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TEXBASE=highbeats
TEXBASE=susceptibility1
BIBFILE=references.bib
REBUTTALBASE=rebuttal3
REBUTTALBASE=
TEXFILE=$(TEXBASE).tex
PDFFILE=$(TEXBASE).pdf
TXTFILE=$(TEXBASE).txt
REVISION=cb626892bd10b2908691bd95767f87c3d6a57dfe
ALLPDFFIGURES=$(shell sed -n -e '/^[^%].*includegraphics/{s/^.*includegraphics.*{\([^}]*\)}.*/\1.pdf/;p}' $(TEXFILE))
PDFFIGURES=$(filter-out toblerone_animated.pdf, $(ALLPDFFIGURES))
PT=$(wildcard *.py)
PYTHONFILES=$(filter-out plotstyle.py myfunctions.py numerical_compar_both.py, $(PT))
PYTHONPDFFILES=$(PYTHONFILES:.py=.pdf)
REVISION=fa4f64c144b4fd95b04bd9d631bd8844bb9c820a
ifdef REBUTTALBASE
REBUTTALTEXFILE=$(REBUTTALBASE).tex
REBUTTALPDFFILE=$(REBUTTALBASE).pdf
endif
REBUTTALREVISION=68900a1
REBUTTALREVISION=
# all ###########################################################
ifdef REBUTTALBASE
all: bib supplement rebuttalbib
all: bib rebuttalbib
else
all: bib supplement
all: bib
endif
# python #########################################################
@ -37,9 +35,9 @@ watchplots :
while true; do ! make -q plots && make plots; sleep 0.5; done
# manuscript #####################################################
# rescue_local_eod manuscript #################################################
bib: $(TEXBASE).bbl
$(TEXBASE).bbl: $(TEXFILE) $(BIBFILE) $(PDFFIGURES)
$(TEXBASE).bbl: $(TEXFILE) $(BIBFILE)
lualatex $(TEXFILE)
bibtex $(TEXBASE)
lualatex $(TEXFILE)
@ -50,17 +48,12 @@ $(TEXBASE).bbl: $(TEXFILE) $(BIBFILE) $(PDFFIGURES)
@sed -n -e '1,/You.ve used/p' $(TEXBASE).blg
pdf: $(PDFFILE)
$(PDFFILE) : $(TEXFILE) $(PDFFIGURES)
$(PDFFILE) : $(TEXFILE)
lualatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && lualatex -interaction=scrollmode $< || true
again :
lualatex $(TEXFILE)
supplement: supplement.pdf
supplement.pdf : supplement.tex
lualatex $<
lualatex $<
# watch files #######################################################
watchpdf :
while true; do ! make -s -q pdf && make pdf; sleep 0.5; done
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lualatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && lualatex -interaction=scrollmode $< || true
watchrebuttal :
while true; do ! make -s -q rebuttal && make rebuttal; sleep 0.5; done
while true; do ! make -q rebuttal && make rebuttal; sleep 0.5; done
rebuttaldiff :
latexdiff-git -r $(REBUTTALREVISION) --append-textcmd="response,issue" --pdf $(REBUTTALTEXFILE)
@ -179,7 +172,7 @@ help :
@echo -e \
"make pdf: make the pdf file of the paper.\n"\
"make bib: run bibtex and make the pdf file of the paper.\n"\
"make again: run lualatex and make the pdf file of the paper,\n"\
"make again: run pdflatex and make the pdf file of the paper,\n"\
" no matter whether you changed the .tex file or not.\n"\
"make watchpdf: make the pdf file of the paper\n"\
" whenever the tex file is modified.\n"\

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susceptibility1.tex
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% Beat combinations
\newcommand{\boneabs}{\ensuremath{|\Delta f_{1}|}}%sum
\newcommand{\btwoabs}{\ensuremath{|\Delta f_{2}|}}%sum
\newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum
%\newcommand{\bsum}{\ensuremath{|\boneabs{} + \btwoabs{}|}}%sum
\newcommand{\bsum}{\ensuremath{\boneabs{} + \btwoabs{}}}%sum
\newcommand{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies
@ -405,6 +406,7 @@
\newcommand{\nli}{PNL\ensuremath{(\fbase{})}}%Fr$_{Burst}$
\newcommand{\nlicorr}{PNL\ensuremath{(\fbasecorr{}})}%Fr$_{Burst}$
\newcommand{\suscept}{$|\chi_{2}|$}
\newcommand{\susceptf}{$|\chi_{2}|(f_1, f_2)$}
\newcommand{\frcolor}{pink lines}
\newcommand{\rec}{\ensuremath{\rm{R}}}%{\ensuremath{con_{R}}}
@ -476,13 +478,15 @@
\end{flushleft}
% Please keep the abstract below 300 words
\section*{Abstract}
\lipsum[1-1]
\section*{Abstract}
Neuronal processing is inherently nonlinear, mechanisms such as the spiking threshold, or recitication during synaptic transmission are central to neuronal computations. Here we address the consequences of nonlinear interactions between two sinewave stimuli in the context of an electrosensory cocktail party in weakly electric fish. In a previous field study, it was observed that an extremely weak intruder signal was detected despite the presence of a much stronger female signal. Modelling studies showed that, in some scenarios, the presence of the strong female signal leads to an improved intruder detection. This was associated with nonlinearities in neuronal processing. Theoretical work has shown that the presence of two independent periodic