Replica-free evaluation of the neuronal population information with mixed continuous and discrete stimuli: from the linear to the asymptotic regime.

Recent studies have explored theoretically the ability of populations of neurons to carry information about a set of stimuli, both in the case of purely discrete or purely continuous stimuli, and in the case of multidimensional continuous angular and discrete correlates, in the presence of additional quenched disorder in the distribution. An analytical expression for the mutual information has been obtained in the limit of large noise by means of the replica trick. Here, we show that the same results can actually be obtained in most cases without the use of replicas, by means of a much simpler expansion of the logarithm. Fitting the theoretical model to real neuronal data, we show that the introduction of correlations in the quenched disorder improves the fit, suggesting a possible role of signal correlations-actually detected in real data-in a redundant code. We show that even in the more difficult analysis of the asymptotic regime, an explicit expression for the mutual information can be obtained without resorting to the replica trick despite the presence of quenched disorder, both with a Gaussian and with a more realistic thresholded-Gaussian model. When the stimuli are mixed continuous and discrete, we find that with both models the information seem to grow logarithmically to infinity with the number of neurons and with the inverse of the noise, even though the exact general dependence cannot be derived explicitly for the thresholded-Gaussian model. In the large noise limit, lower values of information were obtained with the thresholded-Gaussian model, for a fixed value of the noise and of the population size. On the contrary, in the asymptotic regime, with very low values of the noise, a lower information value is obtained with the Gaussian model.