Neural Population Coding Is Optimized by Discrete Tuning Curves

The sigmoidal tuning curve that maximizes the mutual information for a Poisson neuron, or population of Poisson neurons, is obtained. The optimal tuning curve is found to have a discrete structure that results in a quantization of the input signal. The number of quantization levels undergoes a hierarchy of phase transitions as the length of the coding window is varied. We postulate, using the mammalian auditory system as an example, that the presence of a subpopulation structure within a neural population is consistent with an optimal neural code.

Figure 1

The partition boundaries ${\theta }_i$ against the maximum mean spike count $N=T{\nu }_{\max }$. Also shown for $N=2$, 7, 15, and 22 are the optimal $f(x)$ (top insets) and the population distributions (bottom insets). The parameters are $x_{\min }=0$, $x_{\max }=1$, ${\nu }_{\min }=0$, and $x$ uniformly distributed.

Figure 2

The set of optimal firing rates ${\varphi }_i$, $i=0,\dots ,M-1$, for the overall population with $M-1$ subpopulations, as a function of $N$ (top), and the mutual information for the optimal solution (bottom). The dashed line shows the mutual information that would result if binary coding were utilized [14].