everything for 1ms kernels

nonlinearbaseline.tex

@@ -278,7 +278,7 @@ Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by th
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 \section{Abstract}
 Spiking thresholds in neurons or rectification at synapses are essential for neuronal computations rendering neuronal processing inherently nonlinear. Nevertheless, linear response theory has been instrumental for understanding, for example, the impact of noise or neuronal synchrony on signal transmission, or the emergence of oscillatory activity, but is valid only at low stimulus amplitudes or large levels of intrinsic noise. At higher signal-to-noise ratios, however, nonlinear response components become relevant. Theoretical results for leaky integrate-and-fire neurons in the weakly nonlinear regime suggest strong responses at the sum of two input frequencies if one of these frequencies or their sum match the neuron's baseline firing rate.
-We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify these predicted nonlinear responses primarily in low-noise P-units and in more than every second ampullary cell. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions.
+We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify these predicted nonlinear responses primarily in a few low-noise P-units and in more than every second ampullary cell. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions.
 
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 \section{Significance statement}
@@ -333,7 +333,7 @@ The P-unit model parameters and spectral analysis algorithms are available at \u
 The baseline firing rate $r$ was calculated as the number of spikes divided by the duration of the baseline recording (median 32\,s). The coefficient of variation (CV) of the interspike intervals (ISI) is their standard deviation relative to their mean: $\rm{CV}_{\rm base} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle} / \langle ISI \rangle$. If the baseline was recorded several times in a recording, the measures from the longest recording were taken.
 
 \paragraph{White noise analysis} \label{response_modulation}
-When stimulated with band-limited white noise stimuli, neuronal activity is modulated around the average firing rate that is similar to the baseline firing rate and in that way encodes the time-course of the stimulus. For an estimate of the time-dependent firing rate $r(t)$ we convolved each spike train with normalized Gaussian kernels with standard deviation of 2\,ms and averaged the resulting single-trail firing rates over trials. The response modulation quantifies the variation of $r(t)$ computed as the standard deviation in time $\sigma_{s} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t )^2\rangle_t}$, where $\langle \cdot \rangle_t$ denotes averaging over time.
+When stimulated with band-limited white noise stimuli, neuronal activity is modulated around the average firing rate that is similar to the baseline firing rate and in that way encodes the time-course of the stimulus. For an estimate of the time-dependent firing rate $r(t)$ we convolved each spike train with normalized Gaussian kernels with standard deviation of 1\,ms and averaged the resulting single-trail firing rates over trials. The response modulation quantifies the variation of $r(t)$ computed as the standard deviation in time $\sigma_{s} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t )^2\rangle_t}$, where $\langle \cdot \rangle_t$ denotes averaging over time.
 
 \paragraph{Spectral analysis}\label{susceptibility_methods}
 To characterize the relation between the spiking response evoked by white-noise  stimuli, we estimated the first- and second-order susceptibilities in the frequency domain. For this we converted spike times into binary vectors $x_k$ with $\Delta t = 0.5$\,ms wide bins that are set to 2\,kHz where a spike occurred and zero otherwise. Fast Fourier transforms (FFT) $S(\omega)$ and $X(\omega)$ of the stimulus $s_k$ (also down-sampled to a sampling rate of 2\,kHz) and $x_k$, respectively, were computed numerically according to
@@ -650,7 +650,7 @@ In the P-unit models, each model cell contributed with three RAM stimulus presen
 
 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 $U=15312$, $p=7\times 10^{-7}$), because in the measured cells, response modulation is positively correlated with baseline CV (Pearson $R=0.45$, $p=1\times 10^{-19}$), 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 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}).