ms \\\hline
2012-07-03-ak& $10.6$& $1.38$& $-1.32$& $0.001$& $96.05$& $0.01$& $1.18$& $0.12$ \\
2013-01-08-aa& $4.5$& $1.20$& $0.59$& $0.001$& $37.52$& $0.01$& $1.18$& $0.38$ \\
 \hline
  \end{tabular}
  \end{center}
\end{table*}% 2013-01-08-aa %  2012-07-03-ak





\newpage







%Recording at the frequency combinations \bcsum{} and \bcdiff{} \fbasecorr{}at the burst-corrected firing rate

















\appendix 
\setcounter{secnumdepth}{2}
\section{Appendix}

\begin{figure*}[hp]%hp!
\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. \figitem{A} Regular firing low-CV P-unit. \figitem[i]{A} Left: Interspike intervals (ISI) distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast. 
 Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. \figitem{B} Noisy high-CV P-Unit. Panels as in \panel{A}.
  }
\end{figure*}

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