Introduction

Suppose that we have a population $\mathcal{P}$ of an arbitrary number of neurons $N$. Suppose that the population responds to a time dependent scalar variable $x(t)$ by producing a number of spike trains as shown by Fig. 1, i.e.

Figure 1: Population of neurons

$$S(t)=[s_1(t), s_2(t), \cdots, s_N(t)],$$

where

$$s_j(t)=\sum_{k=1}^{n_j(t)}\delta(t-t_k^j)\qquad \forall j\in \{1, 2 \cdots, N\},$$

and $n_j(t)$ is the total number of spikes produced by the $j-$th neuron in the interval $[0, t]$. We allow the possibility of $n_j=0$, in which case $s_j(t)\equiv 0$.

Let us suppose that the population acts with a certain degree of randomness, i.e. the repetition of the same $x(t)$ does not produce identical response. Assuming that the collection of spike trains carries ``enough'' information about $x(t)$ (i.e. $x(t)$ is encoded in the spike trains), we can ask the following question: How would one decode an arbitrary function of $x$ (and in particular $x$ itself) from the spike train observations?