signals can lead to nonlinear interactions. We here extend on this by applying the analysis of the second-order susceptibility to experimentally recorded primary electroreceptor afferents of the active (P-units) and the passive (ampullary cells) electrosensory system. Our combined experimental and modelling approach shows that nonlinear interactions can be found in these cells and depends on the level of intrinsic noise. We can further show that simple white-noise stimulation can be used to quickly access the second-order susceptibility of a system even when the system is driven by the amplitude modulation of a carrier such as the electric organ discharge of weakly electric fish. This method can thus be easily applied to describe nonlinear processing in any sensory modality whether they are driven by direct stimuli or amplitude modulations.
% Please keep the Author Summary between 150 and 200 words
% Use first person. PLOS ONE authors please skip this step.
% Author Summary not valid for PLOS ONE submissions.
\section*{Author summary}
\lipsum[1-1]
Weakly electric fish use their self-generated electric field to detect a wide range of behaviorally relevant stimuli. Intriguingly, they show detection performances of stimuli that are (i) extremely weak and (ii) occur in the background of strong foreground signals, reminiscent of what is often described as the cocktail party problem. Such performances are achieved by boosting the signal detection through nonlinear mechanisms. We here analyze nonlinear encoding in two different populations of primary electrosensory afferences of the weakly electric fish. We derive the rules under which nonlinear effects can be observed in both electrosensory subsystems. In a combined experimental and modelling approach we generalize the approach of nonlinear susceptibility to systems that respond to amplitude modulations of a carrier signal.
\linenumbers
%Nonlinearities contribute to the encoding of the full behaviorally relevant signal range in primary electrosensory afferents.
@ -506,6 +510,7 @@
%\end{frontmatter}
\notejg{Cite Schlungbaum in introduction}
\section*{Introduction}
@ -523,35 +528,38 @@ While the sensory periphery can often be well described by linear models, this i
Recent theoretical and modelling work on leaky integrate-and-fire (LIF) model neurons revealed specific pattern of nonlinear interactions when driven with weak sinewaves\cite{Voronenko2017}. This situation is reminiscent of the real fish's situation in which three electric fish interact. The active electrosensory system is then exposed to the two AMs arising from the interactions with either foreign fish's field. Previous recordings in the natural habitat showed interactions of three animals in which two animals interacted closely (strong signals) and were interrupted by an intruding animal (weak signal). The intruder was detected at spatial distances that led to an extremely faint intruder signal while the strong signal of the other animal is present\cite{Henninger2018}. When the receiver fish with EOD frequency \feod{} is alone, the P-unit fire action potentials as a baseline response $r_{0}$ at a spontaneous baseline rate \fbase{}. Accordingly, a peak at \fbase{} is present in the power spectrum of the neuronal response (\subfigrefb{fig:motivation}{A}, bottom). Phase-locking to the own field also leads to a representation of \feod{} in the P-unit firing rate (see \figref{fig:cells_suscept}\panel{B})\cite{Sinz2020}. The beat frequency when two fish interact is also represented in the respective responses and accordingly, there is a peak in power spectrum at $\Delta f_1$ or $\Delta f_2$ (\subfigrefb{fig:motivation}{B, C}, respectively). When three fish encounter, all their waveforms interfere with both beat frequencies being present in the superimposed signal (\subfigrefb{fig:motivation}{D}, second row). The power spectrum of the neuronal activity contains both previously seen beat peaks, but also nonlinear interaction peaks at the sum frequency \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}, bottom), thus the response of the system is not equal to the input (\subfigrefb{fig:motivation}{D}, top). The appearing difference peak is known as the social envelope\cite{Stamper2012Envelope, Savard2011}. The neuron shown here, clearly encodes the envelope. Whether P-units in general encode envelopes has been subject of controversy, some works do not consider P-units as envelope encoders\cite{Middleton2006}, while others identify some P-units as envelope encoders\cite{Savard2011}.
