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note on FFT methods

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Jan Benda 2025-05-23 19:43:44 +02:00
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nonlinearbaseline.tex
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@ -706,6 +706,7 @@ The average firing rate during stimulation, $r_s = \langle r(t) \rangle_t$, is g
The neuron is driven by the stimulus and thus its spiking response depends on the time course of the stimulus. To characterize the relation between stimulus $s(t)$ and response $x(t)$, we calculated the first- and second-order susceptibilities in the frequency domain. The neuron is driven by the stimulus and thus its spiking response depends on the time course of the stimulus. To characterize the relation between stimulus $s(t)$ and response $x(t)$, we calculated the first- and second-order susceptibilities in the frequency domain.
Fast fourier transforms (FFT) $\tilde s_T(\omega)$ and $\tilde x_T(\omega)$ of $s(t)$ and $x(t)$, respectively, were computed according to $\tilde x_T(\omega) = \int_{0}^{T} \, x(t) e^{- i \omega t}\,dt$ for $T=0.5$\,s long segments with no overlap, resulting in a spectral resolution of 2\,Hz for the experimental data. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. In the experimental data, most stimuli had a duration of 10\,s and were chopped into 20 segments. Spectral measures were computed for single trials of neural responses, series of spike times, \eqnref{eq:spikes}. For spectral parameters used in simulations, see below. Fast fourier transforms (FFT) $\tilde s_T(\omega)$ and $\tilde x_T(\omega)$ of $s(t)$ and $x(t)$, respectively, were computed according to $\tilde x_T(\omega) = \int_{0}^{T} \, x(t) e^{- i \omega t}\,dt$ for $T=0.5$\,s long segments with no overlap, resulting in a spectral resolution of 2\,Hz for the experimental data. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. In the experimental data, most stimuli had a duration of 10\,s and were chopped into 20 segments. Spectral measures were computed for single trials of neural responses, series of spike times, \eqnref{eq:spikes}. For spectral parameters used in simulations, see below.
\notejb{Update FFT parameter. Describe binary spiketrain normalized by dt and plain FFT. Power spectra scaled by dt/nfft, bispectra by dt**2/nfft.}
The power spectrum of the stimulus $s(t)$ was estimated as The power spectrum of the stimulus $s(t)$ was estimated as
\begin{equation} \begin{equation}
@ -734,7 +735,7 @@ describes nonlinear interactions that generate responses at the sum and differen
The second-order susceptibility The second-order susceptibility
\begin{equation} \begin{equation}
\label{eq:susceptibility} \label{eq:susceptibility}
\chi_{2}(\omega_{1}, \omega_{2}) = \frac{S_{xss} (\omega_{1},\omega_{2})}{2\sqrt{S_{ss} (\omega_{1}) S_{ss} (\omega_{2})}} \chi_{2}(\omega_{1}, \omega_{2}) = \frac{S_{xss} (\omega_{1},\omega_{2})}{2 S_{ss} (\omega_{1}) S_{ss} (\omega_{2})}
\end{equation} \end{equation}
normalizes the second-order cross-spectrum by the spectral power at the two stimulus frequencies. normalizes the second-order cross-spectrum by the spectral power at the two stimulus frequencies.
% Applying the Fourier transform this can be rewritten resulting in: % Applying the Fourier transform this can be rewritten resulting in: