diff --git a/nonlinearbaseline.tex b/nonlinearbaseline.tex index c65b68e..767ad9e 100644 --- a/nonlinearbaseline.tex +++ b/nonlinearbaseline.tex @@ -231,156 +231,6 @@ \newcommand{\notebl}[1]{\note[BL]{#1}} \newcommand{\notems}[1]{\note[MS]{#1}} -%%%%% new commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newcommand{\fstim}{\ensuremath{f_{\rm{stim}}}} -\newcommand{\feod}{\ensuremath{f_{\rm{EOD}}}} -\newcommand{\feodsolid}{\ensuremath{f_{\rm{EOD}}}} -\newcommand{\feodhalf}{\ensuremath{f_{\rm{EOD}}/2}} -\newcommand{\fbase}{\ensuremath{f_{\rm{base}}}} -\newcommand{\fbasesolid}{\ensuremath{f_{\rm{base}}}} -\newcommand{\fstimintro}{\ensuremath{\rm{EOD}_{2}}} -\newcommand{\feodintro}{\ensuremath{\rm{EOD}_{1}}} -\newcommand{\ffstimintro}{\ensuremath{f_{2}}} -\newcommand{\ffeodintro}{\ensuremath{f_{1}}} - -\newcommand{\carrierinput}{\ensuremath{y(t)}} - - -\newcommand{\baseval}{134} -\newcommand{\bone}{\ensuremath{\Delta f_{1}}} -\newcommand{\btwo}{\ensuremath{\Delta f_{2}}} -\newcommand{\fone}{$f_{1}$} -\newcommand{\ftwo}{$f_{2}$} -\newcommand{\ff}{$f_{1}$--$f_{2}$} -\newcommand{\auc}{\rm{AUC}} - 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-\newcommand{\rocf}{98}%sum %\Delta -\newcommand{\roci}{36}%sum %\Delta - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% 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{\bdiff}{\ensuremath{|\boneabs{} - \btwoabs{}|}}%diff of both beat frequencies - - -\newcommand{\signalnoise}{$s_\xi(t)$} - -\newcommand{\bsumb}{$\bsum{}=\fbase{}$} -\newcommand{\btwob}{$\Delta f_{2}=\fbase{}$} -\newcommand{\boneb}{$\Delta f_{1}=\fbase{}$} -\newcommand{\bsumbtwo}{$\bsum{}=2 \fbase{}$} -\newcommand{\bsumbc}{$\bsum{}=\fbasecorr{}$} -\newcommand{\bsume}{$\bsum{}=\feod{}$} -\newcommand{\bsumehalf}{$\bsum{}=\feod{}/2$} -\newcommand{\bdiffb}{$\bdiff{}=\fbase{}$}%diff of both beat frequencies -\newcommand{\bdiffbc}{$\bdiff{}=\fbasecorr{}$}%diff of both beat frequencies -\newcommand{\bdiffe}{$\bdiff{}=\feod{}$}%diff of both -\newcommand{\bdiffehalf}{$\bdiff{}=\feod{}/2$}%diff of both beat frequencies - -%%%%%%%%%%%%%%%%%%%%%%% Frequency combinations -\newcommand{\fsum}{\ensuremath{f_{1} + f_{2}}}%sum -\newcommand{\fdiff}{\ensuremath{|f_{1}-f_{2}|}}%diff of - -\newcommand{\fsumb}{$\fsum=\fbase{}$}%su\right m -\newcommand{\ftwob}{$f_{2}=\fbase{}$}%sum -\newcommand{\foneb}{$f_{1}=\fbase{}$}%sum -\newcommand{\ftwobc}{$f_{2}=\fbasecorr{}$}%sum -\newcommand{\fonebc}{$f_{1}=\fbasecorr{}$}%sum -\newcommand{\fsumbtwo}{$\fsum{}=2 \fbase{}$}%sum -\newcommand{\fsumbc}{$\fsum{}=\fbasecorr{}$}%sum -\newcommand{\fsume}{$\fsum{}=\feod{}$}%sum -\newcommand{\fsumehalf}{$\fsum{}=\feod{}/2$}%sum -\newcommand{\fdiffb}{$\fdiff{}=\fbase{}$}%diff of both beat frequencies -\newcommand{\fdiffbc}{$\fdiff{}=\fbasecorr{}$}%diff of both beat frequencies -\newcommand{\fdiffe}{$\fdiff{}=\feod{}$}%diff of both -\newcommand{\fdiffehalf}{$\fdiff{}=\feod{}/2$}%diff of both - -\newcommand{\fctwo}{\ensuremath{f_{\rm{Female}}}}%sum -\newcommand{\fcone}{\ensuremath{f_{\rm{Intruder}}}}%sum -\newcommand{\bctwo}{\ensuremath{\Delta \fctwo{}}}%sum -\newcommand{\bcone}{\ensuremath{\Delta \fcone{}}}%sum -\newcommand{\bcsum}{\ensuremath{|\bctwo{}| + |\bcone{}|}}%sum -\newcommand{\abcsumabbr}{\ensuremath{A(\Delta f_{\rm{Sum}})}}%sum -\newcommand{\abcsum}{\ensuremath{A(\bcsum{})}}%sum - -\newcommand{\abcone}{\ensuremath{A(\bcone{})}}%sum - -\newcommand{\bctwoshort}{\ensuremath{\Delta f_{F}}}%sum -\newcommand{\bconeshort}{\ensuremath{\Delta f_{I}}}%sum -\newcommand{\bcsumshort}{\ensuremath{|\bctwoshort{} + \bconeshort{}|}}%sum -\newcommand{\abcsumshort}{\ensuremath{A(\bcsumshort{})}}%sum -\newcommand{\abconeshort}{\ensuremath{A(|\bconeshort{}|)}}%sum - -\newcommand{\abcsumb}{\ensuremath{A(\bcsum{})=\fbasesolid{}}}%sum -\newcommand{\bcdiff}{\ensuremath{||\bctwo{}| - |\bcone{}||}}%diff of both beat frequencies - -\newcommand{\bcsumb}{\ensuremath{\bcsum{} =\fbasesolid{}}}%su\right m -\newcommand{\bcsumbn}{\ensuremath{\bcsum{} \neq \fbasesolid{}}}%su\right m -\newcommand{\bctwob}{\ensuremath{\bctwo{} =\fbasesolid{}}}%sum -\newcommand{\bconeb}{\ensuremath{\bcone{} =\fbasesolid{}}}%sum -\newcommand{\bcsumbtwo}{\ensuremath{\bcsum{}=2 \fbasesolid{}}}%sum -\newcommand{\bcsumbc}{\ensuremath{\bcsum{}=\fbasecorr{}}}%sum -\newcommand{\bcsume}{\ensuremath{\bcsum{}=f_{\rm{EOD}}}}%sum -\newcommand{\bcsumehalf}{\ensuremath{\bcsum{}=f_{\rm{EOD}}/2}}%sum - -\newcommand{\bcdiffb}{\ensuremath{\bcdiff{}=\fbasesolid{}}}%diff of both beat frequencies -\newcommand{\bcdiffbc}{\ensuremath{\bcdiff{}=\fbasecorr{}}}%diff of both beat frequencies -\newcommand{\bcdiffe}{\ensuremath{\bcdif{}f=f_{\rm{EOD}}}}%diff of both -\newcommand{\bcdiffehalf}{\ensuremath{\bcdiff{}=f_{\rm{EOD}}/2}}%diff of both - -\newcommand{\burstcorr}{\ensuremath{{Corrected}}} -\newcommand{\cvbasecorr}{CV\ensuremath{_{BaseCorrected}}} -\newcommand{\cv}{CV\ensuremath{_{Base}}}%\cvbasecorr{} -\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}}} -\newcommand{\rif}{\ensuremath{\rm{RIF}}} -\newcommand{\ri}{\ensuremath{\rm{RI}}}%sum -\newcommand{\rf}{\ensuremath{\rm{RF}}}%sum -\newcommand{\withfemale}{\rm{ROC}\ensuremath{\rm{_{Female}}}}%sum \textit{ -\newcommand{\wofemale}{\rm{ROC}\ensuremath{\rm{_{NoFemale}}}}%sum -\newcommand{\dwithfemale}{\rm{\ensuremath{\auc_{Female}}}}%sum CV\ensuremath{_{BaseCorrected}} -\newcommand{\dwofemale}{\rm{\ensuremath{\auc_{NoFemale}}}}%sum -\newcommand{\fp}{\ensuremath{\rm{FP}}}%sum -\newcommand{\cd}{\ensuremath{\rm{CD}}}%sum %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -399,14 +249,11 @@ \item nonlinear coding \end{keywords} -\notejb{SI ist ein SNR wenn mit r stimuliert wird! Discuss this when introduced.} % Please keep the abstract below 300 words \section{Abstract} Neuronal processing is inherently nonlinear --- spiking thresholds or rectification at synapses is essential to neuronal computations. Nevertheless, linear response theory has been instrumental in understanding, for example, the impact of noise or neuronal synchrony on signal transmission, or the emergence of oscillatory activity, but is valid only at low stimulus amplitudes or large levels of intrinsic noise. At higher signal-to-noise ratios, however, nonlinear response components become relevant. Theoretical results for leaky integrate-and-fire neurons in the weakly nonlinear regime suggest strong responses at the sum of two input frequencies if these frequencies or their sum match the neuron's baseline firing rate. -We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify the predicted nonlinear responses in low-noise P-units and much stronger in all ampullary cells. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions. - -% Such nonlinear responses boost responses to weak sinusoidal stimuli and are therefore of immediate relevance for wave-type electric fish that are exposed to superpositions of many frequencies in social contexts. +We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify the predicted nonlinear responses in low-noise P-units and much stronger in more than half of the ampullary cells. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions. % Please keep the Author Summary between 150 and 200 words % Use the first person. PLOS ONE authors please skip this step. @@ -428,7 +275,7 @@ We like to think about signal encoding in terms of linear relations with unique The transfer function that describes the linear properties of a system, is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. Including higher-order terms of the Volterra series, the nonlinear nature of spider mechanoreceptors \citep{French2001}, mammalian visual systems \citep{Victor1977, Schanze1997}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated. -Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. \notejg{kann den Gegensatz nicht erkennen..., kann der noise Einschub vielleicht woanders hin oder ist das eine Weiterfuehrung des noise Gedankens? Dann kann das vice versa weg.} Vice versa, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, in which the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where one of two stimulus frequencies equals or both frequencies add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet. \notejg{... one of two stimulus frequencies... includes the case that one equals but the other is something completely different} +Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Also, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, in which the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where one of two stimulus frequencies equals or both frequencies add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet. Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The p-type electroreceptor afferents of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly nonlinear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the receiver's EOD. Nonlinear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation nonlinearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}. @@ -436,16 +283,16 @@ Here we search for such weakly nonlinear responses in electroreceptors of the tw \begin{figure*}[t] \includegraphics[width=\columnwidth]{motivation.pdf} - \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting (cell identifier ``2021-08-03-ac"). Receiver with EOD frequency $\feod{} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both foreign signals have the same strength relative to the own field amplitude (10\,\% contrast). Top row: Sketch of signal processing in the nonlinear system (black box). Second row: Interference of the receiver EOD with the EODs of other fish, bold line highlights the amplitude modulation. Third row: Respective spike trains of the recorded P-unit. Fourth row: Firing rate, estimated by convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: The cell is driven by the self-generated field alone. The baseline firing rate \fbase{} dominates the power spectrum of the firing rate ($f_{base} = 139$\,Hz). \figitem{B} The receiver's EOD and a foreign fish with an EOD frequency $f_{1}=631$\,Hz are present. EOD interference induces an amplitude modulation, referred to as beat. \figitem{C} The receiver and a fish with an EOD frequency $f_{2}=797$\,Hz are present. The resulting beat is faster as the difference between the individual frequencies is larger. \figitem{D} All three fish with the EOD frequencies \feod{}, $f_{1}$ and $f_{2}$ are present, a second-order amplitude modulation occurs, commonly referred to as envelope. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate. + \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting (cell identifier ``2021-08-03-ac"). Receiver with EOD frequency $f_{\rm EOD} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both foreign signals have the same strength relative to the own field amplitude (10\,\% contrast). Top row: Sketch of signal processing in the nonlinear system (black box). Second row: Interference of the receiver EOD with the EODs of other fish, bold line highlights the amplitude modulation. Third row: Respective spike trains of the recorded P-unit. Fourth row: Firing rate, estimated by convolution of the spike trains with a Gaussian kernel ($\sigma = 1$\,ms). Bottom row: Power spectrum of the firing rate. \figitem{A} Baseline condition: The cell is driven by the self-generated field alone. The baseline firing rate $r$ dominates the power spectrum of the firing rate ($f_{base} = 139$\,Hz). \figitem{B} The receiver's EOD and a foreign fish with an EOD frequency $f_{1}=631$\,Hz are present. EOD interference induces an amplitude modulation, referred to as beat. \figitem{C} The receiver and a fish with an EOD frequency $f_{2}=797$\,Hz are present. The resulting beat is faster as the difference between the individual frequencies is larger. \figitem{D} All three fish with the EOD frequencies $f_{\rm EOD}$, $f_1$ and $f_2$ are present. A second-order amplitude modulation occurs, commonly referred to as envelope. Nonlinear peaks occur at the sum and difference of the two beat frequencies in the power spectrum of the firing rate. } \end{figure*} -Theoretical work on leaky integrate-and-fire and conductance-based models suggests a distinct structure of the second-order response function for neurons with low levels of intrinsic noise driven in the supra-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we explored a large set of recordings of P-units and ampullary cells of the active and passive electrosensory systems of the brown ghost knifefish \textit{Apteronotus leptorhynchus} to search for such weakly nonlinear responses in real neurons. Additional simulations of LIF-based models of P-unit spiking put the experimental findings into context. We start with demonstrating the basic concepts using example P-units and models. \notejg{relativ viel Wiederholung des ob gesagten, kuerzen?} +We explored a large set of recordings of P-units and ampullary cells of the active and passive electrosensory systems of the brown ghost knifefish \textit{Apteronotus leptorhynchus} to search for weakly nonlinear responses as they have been predicted by theoretical work \citep{Voronenko2017}. Additional simulations of LIF-based models of P-unit spiking put the experimental findings into context. We start with demonstrating the basic concepts using example P-units and models. \subsection{Nonlinear responses in P-units stimulated with two frequencies} -Without external stimulation, a P-unit is driven by the fish's own EOD alone (with a specific EOD frequency \feod{}) and spontaneously fires action potentials at the baseline rate \fbase{}. Accordingly, the power spectrum of the baseline activity has a peak at \fbase{} (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a} and consequently the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is associated with the clipping of the P-unit's firing rate at zero \citep{Barayeu2023}. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - \feod > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note, $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate. +Without external stimulation, a P-unit is driven by the fish's own EOD alone (with a specific EOD frequency $f_{\rm EOD}$) and spontaneously fires action potentials at the baseline rate $r$. Accordingly, the power spectrum of the baseline activity has a peak at $r$ (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - f_{\rm EOD}$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a} and consequently the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is associated with the clipping of the P-unit's firing rate at zero \citep{Barayeu2023}. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - f_{\rm EOD} > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note, $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate. -When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum $\Delta f_1 + \Delta f_2$ and the difference frequency $\Delta f_2 - \Delta f_1$ (\subfigrefb{fig:motivation}{D}). Thus, the cellular response is not equal to the sum of the responses to the two beats presented separately. These additional peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that, by definition, are absent in linear systems\notejg{should not occur?}. +When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum $\Delta f_1 + \Delta f_2$ and the difference frequency $\Delta f_2 - \Delta f_1$ (\subfigrefb{fig:motivation}{D}). Thus, the cellular response is not equal to the sum of the responses to the two beats presented separately. These additional peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that, by definition, are absent in linear systems. \subsection{Linear and weakly nonlinear regimes} @@ -468,19 +315,19 @@ In the following, however, we are interested in how the nonlinear responses depe \subsection{Nonlinear signal transmission in P-units} -P-units fire action potentials probabilistically phase-locked to the self-generated EOD \citep{Bastian1981a}. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:punit}{A}). In this example, the baseline ISI distribution has a CV$_{\text{base}}$ of 0.49, which is at the center of the P-unit population \citep{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks: the first is located at the baseline firing rate $r$, the second is located at the discharge frequency \feod{} of the electric organ (\subfigref{fig:punit}{B}). +P-units fire action potentials probabilistically phase-locked to the self-generated EOD \citep{Bastian1981a}. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:punit}{A}). In this example, the baseline ISI distribution has a CV$_{\text{base}}$ of 0.49, which is at the center of the P-unit population \citep{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks: the first is located at the baseline firing rate $r$, the second is located at the discharge frequency $f_{\rm EOD}$ of the electric organ (\subfigref{fig:punit}{B}). \begin{figure*}[p] \includegraphics[width=\columnwidth]{punitexamplecell} - \caption{\label{fig:punit} Linear and nonlinear stimulus encoding in example P-units. \figitem{A} Interspike interval (ISI) distribution of a cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field (cell identifier ``2020-10-27-ag''). This cell has a rather high baseline firing rate $r=405$\,Hz and an intermediate CV$_{\text{base}}=0.49$ of its interspike intervals. \figitem{B} Power spectral density of the cell's baseline response with marked peaks at the cell's baseline firing rate $r$ and the fish's EOD frequency $f_{\text{EOD}}$. \figitem{C} Random amplitude modulation (RAM) stimulus (top, red, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit for two different stimulus contrasts (right). The stimulus contrast quantifies the standard deviation of the RAM relative to the fish's EOD amplitude. \figitem{D} Gain of the transfer function (first-order susceptibility), \eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light blue) and 20\,\% contrast (dark blue) RAM stimulation of 5\,s duration. \figitem{E} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both the low and high stimulus contrast. At the lower stimulus contrast an anti-diagonal where the sum of the two stimulus frequencies equals the neuron's baseline frequency clearly sticks out of the noise floor. \figitem{F} At the higher contrast, the anti-diagonal is much weaker. \figitem{G} Second-order susceptibilities projected onto the diagonal (averages over all anti-diagonals of the matrices shown in \panel{E, F}). The anti-diagonals from \panel{E} and \panel{F} show up as a peak close to the cell's baseline firing rate $r$. The susceptibility index, SI($r$), quantifies the height of this peak relative to the values in the vicinity \notejb{See equation XXX}. \figitem{H} ISI distributions (top) and second-order susceptibilities (bottom) of two more example P-units (``2021-06-18-ae'', ``2017-07-18-ai'') showing an anti-diagonal, but not the full expected triangular structure. \figitem{I} Most P-units, however, have a flat second-order susceptibility and consequently their SI($r$) values are close to one (cell identifiers ``2018-08-24-ak'', ``2018-08-14-ac'').} + \caption{\label{fig:punit} Linear and nonlinear stimulus encoding in example P-units. \figitem{A} Interspike interval (ISI) distribution of a cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field (cell identifier ``2020-10-27-ag''). This cell has a rather high baseline firing rate $r=405$\,Hz and an intermediate CV$_{\text{base}}=0.49$ of its interspike intervals. \figitem{B} Power spectral density of the cell's baseline response with marked peaks at the cell's baseline firing rate $r$ and the fish's EOD frequency $f_{\text{EOD}}$. \figitem{C} Random amplitude modulation (RAM) stimulus (top, red, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit for two different stimulus contrasts (right). The stimulus contrast quantifies the standard deviation of the RAM relative to the fish's EOD amplitude. \figitem{D} Gain of the transfer function (first-order susceptibility), \eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light blue) and 20\,\% contrast (dark blue) RAM stimulation of 5\,s duration. \figitem{E} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both the low and high stimulus contrast. At the lower stimulus contrast an anti-diagonal where the sum of the two stimulus frequencies equals the neuron's baseline frequency clearly sticks out of the noise floor. \figitem{F} At the higher contrast, the anti-diagonal is much weaker. \figitem{G} Second-order susceptibilities projected onto the diagonal (averages over all anti-diagonals of the matrices shown in \panel{E, F}). The anti-diagonals from \panel{E} and \panel{F} show up as a peak close to the cell's baseline firing rate $r$. The susceptibility index, SI($r$) \eqnref{siindex}, quantifies the height of this peak relative to the values in the vicinity. \figitem{H} ISI distributions (top) and second-order susceptibilities (bottom) of two more example P-units (``2021-06-18-ae'', ``2017-07-18-ai'') showing an anti-diagonal, but not the full expected triangular structure. \figitem{I} Most P-units, however, have a flat second-order susceptibility and consequently their SI($r$) values are close to one (cell identifiers ``2018-08-24-ak'', ``2018-08-14-ac'').} \end{figure*} Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:punit}{C}, top trace, red line), have been commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate from existing recordings the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with 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 blue for low and high contrast stimuli, respectively, \subfigrefb{fig:punit}{C}). Linear encoding, quantified by the first-order susceptibility or transfer function, \eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:punit}{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 \citep{Benda2005}. -The second-order susceptibility, \eqnref{chi2}, quantifies for each combination of two stimulus frequencies \fone{} and \ftwo{} the amplitude and phase of the stimulus-evoked response at the sum \fsum{} (and also the difference, \subfigrefb{fig:model_full}{A}). 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 stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:punit}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate $r$ \citep{Voronenko2017,Franzen2023}. 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 (\subfigrefb{fig:lifresponse}{B}). +The second-order susceptibility, \eqnref{chi2}, quantifies for each combination of two stimulus frequencies $f_1$ and $f_2$ the amplitude and phase of the stimulus-evoked response at the sum $f_1 + f_2$ (and also the difference, \subfigrefb{fig:model_full}{A}). 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 stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:punit}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies $f_1$ and $f_2$ or their sum $f_1 + f_2$ exactly match the neuron's baseline firing rate $r$ \citep{Voronenko2017,Franzen2023}. 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 (\subfigrefb{fig:lifresponse}{B}). -For the example P-unit, we observe a ridge of elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal, \subfigrefb{fig:punit}{E}). This structure is less prominent for the stronger stimulus (\subfigref{fig:punit}{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 the susceptibility values onto the diagonal (white dashed line) by averaging over the anti-diagonals (\subfigrefb{fig:punit}{G}). At low RAM contrast this projection indeed has a distinct peak close to the neuron's baseline firing rate (\subfigrefb{fig:punit}{G}, dot on top line). For the higher RAM contrast this peak is much smaller and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:punit}{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 \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease. +For the example P-unit, we observe a ridge of elevated second-order susceptibility for the low RAM contrast at $f_1 + f_2 = r$ (yellowish anti-diagonal, \subfigrefb{fig:punit}{E}). This structure is less prominent for the stronger stimulus (\subfigref{fig:punit}{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 the susceptibility values onto the diagonal (white dashed line) by averaging over the anti-diagonals (\subfigrefb{fig:punit}{G}). At low RAM contrast this projection indeed has a distinct peak close to the neuron's baseline firing rate (\subfigrefb{fig:punit}{G}, dot on top line). For the higher RAM contrast this peak is much smaller and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:punit}{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 \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease. Overall we observed in 17\,\% of the 159 P-units ridges where the stimulus frequencies add up to the unit's baseline firing rate. Two more examples are shown in \subfigref{fig:punit}{H}. However, we never observed the full triangular structure expected from theory (\subfigrefb{fig:lifresponse}{B}). In all other P-units, we did not observe any structure in the second-order susceptibility (\subfigrefb{fig:punit}{I}). @@ -492,7 +339,7 @@ Overall we observed in 17\,\% of the 159 P-units ridges where the stimulus frequ \caption{\label{fig:ampullary} Linear and nonlinear stimulus encoding in example ampullary afferents. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity (cell identifier ``2012-05-15-ac''). The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. Ampullary afferents do not respond to the fish's EOD frequency, $f_{\text{EOD}}$ --- a sharp peak at $f_{\text{EOD}}$ is missing. \figitem{C} Band-limited white noise stimulus (top, red, with a cutoff frequency of 150\,Hz) added to the fish's self-generated electric field (no amplitude modulation!) and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \eqnref{linearencoding_methods}, of the responses to stimulation with 5\,\% (light green) and 10\,\% contrast (dark green) of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both stimulus contrasts as indicated. Both show a clear anti-diagonal where the two stimulus frequencies add up to the afferent's baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. \figitem{H} ISI distributions (top) and second-order susceptibilities (bottom) of three more example afferents with clear anti-diagonals (``2010-11-26-an'', ``2010-11-08-aa'', ``2011-02-18-ab''). \figitem{I} Some ampullary afferents do not show any structure in their second-order susceptibility (``2014-01-16-aj'').} \end{figure*} -Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs ($0.06 < \text{CV}_{\text{base}} < 0.22$, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the high-frequency EOD and the ISIs have an unimodal distribution (\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 $r$ and its harmonics. Since the cells do not respond to the self-generated EOD, there is no sharp peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a band-limited white noise stimulus (note: for ampullary afferents this is not an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary afferents exhibit very pronounced ridges in the second-order susceptibility, where $f_1 + f_2$ is equal to $r$ or its harmonics (yellow anti-diagonals in \subfigrefb{fig:ampullary}{E--H}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands get weaker (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at $r$, and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). Some ampullary afferents (27\,\% of 30 afferents), however, do not show any such structure in their second-order susceptibility (\subfigrefb{fig:ampullary}{I}). +Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs ($0.06 < \text{CV}_{\text{base}} < 0.22$, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the high-frequency EOD and the ISIs have an unimodal distribution (\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 $r$ and its harmonics. Since the cells do not respond to the self-generated EOD, there is no sharp peak at $f_{\rm EOD}$ (\subfigrefb{fig:ampullary}{B}). When driven by a band-limited white noise stimulus (note: for ampullary afferents this is not an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary afferents exhibit very pronounced ridges in the second-order susceptibility, where $f_1 + f_2$ is equal to $r$ or its harmonics (yellow anti-diagonals in \subfigrefb{fig:ampullary}{E--H}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands get weaker (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at $r$, and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). Some ampullary afferents (27\,\% of 30 afferents), however, do not show any such structure in their second-order susceptibility (\subfigrefb{fig:ampullary}{I}). \subsection{Model-based estimation of the second-order susceptibility} @@ -500,7 +347,7 @@ In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampull \begin{figure*}[p] \includegraphics[width=\columnwidth]{noisesplit} - \caption{\label{fig:noisesplit} Estimation of second-order susceptibilities. \figitem{A} \suscept{} (right) estimated from $N=198$ 256\,ms long FFT segments of an electrophysiological recording of another P-unit (cell ``2017-07-18-ai'', $r=78$\,Hz, CV$_{\text{base}}=0.22$) driven with a RAM stimulus with contrast 5\,\% (left). \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell ``2017-07-18-ai'') based on the same RAM contrast and number of $N=100$ FFT segments. As in the electrophysiological recording only a weak anti-diagonal is visible. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ FFT segments. Now, the expected triangular structure is revealed. \figitem[iv]{B} Convergence of the \suscept{} estimate as a function of FFT segments. \figitem{C} At a lower stimulus contrast of 1\,\% the estimate did not converge yet even for $10^6$ FFT segments. The triangular structure is not revealed yet. \figitem[i]{D} Same as in \panel[i]{B} but in the \textit{noise split} condition: there is no external RAM signal (red) driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as a signal and is presented as an equivalent amplitude modulation ($s_{\xi}(t)$, orange, 10.6\,\% contrast), while the intrinsic noise is reduced to 10\,\% of its original strength (bottom, see methods for details). \figitem[i]{D} 100 FFT segments are still not sufficient for estimating \suscept{}. \figitem[iii]{D} Simulating one million segments reveals the full expected triangular structure of the second-order susceptibility. \figitem[iv]{D} In the noise-split condition, the \suscept{} estimate converges already at about $10^{4}$ FFT segments.} + \caption{\label{fig:noisesplit} Estimation of second-order susceptibilities. \figitem{A} $|\chi_2(f_1, f_2)|$ (right) estimated from $N=100$ 256\,ms long FFT segments of an electrophysiological recording of another P-unit (cell ``2017-07-18-ai'', $r=78$\,Hz, CV$_{\text{base}}=0.22$) driven with a RAM stimulus with contrast 5\,\% (left). \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} $|\chi_2(f_1, f_2)|$ estimated from simulations of the cell's LIF model counterpart (cell ``2017-07-18-ai'') based on the same RAM contrast and number of $N=100$ FFT segments. As in the electrophysiological recording only a weak anti-diagonal is visible. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ FFT segments. Now, the expected triangular structure is revealed. \figitem[iv]{B} Convergence of the $|\chi_2(f_1, f_2)|$ estimate as a function of FFT segments. \figitem{C} At a lower stimulus contrast of 1\,\% the estimate did not converge yet even for $10^6$ FFT segments. The triangular structure is not revealed yet. \figitem[i]{D} Same as in \panel[i]{B} but in the \textit{noise split} condition: there is no external RAM signal (red) driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as a signal and is presented as an equivalent amplitude modulation ($s_{\xi}(t)$, orange, 10.6\,\% contrast), while the intrinsic noise is reduced to 10\,\% of its original strength (bottom, see methods for details). \figitem[i]{D} 100 FFT segments are still not sufficient for estimating $|\chi_2(f_1, f_2)|$. \figitem[iii]{D} Simulating one million segments reveals the full expected triangular structure of the second-order susceptibility. \figitem[iv]{D} In the noise-split condition, the $|\chi_2(f_1, f_2)|$ estimate converges already at about $10^{4}$ FFT segments.} \end{figure*} One simple reason could be the lack of data, i.e. the estimation of the second-order susceptibility is not good enough. Electrophysiological recordings are limited in time, and therefore only a limited number of trials, here repeated presentations of the same frozen RAM stimulus, are available. In our data set we have 1 to 199 trials (median: 10) of RAM stimuli with a duration ranging from 2 to 5\,s (median: 8\,s), resulting in a total duration of 30 to 400\,s. Using a resolution of 0.5\,ms and FFT segments of 512 samples this yields 105 to 1520 available FFT segments for a specific RAM stimulus. As a consequence, the cross-spectra, \eqnref{crossxss}, 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, dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \citep{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of the P-unit estimated from the same low number of FFT (fast fourier transform) segments as in the experiment ($N=100$, compare faint anti-diagonal in the bottom left corner of the second-order susceptibility in \panel[ii]{A} and \panel[ii]{B} in \figrefb{fig:noisesplit}). @@ -509,7 +356,7 @@ In model simulations we can increase the number of FFT segments beyond what woul At a lower stimulus contrast of 1\,\% (\subfigrefb{fig:noisesplit}{C}), however, one million FFT segments are still not sufficient for the estimate to converge (\figrefb{fig:noisesplit}\,\panel[iv]{C}). Still only a faint anti-diagonal is visible (\figrefb{fig:noisesplit}\,\panel[iii]{C}). -Using a broadband stimulus increases the effective input-noise level. This may linearize signal transmission and suppress potential nonlinear responses \citep{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. As we just have seen, 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 \citep{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{crossxss} and \eqref{chi2}) \citep{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \citep{Lindner2022}, we can split the total noise and consider a fraction of it as a stimulus. This allows us 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 $\sqrt{2D \, c_{\rm{noise}}}\;\xi(t)$ with $c_\text{noise} = 0.1$ \notejb{$c$ is already used for contrast} (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \figrefb{fig:noisesplit}\,\panel[i]{D}). We tuned the amplitude of the RAM stimulus $s_{\xi}(t)$ such that the output firing rate and variability (CV of interspike intervals) 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 and thus the estimate converges already at about ten thousand FFT segments (\figrefb{fig:noisesplit}\,\panel[iv]{D}); (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 FFT segments (\figrefb{fig:noisesplit}\,\panel[iii]{D}), but not for a number of segments comparable to the experiment (\figrefb{fig:noisesplit}\,\panel[ii]{D}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \figrefb{fig:noisesplit}\,\panel[iii]{D}). +Using