Theoretical model of neuronal population coding of stimuli with both continuous and discrete dimensions.

In a recent study, the initial rise of the mutual information between the firing rates of N neurons and a set of p discrete stimuli has been analytically evaluated, under the assumption that neurons fire independently of one another to each stimulus and that each conditional distribution of firing rates is Gaussian. Yet real stimuli or behavioral correlates are high dimensional, with both discrete and continuously varying features. Moreover, the Gaussian approximation implies negative firing rates, which is biologically implausible. Here, we generalize the analysis to the case where the stimulus or behavioral correlate has both a discrete and a continuous dimension, like orientation and shape could be in a visual stimulus, or type and direction in a motor action. The functional relationship between the firing patterns and the continuous correlate is expressed through the tuning curve of the neuron, using two different parameters to modulate its width and its flatness. In the case of large noise, we evaluate the mutual information up to the quadratic approximation as a function of population size. We also show that in the limit of large N and assuming that neurons can discriminate between continuous values with a resolution Delta(theta), the mutual information grows to infinity like ln(1/Delta(theta)) when Delta(theta) goes to zero. Then we consider a more realistic distribution of firing rates, truncated at zero, and we prove that the resulting correction, with respect to the Gaussian firing rates, can be expressed simply as a renormalization of the noise parameter. Finally, we demonstrate the effect of averaging the distribution across the discrete dimension, evaluating the mutual information only with respect to the continuously varying correlate.