Replica symmetric evaluation of the information transfer in a two-layer network in presence of continuous and discrete stimuli

In a previous paper we have evaluated analytically the mutual information between the firing rates of N independent units and a set of multidimensional continuous and discrete stimuli, for a finite population size and in the limit of large noise. Here, we extend the analysis to the case of two interconnected populations, where input units activate output ones via Gaussian weights and a threshold linear transfer function. We evaluate the information carried by a population of M output units, again about continuous and discrete correlates. The mutual information is evaluated solving saddle-point equations under the assumption of replica symmetry, a method that, by taking into account only the term linear in N of the input information, is equivalent to assuming the noise to be large. Within this limitation, we analyze the dependence of the information on the ratio M/N, on the selectivity of the input units and on the level of the output noise. We show analytically, and confirm numerically, that in the limit of a linear transfer function and of a small ratio between output and input noise, the output information approaches asymptotically the information carried in input. Finally, we show that the information loss in output does not depend much on the structure of the stimulus, whether purely continuous, purely discrete or mixed, but only on the position of the threshold nonlinearity, and on the ratio between input and output noise.