a broadband stimulus increases the effective input-noise level. This may linearize signal transmission and suppress potential nonlinear responses \citep{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. As we just have seen, 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 \citep{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{crossxss} and \eqref{chi2}) \citep{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \citep{Lindner2022}, we can split the total noise and consider a fraction of it as a stimulus. This allows us 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 $\sqrt{2D \, \alpha_{\rm noise}}\;\xi(t)$ with $\alpha_{\rm noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \figrefb{fig:noisesplit}\,\panel[i]{D}). We tuned the amplitude of the RAM stimulus $s_{\xi}(t)$ such that the output firing rate and variability (CV of interspike intervals) 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 $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved and thus the estimate converges already at about ten thousand FFT segments (\figrefb{fig:noisesplit}\,\panel[iv]{D}); (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 FFT segments (\figrefb{fig:noisesplit}\,\panel[iii]{D}), but not for a number of segments comparable to the experiment (\figrefb{fig:noisesplit}\,\panel[ii]{D}). In addition to the strong response at $f_1 + f_2 = r$, we now also observe pronounced nonlinear responses at $f_1=r$ and $f_2=r$ (vertical and horizontal lines, \figrefb{fig:noisesplit}\,\panel[iii]{D}). \begin{figure}[p] @@ -605,13 +452,13 @@ The population of ampullary cells is generally more homogeneous, with lower base \begin{figure*}[t] \includegraphics[width=\columnwidth]{model_full.pdf} - \caption{\label{fig:model_full} Second-order susceptibility qualitatively predicts responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ FFT segments of model simulations in the noise-split condition (cell ``2013-01-08-aa''). Dashed white lines mark zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies. \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 2\,\%. Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.} + \caption{\label{fig:model_full} Second-order susceptibility qualitatively predicts responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both positive and negative frequencies. $|\chi_2|(f_1, f_2)$ was estimated from $N=10^6$ FFT segments of model simulations in the noise-split condition (cell ``2013-01-08-aa''). Dashed white lines mark zero frequencies. Nonlinear responses at $f_1 + f_2$ are quantified in the upper right and lower left quadrants. Nonlinear responses at $f_2 - f_1$ are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $r=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies. \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves with frequencies $f_1$ and $f_2$, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 2\,\%. Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match $r$. \figitem{C} The difference between $f_2$ and $f_1$ matches $r$. \figitem{D} Only the first beat frequency matches $r$. \figitem{E} None of the two beat frequencies matches $r$.} \end{figure*} \subsection{Second-order susceptibility explains nonlinear responses to pure sinewave stimuli} Using the RAM stimulation we found pronounced nonlinear responses, in particular in the limit to weak stimulus amplitudes. How do these findings relate to the situation of pure sine wave stimuli with finite amplitudes that approximate the interference of EODs of real animals? For the P-units, the relevant signals are the beat frequencies $f_1$ and $f_2$ 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, $f_2$ was similar to the baseline firing rate, corresponding to the horizontal ridge of the second-order susceptibility (\subfigrefb{fig:lifresponse}{B}). The difference frequency is not covered by the so-far shown part of the second-order susceptibility, in which only the response at the sum of the two stimulus frequencies is addressed. -However, the second-order susceptibility \eqnref{chi2} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full $\chi_2$ matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at the sum $f_1 + f_2$ are shown, whereas in the lower-right and upper-left quadrants responses at the difference $f_2 - f_1$ are shown (\figref{fig:model_full}). The vertical and horizontal lines at $f_1 = r$ and $f_2 = r$ are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} and extend into the upper-left quadrant, where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at $f_2 - f_1$ 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}, \citealp{Schlungbaum2023}). +However, the second-order susceptibility \eqnref{chi2} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full $\chi_2$ matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of $|\chi_2|(f_1, f_2)$, stimulus-evoked responses at the sum $f_1 + f_2$ are shown, whereas in the lower-right and upper-left quadrants responses at the difference $f_2 - f_1$ are shown (\figref{fig:model_full}). The vertical and horizontal lines at $f_1 = r$ and $f_2 = r$ are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} and extend into the upper-left quadrant, where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at $f_2 - f_1$ 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}, \citealp{Schlungbaum2023}). Is it possible to predict nonlinear responses in a three-fish setting based on second-order susceptibility matrices estimated from RAM stimulation in the noise-split configuration (\subfigrefb{fig:model_full}{A})? We test this by stimulating the same model with two weak beats (\subfigrefb{fig:model_full}{B--E}). Qualitatively, the second-order susceptibility predicts the presence and absence of peaks in the response spectrum at the sums and differences of the two beat frequencies. However, a quantitative prediction fails. This is because pure sine waves influence a nonlinear system in different ways than a white-noise stimulus. This will be addressed in detail in a future manuscript. @@ -649,9 +496,6 @@ Noise stimuli have the advantage that a range of frequencies can be measured wit We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, pure sine wave stimulation is spectrally focused and drives the system on the background of the intrinsic noise. This explains why we can observe nonlinear interactions between sine wave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{fig:noisesplit}). -%As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals. -%We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here, in accordance with the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies. - Extracting the AM from the stimulus already requires a (threshold) nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. This nonlinearity, however, does not show up in our estimates of the susceptibilities, because in our analysis we directly relate the AM waveform to the recorded cellular