The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features determine which cells show nonlinear encoding.
The P-unit responses can be partially explained by simple linear filters. The linear relation of the cellular response and the stimulus can be captured by the first-order susceptibility (or transfer function). As in Volterra series, higher-order terms describe the nonlinear interactions. We quantify the nonlinearity of P-unit encoding by estimating the second-order susceptibility from white-noise responses\cite{Nikias1993,Neiman2011fish, Voronenko2017,Egerland2020}. We address whether the theory still holds for neurons that do not encode the foreign signals directly but respond to the AM of a carrier. For this study we re-use a large set of recordings of afferents of both the active (P-units) and the passive electrosensory (ampullary cells) to work out which cellular features enhance nonlinear encoding.
\section*{Results}
Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs) that are called beats (two-fish interaction) and envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units encode such AMs of the underlying EOD carrier in their time-dependent firing rates \cite{Bastian1981a,Walz2014}. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process.
%P-units are heterogeneous in their baseline firing properties \cite{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness.
Theoretical work \cite{Voronenko2017} shows that stochastic leaky integrate-and-fire (LIF) model neurons may show nonlinear stimulus encoding when the model is driven by two cosine signals with specific frequencies. In the context of weakly electric fish, such a setting is part of the animal's everyday life. The sinusoidal electric-organ discharges (EODs) of neighboring animals interfere and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, the P-units, encode such AMs of the underlying EOD carrier in their time-dependent firing rates \cite{Bastian1981a,Walz2014}. P-units are heterogeneous in their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \cite{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. We start with exploring the influence of intrinsic noise on nonlinear encoding in P-units.
\subsection*{Nonlinear signal transmission in low-CV P-units} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
\subsection*{Nonlinear signal transmission in low-CV P-units}
Nonlinear encoding as quantified by the second-order susceptibility is expected to be especially pronounced in cells with weak intrinsic noise, i.e. low baseline CVs \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD \cite{Bastian1981a}. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the baseline ISI distribution has a CV$_{\text{base}}$ of 0.2, which is at the lower end of the P-unit population \cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks: the first is located at the baseline firing rate \fbase, the second is located at the discharge frequency \feod{} of the electric organ and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
\begin{figure*}[!ht]
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{cells_suscept}
\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) of the same P-unit. The stimulus contrast reflects the strength of the AM. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 10\% (light purple) and 20\% contrast (dark purple) RAM stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.}
\end{figure*}
Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_suscept}{C}, top trace, red line) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture nonlinear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). The linear encoding (see \Eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again.
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus driven responses of sensory neurons by means of transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly to fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper?}.
Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines). The reason behind this decrease is that the higher RAM contrast is associated with a simultaneously increase of the stimulus amplitude but also of the total noise in the system. This is important since increased noise is known to linearize signal
transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}.
The second-order susceptibility, \Eqnref{eq:susceptibility}, quantifies for each combination of two stimulus frequencies \fone{} and \ftwo{} amplitude and phase of the stimulus-evoked response at the sum \fsum{} of these two frequencies. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that simulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF driven in the super-threshold regime with two sinewave stimuli, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} excatly match the neuron's baseline firing rate \fbase{} \cite{Voronenko2017}. Only then additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus contrasts (for more details see supplementary information: \nameref*{S1:highcvpunit}).
% DAS GEHOERT IN DIE DISKUSSION:
% To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}).
\begin{figure*}[!ht]
For the low-CV P-unit we observe a band of slightly elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}). This structure vanishes for the stronger stimulus (\subfigref{fig:cells_suscept}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected susceptibility values onto the diagonal by taking the mean over all anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). For the low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:cells_suscept}{G}, dot on top line). For the higher RAM contrast, however, this peak vanishes and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude but also increases the total noise in the system. Increased noise is known to linearize signal transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease.
In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounced nonlinearities even at low stimulus contrasts (\figrefb{fig:cells_suscept_high_CV}).
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{ampullary}
\caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
}
@ -559,23 +567,25 @@ High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus co
\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
Irrespective of the CV, neither P-unit shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green).
Irrespective of the CV, neither of the two example P-units shows the complete expected structure of nonlinear interactions. Electric fish posses an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs (CV$_{\text{base}}=0.08$--$0.22$)\notejb{in the figure the CV is 0.07, below this range!}\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a low-contrast noise stimulus (note: this is not an AM but a stimulus that is added to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projection onto the diagonal looses its distinct peak at \fsum{} and its overal level is reduced (\subfigrefb{fig:ampullary}{G}, bottom).
