Neural Response Function

The Neural Response Function is quite simply just the sum of all Dirac Delta Function’s for a number $n$ neuron spikes over a certain time.

We denote the times in which these spikes occur as a list of times $t$, where a time $i=1,2,\dots ,n$ is $t_i$.

The trial when spikes are recorded is denoted to start at time $0$ and end at a time $T$, where the bounds are:

$$0\le t_i\le T\forall i$$

The Neural Response function is represented as the sum of infinitesimally narrow, idealized spikes in the form of these Dirac Delta Functions.¹

$$\rho (t)=\sum _{i=1}^n\delta (t-t_i)$$

---

This neural response function can then be used to express sums over spikes as integrals over time.

An example, well behaved function could look like:

$$\sum _{i=1}^{n}h(t-t_i)={\int }_{-\infty }^{\infty }d$$

Footnotes

1. Dayan, Peter, and Laurence F. Abbott. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press, 2005.​ ↩