updating intro and discussion
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\documentclass[preprint, 10pt, 3p]{elsarticle}
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\usepackage{natbib}
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\setcitestyle{super,comma,sort&compress}
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%\setcitestyle{super,comma,sort&compress}
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%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[mediumspace,mediumqspace, Gray,amssymb]{SIunits} % \ohm, \micro
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@ -422,36 +422,43 @@ In this work, the influence of nonlinearities on stimulus encoding in the primar
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\section{Introduction}
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% of spiking responses influences the encoding in P-units Nonlinearities can arise not only between cells but also inside single neurons,
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%, as has been shown in modeling studies % In the previous chapter, it was elaborated that the encoding of high beat frequencies requires a nonlinearity at the synapse between the electroreceptors and the afferent P-unit.
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Neuronal systems are inherently nonlinear with nonlinearities being observed in all sensory modalities \citep{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. Rectification is a famous nonlinearity that is assumed to occur in through the transduction machinery of inner hair cells \citep{Peterson2019}, signal rectification in receptor cells \citep{Chacron2000, Chacron2001} or the rheobase of action-potential generation \citep{Middleton2007, Longtin2008}. Nonlinearity can be necessary to explain the behavior of complex cells in the visual system \citep{Adelson1985}, to extract information about the stimulus \citep{Barayeu2023} and to encode stimulus features as up- and down-strokes \citep{Gabbiani1996}.
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In this work, nonlinearities are studied in the gymnotiform weakly electric fish of the species \lepto. These fish are equipped with an electric organ \citep{Salazar2013}, that is constantly active and produces a quasi sinusoidal electric organ discharge (EOD), with a long-term stable and fish-specific frequency (\feod, \citealp{Knudsen1975, Henninger2020}). \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
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[\citealp{Carr1982}], sense the actively generated electric field and its modulations.
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Within a single tuberous electroreceptor organ about 30 primary electroreceptors form ribbon synapses onto the dendrites of the same afferent fiber
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\citep{Szabo1965, Wachtel1966} that projects via the lateral line nerve to the hindbrain. There, the afferent fiber synapses onto pyramidal neurons in the
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electrosensory lateral line lobe (ELL, \citealp{Maler2009a}). The EOD is encoded by the p-type electroreceptor afferents, the P-units. These fire phase-locked to the EOD of the fish with a spike at a similar phase in the EOD cycle, with the phase depending on the receptor location \citep{Hladnik2023}. P-units fire not once in every EOD cycle but in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at multiples of the EOD period. P-units have heterogeneous baseline firing properties with the mean baseline firing rate \fbase{} varying between 50 and 450\,Hz \citep{Grewe2017, Hladnik2023} and the coefficient of variation (CV) of their ISIs varying between 0.2 and 1.7 \citep{Grewe2017, Hladnik2023}.
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The time resolved firing rate of a neuron can be described by the Volterra series where the first term describes the linear contribution an all higher terms the nonlinear contributions \citep{Voronenko2017}. \citet{Voronenko2017} analytically retrieved the second term of the Volterra series, the second-order susceptibility, based on leaky integrate and fire (LIF) models, where the input were two sine waves. There it was demonstrated that the second-order susceptibility could be very strong, but only at specific input frequencies. A triangular nonlinear shape was predicted, with nonlinearities appearing if one of the beat frequencies, the sum or the difference of the beat frequencies was equal to the mean baseline firing rate \fbase{} (\citealp{Voronenko2017}). These effects were especially pronounced if one of the signals had a faint signal amplitude. Such nonlinearities might influence faint signal detection, as it was observed in the field in the framework of the electrosensory cocktail party \citep{Henninger2018}.
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In previous works, P-units have been mainly considered to be linear encoders \citep{Xu1996, Benda2005, Gussin2007, Grewe2017}. Still, P-units exhibit nonlinear effects, especially for strong beat amplitudes \citep{Nelson1997}. Although some P-units can be described by a linear decoder, other P-units require a nonlinear decoder \citep{Chacron2004}. In the following, the focus will be on nonlinear effects in the time-resolved firing rate.
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%Such nonlinearities might appear in a three fish setting as e.g. the electrosensory cocktail party , where they might facilitate the encoding of faint signals
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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}.
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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).