\subsection*{Model-based estimation of the nonlinear structure}
Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. Here we investigate how these discrepancies can be understood.
%Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents.
In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fig:ampullary}), we only observe nonlinear responses on the anti-diagnal of the second-order susceptibility, where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, which is in line with theoretical expectations\cite{Voronenko2017}. However, a pronounced nonlinear response at frequencies \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. Here we investigate how these discrepancies can be understood.
\begin{figure*}[!h]
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 2.5\% contrast. The plot shows that \suscept{} surface for $N=11$. Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). The signal noise component \signalnoise{} and the intrinsic noise are shown in the center and bottom. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but $10^6$ stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise are treated as signal (center trace, \signalnoise) while the intrinsic noise is reduced (see methods for details). Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).
}
\end{figure*}
In the electrophysiological experiments, we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, with parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore responses to only a limited number of trials, repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of repetitions as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal
transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and \signalnoise, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
In simulations of the model we can increase the number of repetitions beyond what would be experimentally possible, here to one million (\subfigrefb{model_and_data}\,\panel[iii]{B}). The estimate of the second-order susceptibility indeed improves. It gets less noisy and the diagonal at \fsum{} is emphasized. However, the expected vertical and horizontal lines at \foneb{} and \ftwob{} are still missing.
Using a broadband stimulus increases the effective input-noise level and this may linearize signal transmission and suppress potential nonlinear responses \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full expected structure of the second-order susceptibility should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and \signalnoise, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}).
Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{C}), that saturate in its peak values for $N>10^5$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of a statistical estimate that is based on 11 trials only. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response.
@ -585,36 +595,40 @@ With high levels of intrinsic noise, we would not expect the nonlinear response
\subsection*{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
We calculated the second-order susceptibility surfaces at \fsum{} by extracting the respective spectral components of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the beat frequencies \bone{} and \btwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs (\figref{fig:motivation}). The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In the example $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}. In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red circle, \subfigrefb{fig:motivation}{D}), that cannot be explained with the so-far shown second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{C}, in which only the nonlinearity at \fsum{} in the response is addressed. % less prominent,
\begin{figure*}[!ht]
We estimated the second-order susceptibility of P-unit responses using RAM stimuli. In particular, we found pronounced nonlinear responses in the limit of weak stimulus amplitudes. How do these findings relate to the situation of two pure sinewave stimuli with finite amplitudes that approximates the interference of EODs of real animals? For the P-units the relevant signals are the beat frequencies \bone{} and \btwo{} that arise from the interference of either of the two foreign EODs with the receiving fish's own EOD (\figref{fig:motivation}). In the introductory example, the response power spectrum showed peaks from nonlinear interactions at the sum of the two beat frequencies (orange marker, \subfigrefb{fig:motivation}{D}) and at the difference between the two beat frequencies (red marker, \subfigrefb{fig:motivation}{D}). In this example, $\Delta f_{2}$ was similar to \fbase{}, corresponding to the horizontal line of the second-order susceptibility estimated for a vanishing external RAM stimulus (\subfigrefb{model_and_data}\,\panel[iii]{C}). In the three-fish example, there was a second nonlinearity at the difference between the two beat frequencies (red dot, \subfigrefb{fig:motivation}{D}), that is not covered by the so-far shown part of the second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iii]{C}), in which only the response at the sum of the two stimulus frequencies is addressed. % less prominent,
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{model_full}
\caption{\label{fig:model_full} \figitem[]{A} Full second-order susceptibility of the model of an electrophysiologically recorded P-unit (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\Eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Baseline firing rate $\fbase{}=120$\,Hz. Absolute value of the model second-order susceptibility, with $N=10^6$ stimulus realizations and the intrinsic noise split (see methods). The position of the orange letters corresponds to the beat frequencies used for the pure sinewave stimulation in the subsequent panels. The position of the red letters correspond to the difference of those beat frequencies. \figitem{B--E} Power spectral density of model responses under pure sinewave stimulation. \figitem{B} The two beat frequencies are in sum equal to \fbase{}. \figitem{C} The difference of the two beat frequencies is equal to \fbase{}. \figitem{D} Only one beat frequency is equal to \fbase{}. \figitem{C} No beat frequencies is equal to \fbase{}.}
\end{figure*}
%section \ref{intrinsicsplit_methods}).\figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B} The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.
However, the second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \Eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants \cite{Voronenko2017} (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for the nonlinearity at \fsum{} and extend into the upper-left quadrant (representing \fdiff) fading out towards more negative $f_1$ frequencies. The \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) can be explained by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}).
