A DIAGRAMMATIC APPROACH TO STUDY THE INFORMATION TRANSFER IN WEAKLY NONLINEAR CHANNELS

In a recent work we have introduced a novel approach to study the effect of weak nonlinearity in the transfer function on the information transmitted by an analogue channel, by means of a perturbative diagrammatic expansion. We extend here the analysis to all orders in perturbation theory, which allows us to release any constraint concerning the magnitude of the expansion parameter and to establish the rules to calculate easily the contribution at any order. As an example we explicitly compute the information up to the second order in nonlinearity, in presence of random Gaussian connectivity and in the limit when the output noise is not small. We analyze the first and second order contributions to the mutual information as a function of the nonlinearity and as a function of the number of output units. We believe that an extensive application of our method via the analysis of the different contributions at distinct orders might be able to fill a gap between well-known analytical results obtained for linear channels and nontrivial treatments which are required to study highly nonlinear channels.