From d4d8011953ef983aace69012a1725dccb3a358e2 Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Tue, 17 Mar 2026 18:31:24 +0100 Subject: [PATCH] added doi for data repository --- nonlinearbaseline.tex | 7 +- submission/figure01.pdf | Bin 0 -> 27915 bytes submission/figure02.pdf | Bin 0 -> 217504 bytes submission/figure03.pdf | Bin 0 -> 34557 bytes submission/figure04.pdf | Bin 0 -> 88670 bytes submission/figure05.pdf | Bin 0 -> 387462 bytes submission/figure06.pdf | Bin 0 -> 335437 bytes submission/figure07.pdf | Bin 0 -> 94025 bytes submission/figure08.pdf | Bin 0 -> 154282 bytes submission/figure09.pdf | Bin 0 -> 79524 bytes submission/figure10.pdf | Bin 0 -> 298346 bytes submission/manuscript.tex | 1090 +++++++++++++++++++++++++++++++++++++ 12 files changed, 1094 insertions(+), 3 deletions(-) create mode 100644 submission/figure01.pdf create mode 100644 submission/figure02.pdf create mode 100644 submission/figure03.pdf create mode 100644 submission/figure04.pdf create mode 100644 submission/figure05.pdf create mode 100644 submission/figure06.pdf create mode 100644 submission/figure07.pdf create mode 100644 submission/figure08.pdf create mode 100644 submission/figure09.pdf create mode 100644 submission/figure10.pdf create mode 100644 submission/manuscript.tex diff --git a/nonlinearbaseline.tex b/nonlinearbaseline.tex index 7d611d0..106538c 100644 --- a/nonlinearbaseline.tex +++ b/nonlinearbaseline.tex @@ -293,8 +293,8 @@ Electric stimuli were attenuated (ATN-01M, \secret{npi-electronics, Tamm, German \subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, nixio \citep{Stoewer2014}, and thunderlab (\secret{\url{https://github.com/bendalab/thunderlab}}). -\paragraph{Code accessibility} -The P-unit model parameters and spectral analysis algorithms are available at \secret{\url{https://github.com/bendalab/punitmodel/tree/v1}}. +\paragraph{Code and data accessibility} +The P-unit model parameters and spectral analysis algorithms are available at \secret{\url{https://github.com/bendalab/punitmodel/tree/v1}}. The electrophysiological data of all the P-units and Ampullary cells analyzed here, the analysis scripts, and the scripts simulating the P-unit model responses are accessible at \secret{\url{https://doi.org/10.12751/g-node.4w8ebq}}. \paragraph{Baseline analysis}\label{baselinemethods} 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. @@ -673,7 +673,8 @@ We have demonstrated pronounced nonlinear responses in primary electrosensory af 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. \paragraph{Founding sources} -Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by the DFG, project number 430157666. +Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by the DFG, project number 430157666, +and by the Open Access Publication Fund of the University of T\"ubingen. %\bibliographystyle{apalike}%alpha}%}%alpha}%apalike} diff --git a/submission/figure01.pdf b/submission/figure01.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fa17a1ca68785d97a9a2711062ec9e338c237358 GIT binary patch literal 27915 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spectrum, suggesting a linear response (\subfigrefb{fig:twobeats}{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. + +\fighere{fig:twobeats} + +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:twobeats}{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} + +The stimuli used in \figref{fig:twobeats} had the same AM contrasts of 10\,\% that evoke rather strong modulations in a P-unit's firing rate response. Whether these definitely not-small stimulus amplitudes fall into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}. + +\fighere{fig:regimes} + +At very low stimulus contrasts (in the example cell less than approximately 1.2\,\% relative to the receiver's EOD amplitude) the spectrum has small peaks at the beat frequencies (\subfigref{fig:regimes}{A,B}, green and purple). The amplitudes of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:regimes}{E}, thin lines), an indication of the linear response at lowest stimulus amplitudes. The largest peak at the receiver's EOD frequency indicates the stochastic locking of the P-unit spikes to the EOD, and its side-peaks are remnants of the two stimulating EOD frequencies \citep{Sinz2020}. + +This linear regime is followed by the weakly nonlinear regime (in the example cell between approximately 1.2\,\% and 2.6\,\% stimulus contrast). In addition to the peaks at the stimulating beat frequencies, peaks at the sum and the difference of the beat frequencies appear in the response spectrum (\subfigref{fig:regimes}{C}, orange and red). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:regimes}{E}, thin lines). Note, that we have chosen $\Delta f_2$ to match the baseline firing rate $r$ of the neuron. + +At higher stimulus amplitudes, the linear response and the weakly-nonlinear response begin to deviate from their linear and quadratic dependency on amplitude (\subfigrefb{fig:regimes}{E}) and additional peaks appear in the response spectrum (\subfigrefb{fig:regimes}{D}). At high stimulus contrasts, additional nonlinearities in the system, in particular clipping of the firing rate, shape the responses. + +For this example, we chose very specific stimulus (beat) frequencies. In the following, however, we are interested in how the nonlinear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime. For the sake of simplicity we will drop the $\Delta$ notation even though P-unit stimuli are beats. + + +\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 $f_{\rm EOD}$ of the electric organ (\subfigref{fig:punit}{B}). + +\fighere{fig:punit} + +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 $f_1$ 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:lifsuscept}{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:lifsuscept}{B}). + +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 in the second-order susceptibility 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:lifsuscept}{B}). In all other P-units, we did not observe any structure in the second-order susceptibility (\subfigrefb{fig:punit}{I}). + + +\subsection{Ampullary afferents exhibit strong nonlinear interactions} + +\fighere{fig:ampullary} + +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} +In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampullary}), we only observe nonlinear responses where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, but not where either of the frequencies alone matches the baseline rate. In the following, we investigate this discrepancy to the theoretical expectations (\figrefb{fig:lifsuscept}, \citealp{Voronenko2017,Franzen2023}). + +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 50\,s (median: 10\,s), total stimulation durations per cell range between 30 and 400\,s. Using a temporal 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}). + +\fighere{fig:noisesplit} + +In model simulations we can increase the number of FFT segments beyond what would be experimentally possible, here to one million (\figrefb{fig:noisesplit}\,\panel[iii]{B}). Then, the estimate of the second-order susceptibility indeed improves. It gets less noisy, the diagonal at $f_1 + f_2 = r$ is emphasized, and the vertical and horizontal ridges at $f_1 = r$ and $f_2 = r$ are revealed. Increasing the number of FFT segments also reduces the order of magnitude of the susceptibility estimate until close to one million the estimate levels out at a low values (\subfigrefb{fig:noisesplit}\,\panel[iv]{B}). + +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 $\sqrt{2D \, \alpha_{\rm noise}}\;\xi(t)$ 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}). + +\subsection{Weakly nonlinear interactions in many model cells} +In the previous section we have shown one example cell for which we find in the corresponding model the expected strong ridges in the second-order susceptibility (\figrefb{fig:noisesplit}\,\panel[iii]{B},\panel[iii]{D}). Using our 39 P-unit models, we now can explore how many P-unit model neurons show such a triangular structure. + +By just looking at the second-order susceptibilities estimated using the noise-split method (first column of \figrefb{fig:modelsusceptcontrasts}) we can readily identify strong triangular patterns in 11 of the 39 model cells (28\,\%, see \figrefb{fig:modelsusceptcontrasts}\,\panel[i]{A}\&\panel[i]{B} for two examples). In another 5 cells (13\,\%) the triangle is much weaker and sits on top of a smooth bump of elevated second-order susceptibility (\figrefb{fig:modelsusceptcontrasts}\,\panel[i]{C} shows an example). The remaining 23 model cells (59\,\%) show no triangle (see \figrefb{fig:modelsusceptcontrasts}\,\panel[i]{D} for an example). + +\fighere{fig:modelsusceptcontrasts} + +This categorization is supported by the susceptibility index, SI($r$), \eqnref{siindex}, which quantifies the height of the ridge where the stimulus frequencies add up to the neuron's baseline firing rate relative to the background. Values above one indicate an elevated ridge. The absence of such a ridge results in values close to one. Indeed, the cells showing only a weak triangle (orange) arise out of values around one and the cells showing strong triangles (red) have consistently SI($r$) values exceeding 1.8 (\figrefb{fig:modelsusceptcontrasts}\,\panel[i]{E}). + +The SI($r$) correlates with the CVs of the cell's baseline interspike intervals ($r=-0.60$, $p<0.001$). The lower the cell's CV$_{\text{base}}$, the higher the SI($r$) value and thus the stronger the triangular structure of its second-order susceptibility. The model cells with the most distinct triangular pattern in their second-order susceptibility are the ones with the lowest CVs, hinting at low intrinsic noise levels. + + +\subsection{Weakly nonlinear interactions vanish for higher stimulus contrasts} +As pointed out above, noise stimuli act as a linearizing background noise and thus may shift the neural signal transmission into a regime free of nonlinearities (see discussion). Sine-wave stimuli, as for example used in the introductory figure~\ref{fig:regimes}, do not have such a linearizing effect. At which amplitudes a noise stimulus effectively linearizes the system, also depends on cellular properties of the neuron, or on the parameters in our leaky integrate-and-fire neuron. This is what we explore now by varying the contrast of RAM stimuli driving our model cells. + +In the model cells we estimated second-order susceptibilities for RAM stimuli with a contrast of 1, 3, and 10\,\%. The estimates for 1\,\% contrast (\figrefb{fig:modelsusceptcontrasts}\,\panel[ii]{E}) were quite similar to the estimates from the noise-split method, corresponding to a stimulus contrast of 0\,\% ($r=0.97$, $p\ll 0.001$). Thus, RAM stimuli with 1\,\% contrast are sufficiently small to not destroy weakly nonlinear interactions by their linearizing effect. At this low contrast, 51\,\% of the model cells have an SI($r$) value greater than 1.2. + +At a RAM contrast of 3\,\% the SI($r$) values become smaller (\figrefb{fig:modelsusceptcontrasts}\,\panel[iii]{E}). Only 7 cells (18\,\%) have SI($r$) values exceeding 1.2. Finally, at 10\,\% the SI($r$) values of all cells drop below 1.2, except for three cells (8\,\%, \figrefb{fig:modelsusceptcontrasts}\,\panel[iv]{E}). The cell shown in \subfigrefb{fig:modelsusceptcontrasts}{A} is one of them. At 10\,\% contrast the SI($r$) values are no longer correlated with the ones in the noise-split configuration ($r=0.32$, $p=0.05$). To summarize, the regime of distinct nonlinear interactions at frequencies matching the baseline firing rate extends in this set of P-unit model cells to stimulus contrasts ranging from a few percents to about 10\,\%. + +\subsection{Weakly nonlinear interactions can be deduced from limited data} +Estimating second-order susceptibilities reliably requires large numbers (millions) of FFT segments (\figrefb{fig:noisesplit}). Electrophysiological measurements, however, suffer from limited recording durations and %hence limited numbers of available FFT segments and +estimating weakly nonlinear interactions from just a few hundred segments appears futile. To what extend are such limited-data estimates still informative? + +\fighere{fig:modelsusceptlown} + +The second-order susceptibility matrices that are based on only 100 segments look flat and noisy, lacking the triangular structure (\subfigref{fig:modelsusceptlown}{B}). The anti-diagonal ridge, however, %where the sum of the stimulus frequencies matches the neuron's baseline firing rate, +seems to be present whenever the converged estimate shows a clear triangular structure (compare \subfigref{fig:modelsusceptlown}{B} and \subfigref{fig:modelsusceptlown}{A}). %The SI($r$) characterizes the height of the ridge in the second-oder susceptibility plane at the neuron's baseline firing rate $r$. +Comparing SI($r$) values based on 100 FFT segments to the ones based on one or ten million segments for all 39 model cells (\subfigrefb{fig:modelsusceptlown}{C}) supports this impression. They correlate quite well at contrasts of 1\,\% and 3\,\% ($r=0.9$, $p\ll 0.001$). At a contrast of 10\,\% this correlation is weaker ($r=0.38$, $p<0.05$), because there are only three cells left with SI($r$) values greater than 1.2. Despite the good correlations, care has to be taken to set a threshold on the SI($r$) values for deciding whether a triangular structure would emerge for a much higher number of segments. Because at low number of segments the estimates are noisier, there could be false positives for a too low threshold. Setting the threshold to 1.8 avoids false positives for the price of a few false negatives. + +Overall, observing SI($r$) values greater than about 1.8, even for a number of FFT segments as low as one hundred, seems to be a reliable indication for a triangular structure in the second-order susceptibility at the corresponding stimulus contrast. Small stimulus contrasts of 1\,\% are less informative, because of their bad signal-to-noise ratio. Intermediate stimulus contrasts around 3\,\% seem to be optimal, because there, most cells still have a triangular structure in their susceptibility and the signal-to-noise ratio is better. At RAM stimulus contrasts of 10\,\% or higher the signal-to-noise ratio is even better, but only few cells remain with weak triangularly shaped susceptibilities that might be missed as a false positives. + +\subsection{Low CVs and weak responses predict weakly nonlinear responses} +Now we are prepared to evaluate our pool of 39 P-unit model cells, 172 P-units, and 30 ampullary afferents recorded in 80 specimen of \textit{Apteronotus leptorhynchus}. For direct comparison across cells we condensed the structure of the second-order susceptibilities into SI($r$) values, \eqnref{siindex}. Both, model and experimental SI($r$) estimates, \eqnref{siindex}, are based on 100 FFT segments. + +\fighere{fig:dataoverview} + +In the P-unit models, each model cell contributed with three RAM stimulus presentations with contrasts of 1, 3, and 10\,\%, resulting in $n=117$ data points. 19 (16\,\%) had SI($r$) values larger than 1.8, indicating the expected ridges at the baseline firing rate in their second-order susceptibility. The lower the cell's baseline CV, i.e. the less intrinsic noise, the higher the SI($r$) (\figrefb{fig:dataoverview}\,\panel[i]{A}). + +The effective stimulus strength also plays a role in predicting the SI($r$) values. We quantify the effect of stimulus strength on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. The lower the response modulation, i.e. the weaker the effective stimulus, the higher the S($r$) (\figrefb{fig:dataoverview}\,\panel[ii]{A}). Although there is a tendency of low stimulus contrasts to evoke lower response modulations, response modulations evoked by each of the three contrasts overlap substantially, emphasizing the strong heterogeneity of the P-units' sensitivity \citep{Grewe2017}. Cells with high SI($r$) values are the ones with baseline firing rate below 200\,Hz (\figrefb{fig:dataoverview}\,\panel[iii]{A}). + +In comparison to the experimentally measured P-unit recordings, the model cells are skewed to lower baseline CVs (Mann-Whitney $U=13986$, $p=3\times 10^{-9}$), because the models are not able to reproduce bursting, which we observe in many P-units and which leads to high CVs. Also the response modulation of the models is skewed to lower values (Mann-Whitney $U=14051$, $p=4\times 10^{-9}$), because in the measured cells, response modulation is positively correlated with baseline CV (Pearson $R=0.34$, $p=1\times 10^{-4}$), i.e. bursting cells are more sensitive. Median baseline firing rate in the models is by 53\,Hz smaller than in the experimental data (Mann-Whitney $U=17034$, $p=0.0002$). + +In the experimentally measured P-units, each of the $172$ cells contributes on average with two RAM stimulus presentations, presented at contrasts ranging from 1 to 20\,\% to the 376 samples. Despite the mentioned differences between the P-unit models and the measured data, the SI($r$) values do not differ between models and data (median of 1.3, Mann-Whitney $U=19702$, $p=0.09$) and also 16\,\% of the samples from all presented stimulus contrasts exceed the threshold of 1.8. The