%\suscept{} values at \fsum{} match the \fsum{} peak seen in the data (\figref{fig:motivation}). the fading of the
However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff) where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}).
Is it possible based on the second-order susceptibility estimated by means of RAM stimuli (\subfigrefb{fig:model_full}{A}) to predict nonlinear responses in a three-fish setting? We can test this by stimulating the same model with two beats with weak amplitudes \notejb{specify the contrast at least in the figure legend!} (\subfigrefb{fig:model_full}{B--E}). If we choose a frequency combination where the sum of the two beat frequencies is equal to the model's baseline firing rate \fbase{}, a peak at the sum of the two beat frequencies appears in the power spectrum of the response (\subfigrefb{fig:model_full}{B}), as expected from \suscept. If instead we choose two beat frequencies that differ by \fbase{}, a peak is present at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one beat frequency is equal to \fbase{}, both a peak at the sum and at the difference frequency is present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met, neither a peak at the sum nor at the difference of the two beat frequencies appears (\subfigrefb{fig:model_full}{E}).
\notejb{Note in the discussion that \suscept{} predicts stimulus evoked responses whereas in the power spectrum we see peaks even if they are unrelated to the stimulus. Like the baseline firing rate.}
Is it possible based on this second-order susceptibility (\subfigrefb{fig:model_full}{A}) to predict nonlinearities in a three-fish setting? We can test this by providing two beats with weak amplitudes to the same model (\subfigrefb{fig:model_full}{B--E}). If we chose a frequency combination where the sum of the two beat frequencies is equal to \fbase{} a nonlinearity can be observed at the sum of the two beat frequencies in the power spectrum of the response (\subfigrefb{fig:model_full}{B}). If instead we choose two beat frequencies which difference is equal to \fbase{}, a nonlinear peak is present at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one beat frequency is equal to \fbase{}, a peak at the sum, as well as at the difference frequency is present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met no nonlinearity is observed in the model, neither at the sum nor at the difference of the two beat frequencies (\subfigrefb{fig:model_full}{E}).
Even though the second-order susceptibilities here were estimated from data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
% TO DISCUSSION:
%Even though the second-order susceptibilities here were estimated from data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C, E}) and ampullary cells (\panel{B, D, F}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
% The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:data_overview_mod}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles -- \figrefb{fig:cells_suscept_high_CV}).
% Several of the recorded neurons contribute with two samples to the population analysis as their responses have been recorded to two different contrast of the same RAM stimulus. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods).
% The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{} values are highlighted with black squares.
}
\end{figure*}
%\Eqnref{response_modulation}
\subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}). The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:data_overview_mod}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles -- \figrefb{fig:cells_suscept_high_CV}). The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds with pronounced nonlinearity thus depends on the baseline CV (i.e. the internal noise), the CV during stimulation and the response strength.
All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For a comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the nonlinearity \nli{} \Eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:cells_suscept}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:data_overview_mod}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:data_overview_mod}{C}).
The effective stimulus strength also plays an important role. We quantify the effect a stimulus has on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. P-units are heterogeneous in their sensitivity, their response modulations to the same stimulus contrast vary a lot \cite{Grewe2017}. Cells with weak responses to a stimulus, be it an insensitive cell or a weak stimulus, have higher \nli{} values and thus a more pronounced ridge in the second-order susceptibility at \fsumb{} in comparison to strongly responding cells that basically have flat second-order susceptibilities (\subfigrefb{fig:data_overview_mod}{E}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and both the CV and response strength during stimulation (effective output noise).
%(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
%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{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
The population of ampullary cells is generally more homogeneous, with lower CVs than P-units. Accordingly, ampullary cells show much higher \nli{} values than P-units (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{} values are highlighted with black squares. Again, we see that cells that are strongly driven by the stimulus are at the bottom of the distribution and have \nli{} values close to one (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation: Those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high nonlinearity values (\subfigrefb{fig:data_overview_mod}{F}).
The population of ampullary cells is generally more homogeneous, with lower baseline CVs than P-units. Accordingly, \nli{} values of ampullary cells are indeed much higher than in P-units by about a factor of ten. Ampullary cells also show a negative correlation with baseline CV. Again, sensitive cells with strong response modulations are at the bottom of the distribution and have \nli{} values close to one (\subfigrefb{fig:data_overview_mod}{B, D}). The weaker the response modulation because of less sensitive cells or weaker stimulus amplitudes the stronger the nonlinear component of a cell's response (\subfigrefb{fig:data_overview_mod}{F}).
%(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
\section*{Discussion}