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In this work, nonlinearities are studied in the gymnotiform weakly electric fish of the species \lepto. These fish are equipped with an electric organ \citep{Salazar2013}, that is constantly active and produces a quasi sinusoidal electric organ discharge (EOD), with a long-term stable and fish-specific frequency (\feod, \citealp{Knudsen1975, Henninger2020}). The EOD is used for electrolocation \citep{Fotowat2013, Nelson1999} and communication \citep{Fotowat2013, Walz2014, Henninger2018}. If two fish 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}.
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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. The beat frequency is defined as the difference between the two EOD frequencies ($\Delta f_{1} = f_{1}-\feod{}$). Whereas the predictions of the nonlinear response theory presented in \citet{Voronenko2017} apply to P-units, where the main driving force are beats and not the whole signal, will be addressed in the following.
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%with the EOD frequencies \feod{} and $f_{1}$
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%When the EODs of two fish interfere this results in an amplitude modulated signal, that is encoded in the electrosensory system of these fish.
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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 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.
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}
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\end{figure*}
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Cutaneous tuberous organs, that are distributed all over the body of these fish
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[\citealp{Carr1982}], sense the actively generated electric field and its modulations.
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Within a single tuberous electroreceptor organ about 30 primary electroreceptors form ribbon synapses onto the dendrites of the same afferent fiber
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\citep{Szabo1965, Wachtel1966} that projects via the lateral line nerve to the hindbrain. There, the afferent fiber synapses onto pyramidal neurons in the
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electrosensory lateral line lobe (ELL, \citealp{Maler2009a}). The EOD is encoded by the p-type electroreceptor afferents, the P-units. 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}.
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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 in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at multiples of the EOD period. 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, see section \ref{baselinemethods}). P-units also represent \feod{}, 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}, top). When three fish encounter, all their waveforms interfere with both beat 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, but also nonlinear peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). The difference of the two beat frequencies is also known as a social envelope \citep{Stamper2012Envelope, Savard2011}, that often emerges as the modulation of two beats in the superimposed signal. The encoding of envelopes in P-units has is a controvercial topic, with some works not considering P-units as envelope encoders \citep{Middleton2006}, while others identify some P-unit populations as successful in encoding envelopes \citep{Savard2011}. In this work the second-order susceptibility 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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%$\Delta f_{1}$ and $\Delta f_{2}$
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%, as e.g. during an electrosensory cocktail party observed the field (\citealp{Henninger2018})
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%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).
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%\lepto{} is sexually dimorphic with females having lower EOD frequencies (500--750\,Hz) and males having higher EOD frequencies (750--1000\,Hz).
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%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).
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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.
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%\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
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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
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The receiver fish \feod{} is fixed, the varied parameters in a three-fish setting are the EOD frequencies \fone{} and \ftwo{} of the encountered fish, 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. Instead white noise stimulation, where all behaviorally relevant frequencies are present at the same time, can be used to access second-order susceptibility \citep{Neiman2011fish}. 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. 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
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%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.
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%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}.
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@ -480,6 +487,12 @@ In this work, the second-order susceptibility in the spiking responses of P-unit
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%influence the occurrence of nonlinearity by comparing the nonlinearity in P-units with other cell populations as the ampullary cells (lower CVs) and consider not only \lepto{} but also the P-units of \eigen{} (lower CV)
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% if the nonlinear frequency combinations occur where they are predicted based on simple LIF models without a carrier.
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\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 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.
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}
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\end{figure*}
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@ -618,6 +631,18 @@ In this chapter, the CV has been identified as an important factor influencing n
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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 \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.
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%Nonlinear effects only for specific frequency combinations
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\subsubsection{The readout from P-units in pyramidal cells is heterogeneous}
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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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The findings in this work are based on a low-CV P-unit model and might require a selective readout from a homogeneous population of low-CV cells to sustain. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citep{Maler2009a} and P-units have heterogeneous baseline properties (CV and \fbasesolid{}) that do not depend on the location of the receptor on the fish body \citep{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 \citep{Beiran2018, Hladnik2023}.
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%\subsubsection{A heterogeneous readout is required to cover all female-intruder combinations}
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A heterogeneous readout might be not only physiologically plausible but also required to address the electrosensory cocktail party for all female-intruder combinations. In this chapter, the improved intruder detection was present only for specific beat frequencies (\figrefb{ROC_with_nonlin}), corresponding to findings from previous literature \citep{Schlungbaum2023}. If pyramidal cells would integrate only from P-units with the same mean baseline firing rate \fbasesolid{} not all female-male encounters, relevant for the context of the electrosensory cocktail party, could be covered (black square, \subfigrefb{ROC_with_nonlin}{C}). Only a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) could lead to a vertical displacement of the improved intruder detection (red diagonals in \subfigrefb{ROC_with_nonlin}{C}). Weather integrating from such a heterogeneous population with different \fbasesolid{} would cover this behaviorally relevant range in the electrosensory cocktail party should be addressed in further studies.
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%
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%Second-order susceptibility was strongest the lower the CV of the use LIF model \citep{Voronenko2017}.
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@ -628,15 +653,11 @@ In this chapter strong nonlinear interactions were found in a subpopulation of l
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\subsubsection{Neuronal delays might deteriorate nonlinear effects}
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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.
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%These low-frequency modulations of the amplitude modulation are
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\subsection{Encoding of secondary envelopes}%($\n{}=49$) Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study \citet{Savard2011}.
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The RAM stimulus used in this chapter is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citep{Stamper2012Envelope}. Low-frequency secondary envelopes are extracted downstream of P-units in the ELL \citep{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citep{Middleton2007}. The encoding of social envelopes can also be attributed to P-units with stronger nonlinearities, lower firing rates and higher CVs \citep{Savard2011}. In this chapter high CVs were associated with increased bursting (\subfigrefb{data_overview}\,\panel[iii]{A}).
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The RAM stimulus used in this chapter is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citep{Stamper2012Envelope}. In previous works it was shown that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citep{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citep{Middleton2007}. This work addressed that only a small class of cells, with very low CVs would encode the social envolope. This small percentage of the low-CV cells would be in line with no P-units found in the work by \citet{Middleton2007}. On the other hand in previous literature the encoding of social envelopes was attributed to a large subpopulationof P-units with stronger nonlinearities, lower firing rates and higher CVs \citep{Savard2011}. The explanation for these findings might be in another baseline properties of P-units, bursting, firing repeated burst packages of spikes interleaved with quiescence (unpublished work).
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\subsection{More fish would decrease second-order susceptibility}%
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@ -42,7 +42,7 @@
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\documentclass[preprint, 10pt, 3p]{elsarticle}
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\usepackage{natbib}
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\setcitestyle{super,comma,sort&compress}
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%\setcitestyle{super,comma,sort&compress}
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%%%%% units %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage[mediumspace,mediumqspace, Gray,amssymb]{SIunits} % \ohm, \micro
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@ -103,12 +103,6 @@
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%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% references to panels of a figure within the caption:
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% references to panels of a figure within the caption:
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\newcommand{\figitem}[2][]{\newline\ifthenelse{\equal{#1}{}}{\textsf{\bfseries #2}}{\textsf{\bfseries #2}}$_{\sf #1}$}
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% references to panels of a figure within the text:
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@ -177,16 +171,6 @@
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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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@ -256,10 +240,6 @@
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%
|
||||
%Cocktailparty combinations
|
||||
|
||||
|
||||
|
||||
|
||||
%\newcommand{\bctwo}{$\Delta f_{Female}$}%sum
|
||||
%\newcommand{\bcone}{$\Delta f_{Intruder}$}%sum
|
||||
%\newcommand{\bcsum}{$|\Delta f_{Female} + \Delta f_{Intruder}|$}%sum
|
||||
@ -315,8 +295,6 @@
|
||||
|
||||
|
||||
%\newcommand{\spnr}{$\langle spikes_{burst}\rangle$}%diff of
|
||||
|
||||
|
||||
\newcommand{\burstcorr}{\ensuremath{{Corrected}}}
|
||||
\newcommand{\cvbasecorr}{CV\ensuremath{_{BaseCorrected}}}
|
||||
\newcommand{\cv}{CV\ensuremath{_{Base}}}%\cvbasecorr{}
|
||||
@ -368,10 +346,8 @@
|
||||
\newcommand{\tabrefb}[1]{\tabb~\tref{#1}}
|
||||
\newcommand{\tabsrefb}[1]{\tabsb~\tref{#1}}
|
||||
|
||||
|
||||
\usepackage{xr}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%% notes
|
||||
@ -405,22 +381,13 @@
|
||||
\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[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.greqe@uni-tuebingen.de}
|
||||
% \ead[url]{home page}
|
||||
|
||||
\author[1]{Alexandra Barayeu\corref{fnd1}}
|
||||