SI($r$) values of the P-unit population correlate weakly with the CV of the baseline ISIs that range from 0.18 to 1.35 (median 0.49). Cells with lower baseline CVs tend to have more pronounced ridges in their second-order susceptibilities than those with higher baseline CVs (\figrefb{fig:dataoverview}\,\panel[i]{B}). + +Samples with weak responses to a stimulus, due to low sensitivity or a weak stimulus, have higher SI($r$) values in comparison to strongly responding cells, most of them having flat second-order susceptibilities (\figrefb{fig:dataoverview}\,\panel[ii]{B}). P-units with low or high baseline firing rates can have large SI($r$) (\figrefb{fig:dataoverview}\,\panel[iii]{B}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and the response strength during stimulation (effective output noise). + +The population of ampullary cells is more homogeneous, with generally lower baseline CVs than P-units (Mann-Whitney $U=33464$, $p=9\times 10^{-49}$). Accordingly, SI($r$) values of ampullary cells (median 2.3) are indeed higher than in P-units (median 1.3, Mann-Whitney $U=6450$, $p=2\times 10^{-19}$). 52 samples (58\,\%) with SI($r$) values greater than 1.8 would have a triangular structure in their second-order susceptibilities. Ampullary cells also show a negative correlation with baseline CV, despite their narrow distribution of CVs ranging from 0.03 to 0.15 (median 0.09) (\figrefb{fig:dataoverview}\,\panel[i]{C}). Again, sensitive cells with stronger response modulations are at the bottom of the SI($r$) distribution with values close to one (\figrefb{fig:dataoverview}\,\panel[ii]{C}). Similar to P-units, the baseline firing rate does not predict SI($r$) values (\figrefb{fig:dataoverview}\,\panel[iii]{C}). + +\section{Discussion} + +Theoretical work \citep{Voronenko2017,Franzen2023} studied analytically and numerically the weakly-nonlinear responses of spike generating LIF and theta model neurons driven by two sine waves with distinct frequencies. We here investigated such nonlinear responses in two types of electroreceptor afferents that differ in their intrinsic noise levels \citep{Grewe2017} using band-limited white-noise stimuli to estimate second-order susceptibilities. Following \citet{Voronenko2017} we expected to observe distinct ridges in the second-order susceptibility where either of the stimulus frequencies alone or their sum matches the baseline firing rate. We find traces of these nonlinear responses in the majority of ampullary afferents. In P-units, however, only a minority of the recorded cells, i.e. those characterized by low intrinsic noise levels and low output noise, show signs of such nonlinear responses. Complementary model simulations demonstrate in the limit of high numbers of FFT segments, that the estimates from the electrophysiological data are indeed indicative of the theoretically expected triangular structure of supra-threshold weakly nonlinear responses. With this, we provide evidence for weakly-nonlinear responses of a spike generator at low intrinsic noise levels or low stimulus amplitudes in real sensory neurons. + +\subsection{Intrinsic noise limits nonlinear responses} +The weakly nonlinear regime with its triangular pattern of elevated second-order susceptibility resides between the linear and a stochastic mode-locking regime. Too strong intrinsic noise linearizes the system and wipes out the structure of the second-order susceptibility (\citealp{Voronenko2017}, \subfigref{fig:lifsuscept}{B}). The CV of the baseline interspike interval is a proxy for the intrinsic noise in the cells (\citealp{Vilela2009}, note however the effect of coherence resonance for excitable systems close to a bifurcation, \citealp{Pikovsky1997, Lindner2004}). In both cell types, we observe a negative correlation between the second-order susceptibility at $f_1 + f_2 = r$ and the baseline CV (\figrefb{fig:dataoverview}), indicating that it is the level of intrinsic noise that shapes nonlinear responses. Still, only 18\,\% of the P-units analyzed in this study show relevant nonlinear responses. On the other hand, the majority (74\,\%) of the ampullary cells show nonlinear responses as they have generally lower CVs (median of 0.09). + + These findings are in line with previous theoretical and experimental studies showing the linearizing effects of noise in other nonlinear systems \citep{Roddey2000, Chialvo1997}. Increased intrinsic noise has been demonstrated to increase the CV and to reduce nonlinear phase-locking in vestibular afferents \citep{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL \citep{Chacron2006}. Only in cells with sufficiently low levels of intrinsic noise, weakly nonlinear responses can be observed. + +\subsection{Linearization by white-noise stimulation} +Not only the intrinsic noise but also the stimulation with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\figrefb{fig:dataoverview}\,\panel[ii]{B}) and direct stimulation in ampullary cells (\figrefb{fig:dataoverview}\,\panel[ii]{C}). The stronger the effective stimulus, the less pronounced are the ridges in the second-order susceptibility (see \subfigref{fig:punit}{E\&F} for a P-unit example and \subfigref{fig:ampullary}{E\&F} for an ampullary cell). This linearizing effect of noise stimuli limits the weakly nonlinear regime to small stimulus amplitudes. At higher stimulus amplitudes, however, other nonlinearities of the system eventually show up in the second-order susceptibility. + +In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. 41\,\% of the model cells then show the full nonlinear structure (\figref{fig:modelsusceptcontrasts}\,\panel[i]{E}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}. Previous studies on second-order nonlinearities have not observed the weakly nonlinear regime, probably because of the linearizing effects of strong noise stimuli \citep{Victor1977, Schanze1997, Neiman2011}. + +\subsection{Characterizing nonlinear coding from limited experimental data} +Estimating the Volterra series from limited experimental data is usually restricted to the first two or three orders, which might not be sufficient for a proper prediction of the neuronal response \citep{French2001}. As we have demonstrated, a proper estimation of just the second-order susceptibility in the weakly nonlinear regime is challenging in electrophysiological experiments (\figrefb{fig:modelsusceptlown}). Our minimum of 100 FFT segments corresponds to just 26\,s of stimulation and thus is not a challenge. However, the estimates of the second-order susceptibilities start to converge only beyond 10\,000 FFT segments, corresponding to 43\,min of stimulation. Often, however, more than one million segments (corresponding to 71 hours of recording), were needed, which is clearly out of reach. We have demonstrated that even in non-converged estimates based on short recordings, the presence of a anti-diagonal ridge is sufficient to predict a triangular pattern in a converged estimate. + +Making assumptions about the nonlinearities in a system also reduces the amount of data needed for parameter estimation. In particular, models combining linear filtering with static nonlinearities \citep{Chichilnisky2001} have been successful in capturing functionally relevant neuronal computations in visual \citep{Gollisch2009} as well as auditory systems \citep{Clemens2013}. Linear methods based on backward models for estimating the stimulus from neuronal responses, have been extensively used to quantify information transmission in neural systems \citep{Theunissen1996, Borst1999, Wessel1996, Machens2001}, because backward models do not need to generate action potentials that involve strong nonlinearities \citep{Rieke1999}. + +\subsection{Nonlinear encoding in ampullary cells} + +The afferents of the passive electrosensory system, the ampullary cells, exhibit strong second-order susceptibilities (\figref{fig:dataoverview}). Ampullary cells more or less directly translate external low-frequency electric fields into afferent spikes, much like in the standard LIF and theta models used by Lindner and colleagues \citep{Voronenko2017,Franzen2023}. Indeed, we observe in the ampullary cells similar second-order nonlinearities as in LIF models. + +Ampullary stimuli originate from the muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. For a single prey item such as \textit{Daphnia}, these potentials are often periodic but the simultaneous activity of a swarm of prey resembles Gaussian white noise \citep{Neiman2011fish}. Linear and nonlinear encoding in ampullary cells has been studied in great detail in the paddlefish \citep{Neiman2011fish}. The power spectrum of the baseline response shows two main peaks: One peak at the baseline firing frequency, a second one at the oscillation frequency of primary receptor cells in the epithelium, plus interactions of both. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, instead, nonlinear responses are very pronounced there. + +Ampullary stimulus encoding is somewhat different in \textit{A. leptorhynchus}. The power spectrum of the spontaneous response is dominated by only the baseline firing rate and its harmonics, a second oscillator is not visible. The baseline firing frequency, however, is outside the linear coding range \citep{Grewe2017} while it is within the linear coding range in paddlefish \citep{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullary cells increases with stimulus intensity while it disappears in our case (\figrefb{fig:dataoverview}~\panel[ii]{C}) indicating that paddlefish data have been recorded above the weakly-nonlinear regime. + +The population of ampullary cells is very homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties \citep{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, the resulting nonlinear response appears at the baseline rate that is similar in the full population of ampullary cells and that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract such nonlinear responses. How such nonlinear effects might influence prey detection should be addressed in future studies. + +\subsection{Nonlinear encoding in P-units} + +In contrast to the ampullary cells, P-units respond to the amplitude modulation of the self-generated EOD. Extracting the AM 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. + +Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. The natural stimuli encoded by P-units are, however, periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Such interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities observed under noise stimulation for the encoding of distinct frequencies? Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, a 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 and substantial power (\figrefb{fig:twobeats}) although these interactions vanish when stimulating with noise stimuli of similar contrast (\figrefb{fig:modelsusceptcontrasts}). + +The encoding of secondary AMs or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires another nonlinearity in addition to the one needed for extracting the AM. Initially, this nonlinearity was 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. + +The weakly nonlinear interactions in low-CV P-units could facilitate the detectability of faint signals during three-animal interactions as found in courtship contexts in the wild \citep{Henninger2018}. The detection of a faint, distant intruder male could be improved by the presence of a nearby strong female stimulus, because of the nonlinear interaction terms \citep{Schlungbaum2023}. This boosting effect is, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The population of P-units is very heterogeneous in their baseline firing rates and CVs (50--450\,Hz and 0.1--1.4, respectively, \subfigref{fig:dataoverview}{B}, \citealp{Grewe2017, Hladnik2023}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions \citep{Henninger2018, Henninger2020}, while the number of P-units showing weakly nonlinear responses is small. Whether and how this information is specifically maintained and read out by pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain onto which P-units converge \citep{Krahe2014, Maler2009a} is an open question. + +Electric fish are able to slowly modulate their EOD frequency, as for example during the so-called jamming-avoidance-response \citep{Fortune2020}. Such behaviors modify the resulting beat frequency by a few Hertz. This could in principle increase the chance that the now slowly changing beat frequency matches at some point the baseline firing rate of a P-unit, where the weakly nonlinear responses then enhance the detectability of another conspecific \citep{Schlungbaum2023}. Furthermore, transient changes in EOD frequency on timescales of tens of milliseconds up to a few seconds are known as chirps and rises, respectively, and are involved in courtship and aggression behaviors \citep{Henninger2018, Raab2021}. How the encoding of such transient frequency modulations is affected by the nonlinearities described here is another open question, since the presented analysis focuses on stationary signals. + +\subsection{Conclusions} + +We have demonstrated pronounced nonlinear responses in primary electrosensory afferents at weak stimulus amplitudes and sufficiently low intrinsic noise levels. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. The resulting nonlinear components introduce spectral components not present in the original stimulus, that may provide an edge in the context of signal detection problems at stimulus amplitudes close to threshold \citep{Schlungbaum2023}. Electrosensory afferents share an evolutionary history with hair cells \citep{Baker2019} and share many response properties with mammalian auditory nerve fibers \citep{Barayeu2023, Joris2004}. Thus, we expect weakly nonlinear responses for near-threshold stimulation in auditory nerve fibers as well. These could boost or distort responses to two simultaneously presented tones and thus might play a role in forming perception of music. + +\paragraph{Acknowledgements} +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. + +\paragraph{Founding sources} +Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by the DFG, project number 430157666, +and by the Open Access Publication Fund of the University of T\"ubingen. + + +%\bibliography{journalsabbrv,references} +\begin{thebibliography}{} + +\bibitem[\protect\citeauthoryear{Aertsen and Johannesma}{1981}]{Aertsen1981} +Aertsen AMHJ, Johannesma PIM (1981) +\newblock A comparison of the spectro-temporal sensitivity of neurons to tonal + and natural stimuli. +\newblock {\em Biol. 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Coding of time-varying electric field amplitude modulations in a + wave-type electric fish. +\newblock {\em J Neurophysiol}~75:\mbox{2280--2293}. + +\bibitem[\protect\citeauthoryear{Xu \bgroup et al.\egroup }{1996}]{Xu1996} +Xu Z, Payne JR, Nelson ME (1996) +\newblock Logarithmic time course of sensory adaptation in electrosensory + afferent nerve fibers in a weakly electric fish. +\newblock {\em J Neurophysiol}~76:\mbox{2020--2032}. + +\bibitem[\protect\citeauthoryear{Yu \bgroup et al.\egroup }{2005}]{Yu2005} +Yu N, Hup\'e G, Garfinkle C, Lewis JE, Longtin A (2005) +\newblock Coding conspecific identity and motion in the electric sense. +\newblock {\em PLoS Comput Biol}~8:\mbox{e1002564}. + +\bibitem[\protect\citeauthoryear{Yu and Lewis}{1989}]{Yu1989} +Yu X, Lewis E (1989) +\newblock Studies with spike initiators: linearization by noise allows + continuous signal modulation in neural networks. +\newblock {\em IEEE Transactions on Biomedical Engineering}~36:\mbox{36--43}. + +\end{thebibliography} + + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure01} % {lifsuscept} + \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. $r$ and $2r$ mark the neuron's baseline firing rate and its harmonic (dashed vertical lines). \figitem{B} Magnitude of the second-order (nonlinear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because Fourier spectra have positive and negative frequencies, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.} +\end{figure*} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure02} % {flowchart} + \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 $r$, $f_{EOD} - r$, $f_{EOD}$, and $f_{EOD} + r$. With RAM stimulation (\panel[ii]{E}), the peak at the baseline firing rate, $r$, is washed out.