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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.
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
\begin{equation}
@ -734,7 +735,7 @@ describes nonlinear interactions that generate responses at the sum and differen
The second-order susceptibility
\begin{equation}
\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}
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: