The activity of a neuron at time \(t\) typically depends on the behavior of the stimulus over a period of time, starting a few hundred milliseconds prior to \( t \) and ending perhaps tens of milliseconds before \( t \) [1]. Assuming (i) linearity, (ii) time invariance, and (iii) local spatial homogeneity, the response for a cell located at position \( \mathbf{r}_{c} \) in the visual field, can be written as [2]

\begin{equation} R( \mathbf{r}_{c}, t) = R_{0} +\displaystyle \int_{\tau} \iint_\mathbf{r} W ( \mathbf{r}_{c} - \mathbf{r} , \tau ) S( \mathbf{r},t- \tau ) \mathrm{d}^2 \mathbf{r} \mathrm{d}\tau, \end{equation}

where the impulse-response function \( {W}(\mathbf{r}, \tau)\) describes the strength with which a stimulus delivered at position \( \mathbf{r}\) at time \( t- \tau \) affects the response of a neuron located at position \( \mathbf{r}_{c} \) at time \( t \). Since an input event cannot have effects in the past, it follows then that \( {W}(\mathbf{r},\tau < 0) =0\). Thus, the lower integration boundary could also be set to \(\tau=0\).

The integral above is a convolution between the stimulus and impulse-response function, i.e.

\begin{equation} R(\mathbf{r}_{c}, t) = R_{0}+ {W} \otimes S. \end{equation}

From the convolution theorem it follows that the integral in equation above can be reformulated as an integral over temporal and spatial frequencies:

\begin{equation} R(\mathbf{r}_{c}, t) = R_{0}+ \frac{1}{(2\pi)^3} \displaystyle\iint_\mathbf{k}\int_\omega e^{i(\mathbf{k\cdot \mathbf{r}_{c}} - \omega t)} {\widetilde{W}}(\mathbf{k}, \omega) \widetilde{S} (\mathbf{k}, \omega) \mathrm{d}^2\mathbf{k}, \end{equation}

where \( \widetilde{W} \) and \( \widetilde{S} \) are the complex Fourier transforms of the impulse-response function \( W\) and stimulus \(S\), respectively. The complex Fourier transform we use, and its inverse, are given by

\begin{equation} \tilde{y}(\mathbf{k},\omega) = \displaystyle\int_{t} \iint_\mathbf{r} e^{-i(\mathbf{k\cdot r} - \omega t)}y(\mathbf{r}, t) \mathrm{d}^2\mathbf{r}\;\mathrm{d}t, \end{equation}

\begin{equation} y(\mathbf{r}, t) = \frac{1}{(2\pi)^3} \displaystyle\int_{\omega} \iint_\mathbf{k} e^{i(\mathbf{k \cdot r} - \omega t)}\tilde{y}(\mathbf{k}, \omega) \mathrm{d}^2\mathbf{k}\;\mathrm{d}\omega. \end{equation}

References

[1] Dayan, Peter, and Laurence F. Abbott. Theoretical neuroscience. Vol. 10. Cambridge, MA: MIT Press, 2001.

[2] Plesser, H. E., and G. T. Einevoll. "Linear mechanistic models for the dorsal lateral geniculate nucleus of cat probed using drifting-grating stimuli." Network: Computation in Neural Systems 13.4 (2002): 503-530.