FFT parameter for figure 4

nonlinearbaseline.tex

@@ -341,7 +341,7 @@ To characterize the relation between the spiking response evoked by white-noise
   \label{fourier}
   X(\omega) = \sum_{k=0}^{N-1} \, x_k e^{- i \omega k}
 \end{equation}
-for $N=512$ long segments of $T=N \Delta t = 256$\,ms duration with no overlap, resulting in a spectral resolution of about 4\,Hz. Note, that for a real Fourier integral a factor $\Delta t$ is missing. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. Also note, that the Fourier transform of a signal results in both positive as well as negative frequencies. This is because the signal is decomposed into complex exponentials, not sine or cosine waves. To get back a real-valued sine or cosine wave, the components at positive and negative frequencies need to be combined. For example, the sum $e^{-\omega t} + e^{+\omega t}$ of two complex exponentials at freqiencies $\pm\omega$ is $2\cos(\omega t)$.   
+for $N=512$ long segments of $T=N \Delta t = 256$\,ms duration with no overlap, resulting in a spectral resolution of about 4\,Hz. In \figref{fig:regimes} we used $N=2^{18}$ and $\Delta t=0.1$\,ms. Note, that for a real Fourier integral a factor $\Delta t$ is missing. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. Also note, that the Fourier transform of a signal results in both positive as well as negative frequencies. This is because the signal is decomposed into complex exponentials, not sine or cosine waves. To get back a real-valued sine or cosine wave, the components at positive and negative frequencies need to be combined. For example, the sum $e^{-\omega t} + e^{+\omega t}$ of two complex exponentials at freqiencies $\pm\omega$ is $2\cos(\omega t)$.   
 
 In the experimental data the duration of the noise stimuli varied and they were presented once or repeatedly (frozen noise). For the analysis we discarded the responses within the initial 200\,ms of stimulation in each trial. To make the recordings comparable we always used the first 100 segments from as many trials as needed for the following analysis.
 
@@ -501,7 +501,7 @@ When stimulating with both foreign signals simultaneously, additional peaks appe
 
 \begin{figure*}[p]
   \includegraphics[width=\columnwidth]{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. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top: 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. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster. \figitem{A} At low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple circles), the latter enhancing the peak at baseline firig 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 own EOD. The peak at $f_{EOD}$ is also flanked by harmonics of $\Delta f_1$ (gray triangles).  \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies become larger. In addition, a peak at $f_{EOD} - \Delta f_2 $ appears (blue diamond). \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the stimulus frequencies (orange and red). Additional peaks appear at harmonics of $\Delta f_1$ (small green circles) and at $f_{EOD} - \Delta f_2 \pm \Delta f_1$ (purple squares). \figitem{D} At higher stimulus contrasts further peaks appear in the power spectrum. Most of them are interactions of $f_{EOD}$ and $\Delta f_2$ with harmonics of $\Delta f_1$ ($k$ is an integer). \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.}
+  \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. $\Delta f_2$ 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 trains in response to a 26.2\,s long stimulus. \figitem{A} At low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple circles), the latter enhancing the peak at baseline firig 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 own EOD. The peak at $f_{EOD}$ is also flanked by harmonics of $\Delta f_1$ (gray triangles).  \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies become larger. In addition, a peak at $f_{EOD} - \Delta f_2 $ appears (blue diamond). \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the stimulus frequencies (orange and red). Additional peaks appear at harmonics of $\Delta f_1$ (small green circles) and at $f_{EOD} - \Delta f_2 \pm \Delta f_1$ (purple squares). \figitem{D} At higher stimulus contrasts further peaks appear in the power spectrum. Most of them are interactions of $f_{EOD}$ and $\Delta f_2$ with harmonics of $\Delta f_1$ ($k$ is an integer). \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*}
 
 The stimuli used in \figref{fig:twobeats} had the same not-small amplitude resulting in AM contrasts of 10\,\% that evoke strong modulations in a P-unit's firing rate response. Whether this stimulus condition falls 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}.