Generalized Thermostatistics and the Nonequilibrium Landscape Description of Neural Network Dynamics

Generalized thermostatistical formalisms arising from extensions or generalizations of the standard logarithmic entropy are attracting considerable attention nowadays, specially associated with the study of complex systems. Probability distributions optimizing non-standard entropies are common in Nature and are observed in diverse types of complex systems, including those relevant to neuroscience and artificial intelligence (AI). Nonlinear Fokker-Planck dynamics constitutes one of themain mechanisms that can give rise to these distributions. We present a family of nonlinear Fokker-Planck equations associated with general, continuous, neural network, dynamical models for associative memory. