From 8e1ff9482502a45a9e4f11ab49bb950cc1c20464 Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Mon, 2 Feb 2026 09:29:13 +0100 Subject: [PATCH] Benjamins comments --- nonlinearbaseline.tex | 6 +++--- regimes.py | 4 ++-- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/nonlinearbaseline.tex b/nonlinearbaseline.tex index cb4e53e..659fd57 100644 --- a/nonlinearbaseline.tex +++ b/nonlinearbaseline.tex @@ -554,7 +554,7 @@ In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampull 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}). -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 $f_ + f_2 = r$ 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}). +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}). @@ -564,7 +564,7 @@ In the model, however, we know the time course of the intrinsic noise and can us \begin{figure}[p] \includegraphics[width=\columnwidth]{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 less cells show weakly nonlinear interactions as expressed by the triangular structure in the second-order susceptibility.} + \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} \subsection{Weakly nonlinear interactions in many model cells} @@ -660,7 +660,7 @@ The population of ampullary cells is more homogeneous, with generally lower base \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. +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). diff --git a/regimes.py b/regimes.py index 780084b..23b70b4 100644 --- a/regimes.py +++ b/regimes.py @@ -14,7 +14,7 @@ alphas = [0.002, 0.01, 0.03, 0.06] rmax = 500 amax = 60 cthresh1 = 1.2 -cthresh2 = 3.5 +cthresh2 = 3.05 #model_cell = '2018-05-08-ab-invivo-1' # 116, CV=0.68 #alphas = [0.002, 0.008, 0.025, 0.05] @@ -381,7 +381,7 @@ def plot_peaks(ax, s, alphas, contrasts, powerf1, powerf2, powerfsum, yoffs = 35 if amax == 60 else 31 ax.text(cthresh1/2, yoffs, 'linear\nregime', ha='center', va='center') - ax.text((cthresh1 + cthresh2)/2, yoffs, 'weakly\nnonlinear\nregime', + ax.text(2.2, yoffs, 'weakly\nnonlinear\nregime', ha='center', va='center') if amax == 60: ax.text(5.5, yoffs, 'strongly\nnonlinear\nregime',