Detection Problem

Let $O(t)$ be the observed response of the population $\mathcal{P}$ to a time varying scalar function $x(t)$. In general $O(t)$ can be any set of features of the response $S(t)$. Using Bayes' rule one can easily write

$$P(x(t) \vert O(t))=\frac{P(O(t) \vert x(t)) P(x(t))}{P(O(t))}$$ (1)

If $x(t)$ takes values from a discrete set of functions $\{x_i(t)\}_{1}^M$, we can formulate the following detection problem

$$x(t)=\{x_m(t) \vert x_m(t)=\arg \max_i{P(x_i(t) \vert O(t))}\}$$ (2)

The idea given by (2) is not a new one and has been extensively used for various applications. For example, if

$$O(t)=[n_1(t), n_2(t), \cdots, n_N(t)]$$

where $n_j(t)$ is the number of spikes fired by $j-$th neuron in time interval $[0, t]$, one has so-called rate decoding. Since neurons fire with a considerable degree of variability, one might argue that the precise spike timings are not important in the encoding process, and that the idea of rate decoding is justified. However, this encoding/decoding scheme is not very efficient, especially if the firing rates of neurons within the population are not sufficiently high. Assuming the signals are encoded in a sequence of spike times $\{t_k^j\}_{1}^{n_j}$ increases the information capacity of the population $\mathcal{P}$ tremendously. Here, we propose the decoding method that is based on full statistical description of the population response, namely we assume that $O(t)=S(t)$.