} +\end{figure*} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure03} % {twobeats} + \caption{\label{fig:twobeats} Nonlinearity in an electrophysiologically recorded P-unit of \textit{A. leptorhynchus} 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: Superposition of the receiver EOD with the EODs of other fish, colored line highlights the amplitude modulation. Third row: Three trials of spike trains of the recorded P-unit. Fourth row: Firing rate, estimated by convolution of the spike trains with a Gaussian kernel. Bottom row: Power spectrum of the spike trains. \figitem{A} Baseline condition: The cell is driven by the self-generated field alone. The baseline firing rate $r = 139$\,Hz dominates the power spectrum (blue circle). \figitem{B} The receiver's EOD and a single conspecific with an EOD frequency $f_{1}=631$\,Hz are present. Superposition of the two EODs induces a periodic amplitude modulation, referred to as beat, with beat frequency $\Delta f_1=33$\,Hz. The P-unit strongly responds to this beat (purple). \figitem{C} The receiver and a fish with an EOD frequency $f_{2}=797$\,Hz are present. The resulting beat $\Delta f_2=133$\,Hz is faster as the difference between the EOD frequencies is larger. The P-unit response to this faster beat is weaker (green). \figitem{D} All three fish with EOD frequencies $f_{\rm EOD}$, $f_1$, and $f_2$ are present. Additional peaks occur in the power spectrum of the spike response at the sum (orange) and difference (red) of the two beat frequencies, indicating nonlinear interactions between the two frequencies in the P-unit. Note, the spectrum of the raw signal (top row, gray) has power only at the three EOD frequencies $f_{EOD}$, $f_1$, and $f_2$.} +\end{figure*} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure04} % {regimes} + \caption{\label{fig:regimes} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on stimulus amplitudes. The model P-unit (identifier ``2018-05-08-ad'') was stimulated with two sine waves of equal amplitude (contrast) at difference frequencies $\Delta f_1=40$\,Hz and $\Delta f_2=228$\,Hz relative the receiver's EOD frequency $f_{EOD}=656$\,Hz. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top row: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Second row: Spike raster of the model P-unit response. Third row: average firing rate and, bottom row: power spectrum estimated from 1000 trials of spike train responses. \figitem{A} At low stimulus contrasts the response to the two beat frequencies is linear. These frequencies are present in the response spectrum (green and purple circles), the peak at $\Delta f_2$ (purple) enhances the peak at baseline firing rate (blue). The largest peak, however, is always the one at the EOD frequency of the receiver (black), reflecting the locking of P-unit spikes to the receiver's own EOD. The peak at $f_{EOD}$ is also flanked by harmonics of $\Delta f_1$ (gray triangles), where $f_{EOD} + \Delta f_1 = f_1$ is the EOD frequency of the first fish. \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies become larger. In addition, two peaks at $f_{EOD} - \Delta f_2$ and $f_{EOD} + \Delta f_2$ appear (blue diamonds), the latter is the EOD frequency of the second fish, $f_2$. These peaks indicate nonlinear processes acting on the three EOD frequencies from which the beat frequencies are generated. \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the beat frequencies (orange and red circles). Similarly, side peaks appear at $\pm \Delta f_1$ around $f_{EOD} \pm \Delta f_2$ (purple squares) as well as harmonics of $\Delta f_1$ (small green circles). \figitem{D} At higher stimulus contrasts many more peaks appear in the power spectrum. Most of them are interactions of $f_{EOD}$ and $\Delta f_2$ with harmonics of $\Delta f_1$ ($k=1, 2, ...$), since the cell responds strongest to $\Delta f_1$. \figitem{E} Amplitude of the linear (at $\Delta f_1$ and $\Delta f_2$) and nonlinear (at $\Delta f_2 - \Delta f_1$ and $\Delta f_1 + \Delta f_2$) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively. In the linear regime, below a stimulus contrast of about 1.2\,\% (left vertical line), the only peaks in the response spectrum are at the stimulus frequencies. In the weakly nonlinear regime up to a contrast of about 3.5\,\% peaks arise at the sum and the difference of the two stimulus frequencies. At stronger stimulation the amplitudes of these nonlinear responses deviate from the quadratic dependency on stimulus contrast.} +\end{figure*} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure05} % {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 the low 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 in the absolute value of the second-order susceptibility 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*} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure06} % {ampullaryexamplecell} + \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*} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure07} % {noisesplit} + \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*} + +\begin{figure}[p] + %\includegraphics[width=\columnwidth]{figure08} % {modelsusceptcontrasts} + \caption{\label{fig:modelsusceptcontrasts}Dependence of second order susceptibility on stimulus contrast. \figitem{A} Second-order susceptibilities estimated for increasing stimulus contrasts of $c=0, 1, 3$ and $10$\,\% as indicated ($N=10^7$ FFT segments for $c=1$\,\%, $N=10^6$ segments for all other contrasts). $c=0$\,\% refers to the noise-split configuration (limit to vanishing external RAM signal, see \subfigrefb{fig:noisesplit}{D}). Shown are simulations of the P-unit model cell ``2017-07-18-ai'') with a baseline firing rate of $82$\,Hz and CV$_{\text{base}}=0.23$. The cell shows a clear triangular pattern in its second-order susceptibility even up to a contrast of $10$\,\%. Note, that for $c=1$\,\% (\panel[ii]{D}), the estimate did not converge yet. \figitem{B} Cell ``2012-12-13-ao'' (baseline firing rate of $146$\,Hz, CV$=0.23$) also has strong interactions at its baseline firing rate that survive up to a stimulus contrast of $3$\,\%. \figitem{C} Model cell ``2012-12-20-ac'' (baseline firing rate of $212$\,Hz, CV$=0.26$) shows a weak triangular structure in the second-order susceptibility that vanishes for stimulus contrasts larger than $1$\,\%. \figitem{D} Cell ``2013-01-08-ab'' (baseline firing rate of $218$\,Hz, CV$=0.55$) does not show any triangular pattern in its second-order susceptibility. Nevertheless, interactions between low stimulus frequencies become substantial at higher