Neuronal response as a LTI system

A neuronal system can be approximated by a linear, time-invariant system (LTI) with the relevant stimulus s(t) as the input and the observed spike train x(t) as the output:

$$x(t)=h(t)*s(t)=\int_{t=-\infty}^{\infty} x(t-\tau)s(\tau)d\tau$$

Here the neuron responds to the stimulus s(t) in a particular way specified by the impulse response function h(t).

h(t) can be obtained experimentally by observing the spike train x(t) as the response to the stimulus s(t) by

$$H(\omega)=\frac{S_{sx}(\omega)}{S_{ss}(\omega)}$$

Specially if the stimulus is white or uncorrelated

$$r_{ss}(t)=\frac{1}{\sigma^2} \delta(t)$$

its power spectrum becomes a constant

$$S_{ss}(\omega)=\frac{1}{\sigma^2}$$

and the response spike train has N spikes in time window T

$$x(t)=\sum_{k=1}^N \delta(t-t_k)$$

then the transfer function $H(\omega)$ can be obtained as

$$H(\omega)=\frac{S_{sx}(\omega)}{S_{ss}(\omega)}$$

or in time domain

$$\frac{1}{\sigma^2}r_{sx}(t) =\frac{1}{\sigma^2} \int_{-\infty}^{\infty}s(\tau-t)x(\tau) d\tau$$ h(t) =

$$\frac{1}{\sigma^2} \int_{-\infty}^{\infty}s(\tau-t) \sum_{k=1}^N \delta(\tau-t_k) d\tau =\frac{1}{\sigma^2} \sum_{k=1}^N s(t_k-t)$$ =

This is the reverse-correlation formula of h which states that the linear impulse response function of a neuron can be recovered by a simple spike-triggered average of the stimulus preceding the spikes.