diff --git a/nonlinearbaseline.tex b/nonlinearbaseline.tex index c801144..af96985 100644 --- a/nonlinearbaseline.tex +++ b/nonlinearbaseline.tex @@ -452,19 +452,6 @@ Samples with weak responses to a stimulus, due to low sensitivity or a weak stim 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}). -\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. $|\chi_2|(f_1, f_2)$ 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 \notejg{with finite amplitudes, needed?} 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:lifsuscept}{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 $|\chi_2|(f_1, f_2)$, 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. -\notejg{This section is not unimportant but opens many problems, e.g. the quantitative mismatch. Do we need it? Could move essential parts to the discussion, i.e. the f1-f2. have the full spectrum in figure 1 and ``gloss'' over the quality of the prediction.} - \section{Discussion} Theoretical work \citep{Voronenko2017,Franzen2023} derived analytical expressions for 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. @@ -485,6 +472,7 @@ Estimating the Volterra series from limited experimental data is usually restric 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 items 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. @@ -494,10 +482,9 @@ Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectru 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} -\notejg{This section is a bit wild. Parts are repetitions of things that have been said earlier already. Other parts are nice but there is no real flow. We could/should merge is with the last section of the results section.} -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? -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}). +\notejg{This section is a bit wild. Parts are repetitions of things that have been said earlier already. Other parts are nice but there is no real flow. We could/should merge is with the last section of the results section.} +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, 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 these interactions vanish when stimulating with high-contrast noise stimuli (\figrefb{fig:modelsusceptcontrasts}). 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.