responses. Encoding the time-course of the AM, however, has been shown to be linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. In contrast, we here have demonstrated nonlinear interactions originating from the spike generator for broad-band noise stimuli with small amplitudes and for stimulation with two distinct frequencies. Both settings have not been studied yet. The encoding of secondary or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires an additional nonlinearity in the system that was initially attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information whenever the firing rate saturates at zero or the maximum rate at the EOD frequency \citep{Savard2011}. Based on our work, we predict that P-units with low CVs encode the social envelopes even under weak stimulation, whenever the resulting beat frequencies match or add up to the baseline firing rate. Then difference frequencies show up in the response spectrum that characterize the slow envelope. @@ -664,8 +508,8 @@ We have demonstrated pronounced nonlinear responses in primary electrosensory af \section{Acknowledgements} -\notejb{Supported by DFG SPP XXX} -Tim Hladnik, Henriette Walz, Franziska Kuempfbeck, Fabian Sinz, Laura Seidler, Eva Vennemann, and Ibrahim Tunc recorded data. +Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by the DFG, project number 430157666. +We thank Tim Hladnik, Henriette Walz, Franziska Kuempfbeck, Fabian Sinz, Laura Seidler, Eva Vennemann, and Ibrahim Tunc for the data they recorded over the years in our lab \section{Methods} @@ -698,7 +542,7 @@ The fish were stimulated with band-limited white noise stimuli with a cut-off fr The baseline firing rate $r$ was calculated as the number of spikes divided by the duration of the baseline recording (median 32\,s). The coefficient of variation (CV) of the interspike intervals (ISI) is their standard deviation relative to their mean: $\rm{CV}_{\rm base} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle} / \langle ISI \rangle$. If the baseline was recorded several times in a recording, the measures from the longest recording were taken. \paragraph{White noise analysis} \label{response_modulation} -When stimulated with band-limited white noise stimuli, neuronal activity is modulated around the average firing rate that is similar to the baseline firing rate and in that way encodes the time-course of the stimulus. For an estimate of the time-dependent firing rate $r(t)$ we convolved each spike train with normalized Gaussian kernels with standard deviation of 2\,ms and averaged the resulting single-trail firing rates over trials. The response modulation quantifies the variation of $r(t)$ computed as the standard deviation in time $\sigma_{s} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t )^2\rangle_t}$, where $\langle \cdot \rangle_t$ denotes averaging over time. +When stimulated with band-limited white noise stimuli, neuronal activity is modulated around the average firing rate that is similar to the baseline firing rate and in that way encodes the time-course of the stimulus. For an estimate of the time-dependent firing rate $r(t)$ we convolved each spike train with normalized Gaussian kernels with standard deviation of 2\,ms, if not mentioned otherwise, and averaged the resulting single-trail firing rates over trials. The response modulation quantifies the variation of $r(t)$ computed as the standard deviation in time $\sigma_{s} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t )^2\rangle_t}$, where $\langle \cdot \rangle_t$ denotes averaging over time. \paragraph{Spectral analysis}\label{susceptibility_methods} To characterize the relation between the spiking response evoked by white-noise stimuli, we estimated the first- and second-order susceptibilities in the frequency domain. For this we converted spike times into binary vectors $x_k$ with $\Delta t = 0.5$\,ms wide bins that are set to 2\,kHz where a spike occurred and zero otherwise. Fast Fourier transforms (FFT) $S(\omega)$ and $X(\omega)$ of the stimulus $s_k$ (also down-sampled to a sampling rate of 2\,kHz) and $x_k$, respectively, were computed numerically according to @@ -750,12 +594,12 @@ Throughout the manuscript we only show the absolute values of the complex-valued We expected to see a sharp ridge in the second-order susceptibility at $\omega_1 + \omega_2 = r$ \citep{Voronenko2017,Franzen2023}. To characterize this in a single number we computed a susceptibility index. First, we projected the absolute values of the second-order susceptibility matrix onto the diagonal by averaging over anti-diagonal elements. In this projection $D(f)$ we took the position of the maximum \begin{equation} \label{dmax} - f_{\rm peak} = {\rm argmax} D(\fbase{}-50\,\rm{Hz} \leq f \leq r + 50\,\rm{Hz}) + f_{\rm peak} = {\rm argmax} D(r - 50\,\rm{Hz} \leq f \leq r + 50\,\rm{Hz}) \end{equation} within $\pm 50$\,Hz of the neuron's baseline firing rate $r$ as the position of the expected peak. For an estimate of the noise-floor surrounding this peak we averaged over 10\,Hz wide windows 10\,Hz to the left and right of the peak: \begin{equation} \label{dref} - D_{\rm ref} = \frac{1}{2}\left(\langle D(f_{\rm peak}-20\,\rm{Hz} \leq f \leq f_{\rm peak}-10\,\rm{Hz})\rangle_f + \langle D(f_{\rm peak}+10\,\rm{Hz} \leq f \leq f_{\rm peak}+20\,\rm{Hz}))\rangle_f\right) + D_{\rm ref} = \frac{1}{2}\left(\langle D(f_{\rm peak}-20\,\rm{Hz} \leq f \leq f_{\rm peak}-10\,\rm{Hz})\rangle_f + \langle D(f_{\rm peak}+10\,\rm{Hz} \leq f \leq f_{\rm peak}+20\,\rm{Hz}))\rangle_f \right) \end{equation} The size of the peak relative to this reference is then the susceptibility index \begin{equation} @@ -768,7 +612,7 @@ Values larger than one indicate a sharp ridge in the susceptibility matrix close \begin{figure*}[t] \includegraphics[width=\columnwidth]{flowchart.pdf} \caption{\label{flowchart} - Architecture of the P-unit model. Each row illustrates subsequent processing steps for three different stimulation regimes: (i) baseline activity without external stimulus, only the fish's self-generated EOD (the carrier, \eqnref{eq:eod}) is present. (ii) RAM stimulation, \eqnref{eq:ram_equation}. The amplitude of the EOD carrier is modulated with a weak (2\,\% contrast) band-limited white-noise stimulus. (iii) Noise split, \eqnsref{eq:ram_split}--\eqref{eq:Noise_split_intrinsic}, where 90\,\% of the intrinsic noise is replaced by a RAM stimulus, whose amplitude is scaled to maintain the mean firing rate and the CV of the ISIs of the model's baseline activity. As an example, simulations of the model for cell ``2012-07-03-ak'' are shown. \figitem{A} The stimuli are thresholded, \eqnref{eq:threshold2}, by setting all negative values to zero. \figitem{B} Subsequent low-pass filtering, \eqnref{eq:dendrite}, attenuates the carrier and carves out the AM signal. \figitem{C} Intrinsic Gaussian white-noise is added to the signals shown in \panel{B}. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model, \eqnsref{eq:LIF}--\eqref{spikethresh}, in response to the sum of \panel{B} and \panel{C}. \figitem{E} Power spectra of the LIF neuron's spiking activity. Both, baseline activity (\panel[i]{E}) and noise split (\panel[iii]{E}), have the same peaks in the response spectrum at $\fbase$, $f_{EOD} - \fbase$, $f_{EOD}$, and $f_{EOD} + \fbase$. With RAM stimulation (\panel[ii]{E}), the peak at the baseline firing rate, $\fbase$, is washed out.} + Architecture of the P-unit model. Each row illustrates subsequent processing steps for three different stimulation regimes: (i) baseline activity without external stimulus, only the fish's self-generated EOD (the carrier, \eqnref{eq:eod}) is present. (ii) RAM stimulation, \eqnref{eq:ram_equation}. The amplitude of the EOD carrier is modulated with a weak (2\,\% contrast) band-limited white-noise stimulus. (iii) Noise split, \eqnsref{eq:ram_split}--\eqref{eq:Noise_split_intrinsic}, where 90\,\% of the intrinsic noise is replaced by a RAM stimulus, whose amplitude is scaled to maintain the mean firing rate and the CV of the ISIs of the model's baseline activity. As an example, simulations of the model for cell ``2012-07-03-ak'' are shown. \figitem{A} The stimuli are thresholded, \eqnref{eq:threshold2}, by setting all negative values to zero. \figitem{B} Subsequent low-pass filtering, \eqnref{eq:dendrite}, attenuates the carrier and carves out the AM signal. \figitem{C} Intrinsic Gaussian white-noise is added to the signals shown in \panel{B}. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model, \eqnsref{eq:LIF}--\eqref{spikethresh}, in response to the sum of \panel{B} and \panel{C}. \figitem{E} Power spectra of the LIF neuron's spiking activity. Both, baseline activity (\panel[i]{E}) and noise split (\panel[iii]{E}), have the same peaks in the response spectrum at $r$, $f_{EOD} - r$, $f_{EOD}$, and $f_{EOD} + r$. With RAM stimulation (\panel[ii]{E}), the peak at the baseline firing rate, $r$, is washed out.} \end{figure*} \subsection{Leaky integrate-and-fire models for P-units}\label{lifmethods} @@ -776,14 +620,14 @@ Values larger than one indicate a sharp ridge in the susceptibility matrix close Modified leaky integrate-and-fire (LIF) models were constructed to reproduce the specific firing properties of P-units \citep{Chacron2001, Sinz2020, Barayeu2023}. The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD, modeled as a cosine wave \begin{equation} \label{eq:eod} - \carrierinput = y_{EOD}(t) = \cos(2\pi f_{EOD} t) + y(t) = y_{EOD}(t) = \cos(2\pi f_{EOD} t) \end{equation} with EOD frequency $f_{EOD}$ and an amplitude of one. To mimic the interaction with other fish, the EODs of a second or third fish with EOD frequencies $f_1$ and $f_2$, respectively, were added to the normalized EOD, \eqnref{eq:eod}, of the receiving fish according to their contrasts, $c_1$ and $c_2$ at the position of the receiving fish: \begin{equation} \label{eq:modelbeats} - \carrierinput = \cos(2\pi f_{EOD} t) + c_1 \cos(2\pi f_1 t) + c_2\cos(2\pi f_2 t) + y(t) = \cos(2\pi f_{EOD} t) + c_1 \cos(2\pi f_1 t) + c_2\cos(2\pi f_2 t) \end{equation} For two fish, $c_2 = 0$. @@ -794,15 +638,15 @@ Random amplitude modulations (RAMs) were simulated by first generating the AM as \end{equation} The contrast $c$ of the RAM is the standard deviation of the RAM relative to the amplitude of the receiving fish. -First, the input \carrierinput{} is thresholded by setting negative values to zero: +First, the input $y(t)$ is thresholded by setting negative values to zero: \begin{equation} \label{eq:threshold2} - \lfloor \carrierinput \rfloor_0 = \left\{ \begin{array}{rcl} \carrierinput & ; & \carrierinput \ge 0 \\ 0 & ; & \carrierinput < 0 \end{array} \right. + \lfloor y(t) \rfloor_0 = \left\{ \begin{array}{rcl} y(t) & ; & y(t) \ge 0 \\ 0 & ; & y(t) < 0 \end{array} \right. \end{equation} (\subfigrefb{flowchart}{A}). This thresholds models the transfer function of the synapses between the primary receptor cells and the afferent. Together with a low-pass filter \begin{equation} \label{eq:dendrite} - \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor \carrierinput \rfloor_{0} + \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor y(t) \rfloor_{0} \end{equation} the threshold operation is required for extracting the amplitude modulation from the input \citep{Barayeu2024}. The low-pass filter models passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B}) and $\tau_{d}$ is the membrane time constant of the dendrite. Dendritic low-pass filtering was also necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. @@ -833,14 +677,14 @@ The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\t \subsection{Noise split} \label{intrinsicsplit_methods} -Based on the Furutsu-Novikov theorem \citep{Furutsu1963,Novikov1965,Lindner2022,Egerland2020}, we split the total noise, $\sqrt{2D}\;\xi(t)$, of a LIF model, \eqnref{eq:LIF}, into two parts. The first part is the intrinsic noise term, $\sqrt{2D \, c_{\rm{noise}}}\;\xi(t)$, whose strength is reduced by a factor $c_{\rm{noise}}=0.1$ (\subfigrefb{flowchart}\,\panel[iii]{C}). The second part replaces the now missing intrinsic noise by a driving input signal $s_{\xi}(t)$, a RAM stimulus with frequencies up to 300\,Hz (\subfigrefb{flowchart}\,\panel[iii]{A}). The LIF model with splitted noise then reads +Based on the Furutsu-Novikov theorem \citep{Furutsu1963,Novikov1965,Lindner2022,Egerland2020}, we split the total noise, $\sqrt{2D}\;\xi(t)$, of a LIF model, \eqnref{eq:LIF}, into two parts. The first part is the intrinsic noise term, $\sqrt{2D \, \alpha_{\rm noise}}\;\xi(t)$, whose strength is reduced by a factor $\alpha_{\rm noise}=0.1$ (\subfigrefb{flowchart}\,\panel[iii]{C}). The second part replaces the now missing intrinsic noise by a driving input signal $s_{\xi}(t)$, a RAM stimulus with frequencies up to 300\,Hz (\subfigrefb{flowchart}\,\panel[iii]{A}). The LIF model with noise split then reads \begin{eqnarray} \label{eq:ram_split} y(t) & = & (1+ s_\xi(t)) \cos(2\pi f_{EOD} t) \\ \label{eq:Noise_split_intrinsic_dendrite} \tau_{d} \frac{d V_{d}}{d t} & = & -V_{d}+ \lfloor y(t) \rfloor_{0} \\ \label{eq:Noise_split_intrinsic} - \tau_{m} \frac{d V_{m}}{d t} & = & - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{\rm{noise}}}\;\xi(t) + \tau_{m} \frac{d V_{m}}{d t} & = & - V_{m} + \mu + \beta V_{d} - A + \sqrt{2D \, \alpha_{\rm noise}}\;\xi(t) \end{eqnarray} Both, the reduced intrinsic noise and the RAM stimulus, need to replace the original intrinsic noise. Because the RAM stimulus is band-limited and undergoes some transformations before it is added to the reduced intrinsic noise, it is not \textit{a priori} clear, what the amplitude of the RAM should be. We bisected the amplitude of $s_\xi(t)$, until the CV of the resulting interspike intervals matched the one of the original model's baseline activity. The second-order cross-spectra, \eqnref{crossxss}, were computed between the RAM stimulus $s_{\xi}(t)$ and the spike train $x(t)$ it evoked. In this way, the effective signal-to-noise ratio can be increased while maintaining the total noise in the system.