%\ead{alexandra.rudnaya@uni-tuebingen.de}
|
||||
\author[2,3]{Maria Schlungbaum}
|
||||
\author[2,3]{Benjamin Lindner}
|
||||
\author[1,4,5]{Jan Benda}
|
||||
\author[1]{Jan Grewe\corref{cor1}}
|
||||
\ead{jan.greqe@uni-tuebingen.de}
|
||||
\ead{jan.grewe@uni-tuebingen.de}
|
||||
% \ead[url]{home page}
|
||||
|
||||
\cortext[cor1]{Corresponding author}
|
||||
@ -447,52 +414,51 @@
|
||||
%Nonlinearities facilitate the encoding of a wide range of beat frequencies and amplitudes in primary electrosensory afferents
|
||||
|
||||
|
||||
%\newpage
|
||||
%\newpage
|
||||
%\cleardoublepage
|
||||
|
||||
|
||||
\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}
|
||||
|
||||
\end{frontmatter}
|
||||
Neuronal systems are inherently nonlinear with nonlinearities being observed in all sensory modalities \citep{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. Rectification is a famous nonlinearity that is assumed to occur in through the transduction machinery of inner hair cells \citep{Peterson2019}, signal rectification in receptor cells \citep{Chacron2000, Chacron2001} or the rheobase of action-potential generation \citep{Middleton2007, Longtin2008}. Nonlinearity can be necessary to explain the behavior of complex cells in the visual system \citep{Adelson1985}, to extract information about the stimulus \citep{Barayeu2023} and to encode stimulus features as up- and down-strokes \citep{Gabbiani1996}.
|
||||
|
||||
|
||||
\section{Introduction}%\label{chapter2}
|
||||
The time resolved firing rate of a neuron can be described by the Volterra series where the first term describes the linear contribution an all higher terms the nonlinear contributions \citep{Voronenko2017}. \citet{Voronenko2017} analytically retrieved the second term of the Volterra series, the second-order susceptibility, based on leaky integrate and fire (LIF) models, where the input were two sine waves. There it was demonstrated that the second-order susceptibility could be very strong, but only at specific input frequencies. A triangular nonlinear shape was predicted, with nonlinearities appearing if one of the beat frequencies, the sum or the difference of the beat frequencies was equal to the mean baseline firing rate \fbase{} (\citealp{Voronenko2017}). These effects were especially pronounced if one of the signals had a faint signal amplitude. Such nonlinearities might influence faint signal detection, as it was observed in the field in the framework of the electrosensory cocktail party \citep{Henninger2018}.
|
||||
|
||||
% 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.
|
||||
%Such nonlinearities might appear in a three fish setting as e.g. the electrosensory cocktail party , where they might facilitate the encoding of faint signals
|
||||
|
||||
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 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}). The EOD is used for electrolocation \citep{Fotowat2013, Nelson1999} and communication \citep{Fotowat2013, Walz2014, Henninger2018}. If two fish 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. The beat frequency is defined as the difference between the two EOD frequencies ($\Delta f_{1} = f_{1}-\feod{}$). Whereas the 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.
|
||||
|
||||
%with the EOD frequencies \feod{} and $f_{1}$
|
||||
%When the EODs of two fish interfere this results in an amplitude modulated signal, that is encoded in the electrosensory system of these fish.
|
||||
|
||||
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).
|
||||
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. 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}.
|
||||
|
||||
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 in a probabilistic manner, resulting in an interspike interval (ISI) distribution with maxima at multiples of the EOD period. 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, see section \ref{baselinemethods}). P-units also represent \feod{}, 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}, top). When three fish encounter, all their waveforms interfere with both beat 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, but also nonlinear peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{motivation}{D}, bottom). The difference of the two beat frequencies is also known as a social envelope \citep{Stamper2012Envelope, Savard2011}, that often emerges as the modulation of two beats in the superimposed signal. The encoding of envelopes in P-units has is a controvercial topic, with some works not considering P-units as envelope encoders \citep{Middleton2006}, while others identify some P-unit populations as successful in encoding envelopes \citep{Savard2011}. In this work the second-order susceptibility 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.
|
||||
|
||||
%$\Delta f_{1}$ and $\Delta f_{2}$
|
||||
%, as e.g. during an electrosensory cocktail party observed the field (\citealp{Henninger2018})
|
||||
%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).
|
||||
%\lepto{} is sexually dimorphic with females having lower EOD frequencies (500--750\,Hz) and males having higher EOD frequencies (750--1000\,Hz).
|
||||
|
||||
\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*}
|
||||
%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).
|
||||
|
||||
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
|
||||
The receiver fish \feod{} is fixed, the varied parameters in a three-fish setting are the EOD frequencies \fone{} and \ftwo{} of the encountered fish, 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. Instead white noise stimulation, where all behaviorally relevant frequencies are present at the same time, can be used to access second-order susceptibility \citep{Neiman2011fish}. 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. 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
|
||||
|
||||
|
||||
%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.
|
||||
%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}.
|
||||
|
||||
|
||||
@ -521,6 +487,12 @@ In this work, the second-order susceptibility in the spiking responses of P-unit
|
||||
%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.
|
||||
|
||||
\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*}
|
||||
|
||||
|
||||
|
||||
|
||||
@ -659,6 +631,18 @@ In this chapter, the CV has been identified as an important factor influencing n
|
||||
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.
|
||||
|
||||
|
||||
%Nonlinear effects only for specific frequency combinations
|
||||
\subsubsection{The readout from P-units in pyramidal cells is heterogeneous}
|
||||
|
||||
%is required to cover all female-intruder combinations} %gain_via_mean das ist der Frequenz plane faktor
|
||||
%associated with nonlinear effects as \fdiffb{} and \fsumb{} were with the same firing rate low-CV P-units
|
||||
%and \figrefb{ROC_with_nonlin}
|
||||
|
||||
The findings in this work are based on a low-CV P-unit model and might require a selective readout from a homogeneous population of low-CV cells to sustain. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citep{Maler2009a} and P-units have heterogeneous baseline properties (CV and \fbasesolid{}) that do not depend on the location of the receptor on the fish body \citep{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 \citep{Beiran2018, Hladnik2023}.
|
||||
|
||||
|
||||
%\subsubsection{A heterogeneous readout is required to cover all female-intruder combinations}
|
||||
A heterogeneous readout might be not only physiologically plausible but also required to address the electrosensory cocktail party for all female-intruder combinations. In this chapter, the improved intruder detection was present only for specific beat frequencies (\figrefb{ROC_with_nonlin}), corresponding to findings from previous literature \citep{Schlungbaum2023}. If pyramidal cells would integrate only from P-units with the same mean baseline firing rate \fbasesolid{} not all female-male encounters, relevant for the context of the electrosensory cocktail party, could be covered (black square, \subfigrefb{ROC_with_nonlin}{C}). Only a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) could lead to a vertical displacement of the improved intruder detection (red diagonals in \subfigrefb{ROC_with_nonlin}{C}). Weather integrating from such a heterogeneous population with different \fbasesolid{} would cover this behaviorally relevant range in the electrosensory cocktail party should be addressed in further studies.
|
||||
|
||||
%
|
||||
%Second-order susceptibility was strongest the lower the CV of the use LIF model \citep{Voronenko2017}.
|
||||
@ -669,15 +653,11 @@ In this chapter strong nonlinear interactions were found in a subpopulation of l
|
||||
|
||||
|
||||
|
||||
\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.
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%These low-frequency modulations of the amplitude modulation are
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\subsection{Encoding of secondary envelopes}%($\n{}=49$) Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study \citet{Savard2011}.
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The RAM stimulus used in this chapter is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citep{Stamper2012Envelope}. Low-frequency secondary envelopes are extracted downstream of P-units in the ELL \citep{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citep{Middleton2007}. The encoding of social envelopes can also be attributed to P-units with stronger nonlinearities, lower firing rates and higher CVs \citep{Savard2011}. In this chapter high CVs were associated with increased bursting (\subfigrefb{data_overview}\,\panel[iii]{A}).
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The RAM stimulus used in this chapter is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citep{Stamper2012Envelope}. In previous works it was shown that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citep{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citep{Middleton2007}. This work addressed that only a small class of cells, with very low CVs would encode the social envolope. This small percentage of the low-CV cells would be in line with no P-units found in the work by \citet{Middleton2007}. On the other hand in previous literature the encoding of social envelopes was attributed to a large subpopulationof P-units with stronger nonlinearities, lower firing rates and higher CVs \citep{Savard2011}. The explanation for these findings might be in another baseline properties of P-units, bursting, firing repeated burst packages of spikes interleaved with quiescence (unpublished work).
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\subsection{More fish would decrease second-order susceptibility}%
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