contrasts. \figitem{E} The presence of an elevated second-order susceptibility where the stimulus frequency add up to the neuron's baseline frequency, can be identified by the susceptibility index (SI($r$), \eqnref{siindex}) greater than one (horizontal black line). The SI($r$) (density to the right) is plotted as a function of the model neuron's baseline CV for all $39$ model cells. Model cells have been visually categorized based on the strong (11 cells) or weak (5 cells) presence of a triangular pattern in their second-order susceptibility estimated in the noise-split configuration (legend). The cells from \panel{A--D} are marked by black circles. Pearson's correlation coefficients $r$, the corresponding significance level $p$ and regression line (dashed gray line) are indicated. The higher the stimulus contrast, the fewer cells show weakly nonlinear interactions as expressed by the triangular structure in the second-order susceptibility.} +\end{figure} + +\begin{figure}[p] + %\includegraphics[width=\columnwidth]{figure09} % {modelsusceptlown} + \caption{\label{fig:modelsusceptlown}Inferring the triangular structure of the second-order susceptibility from limited data. \figitem{A} Reliably estimating the structure of the second-order susceptibility requires a high number of FFT segments $N$ in the order of one or even ten millions. As an example, susceptibilities of the model cell ``2012-12-21-ak'' (baseline firing rate of 157\,Hz, CV=0.15) are shown for the noise-split configuration ($c=0$\,\%) and RAM stimulus contrasts of $c=1$, $3$, and $10$\,\% as indicated. For contrasts below $10$\,\% this cell shows a nice triangular pattern in its susceptibilities, quite similar to the introductory example of a LIF in \figrefb{fig:lifsuscept}. \figitem{B} However, with limited data of $N=100$ trials the susceptibility estimates are noisy and show much less structure, except for the anti-diagonal at the cell's baseline firing rate. The SI($r$) quantifies the height of this ridge where the two stimulus frequencies add up to the neuron's baseline firing rate. \figitem{C} Correlations between the estimates of SI($r$) based on 100 FFT segments (density to the right) with the converged ones based on one or ten million segments at a given stimulus contrast for all $n=39$ model cells. The black circle marks the model cell shown in \panel{A} and \panel{B}. The black diagonal line is the identity line and the dashed line is a linear regression. The correlation coefficient and corresponding significance level are indicated in the top left corner. The thin vertical line is a threshold at 1.2, the thin horizontal line a threshold at 1.8. The number of cells within each of the resulting four quadrants denote the false positives (top left), true positives (top right), true negatives (bottom left), and false negatives (bottom right) for predicting a triangular structure in the converged susceptibility estimate from the estimates based on only 100 segments.} +\end{figure} + +\begin{figure*}[p] + %\includegraphics[width=\columnwidth]{figure10} % {dataoverview} + \caption{\label{fig:dataoverview} Nonlinear responses in P-units and + ampullary afferents. The second-order susceptibility is condensed + into the susceptibility index, SI($r$) \eqnref{siindex}, that + quantifies the relative amplitude of the ridge where the two + stimulus frequencies add up to the cell's baseline firing rate $r$ + (see \subfigrefb{fig:punit}{G}). In both the models and the + experimental data, the SI($r$) was estimated based on 100 FFT + segments. The SI($r$) is plotted against the cells' CV of its + baseline interspike intervals (left column), the response + modulation (standard deviation of firing rate evoked by the + band-limited white-noise stimulus) --- a measure of effective + stimulus strength (center column), and the cell's baseline firing + rate (right column). Pearson's correlation coefficient $R$, the + corresponding significance level $p$ and the number of samples $n$ + are indicated. Kernel-density estimates of the distributions of + the displayed quantities are plotted on top and right. Data points + are color coded by stimulus contrasts, which are listed together + with the corresponding number of samples in the legend to the + right. The horizontal dashed line marks a threshold for SI($r$) + values at 1.8 and the percentages to the right denote the + fractions of samples above and below this threshold. \figitem{A} + The SI($r$) of all 39 model P-units measured with RAM stimuli with + a cutoff frequency of 300\,Hz. The black square marks the cell + from \subfigrefb{fig:noisesplit}{C}, the circles the four cells + shown in \subfigref{fig:modelsusceptcontrasts}{A--D}, and the + triangle the cell from \subfigref{fig:modelsusceptlown}{A--B}. + \figitem{B} Electrophysiological data from 172 P-units. Each cell + contributes on average with 2 (min. 1, max. 10) RAM stimulus + presentations to the $n=376$ data points. The RAMs had cutoff + frequencies of 300\,Hz (352 samples) and 400\,Hz (24 samples). The + two black triangles mark the responses of the example P-unit from + \subfigrefb{fig:punit}{E,F}, the circles the other four examples + from \subfigrefb{fig:punit}{H}, and the triangle the unit from + \subfigrefb{fig:noisesplit}{A}. \figitem{C} Recordings from 30 + ampullary afferents, each contributing on average 3 (min. 1, + max. 7) RAM stimulus presentations to $n=89$ data points. Stimuli + had a cutoff frequency of 150\,Hz. The two black triangles mark + the responses of the example ampullary afferent from + \subfigrefb{fig:ampullary}{E,F}, and the circles the other four + examples from \subfigrefb{fig:ampullary}{H}.} +\end{figure*} + +\end{document}

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