Dynamical self-organization and formation of cortical maps

S. Amari (Dynamic Interactions in Neural Networks: Models and Data, M.A. Arbid and S. Amari, eds, p.15-34, New York, Springer Verlag, 1989) proposed a general mathematical formulation for the self-organization of synaptic efficacies and response fields under the influence of external stimuli. In the present work, these results are extended to neurons with sigmoidal input-output functions. The existence conditions of the map in the absence of recurrent connections and afferent autocorrelations are derived. Then, by utilizing the mathematically convenient δ-function expansion for well-localized yet otherwise arbitrary recurrent connections, the master equation is reduced into a partial differential equation guaranteed to possess analytic solutions with given boundary conditions (as addressed in the corresponding Cauchy problem). The receptive fields and the excitable regions of the map can thus be studied by standard methods of second-order partial differential equations of elliptic or hyperbolic type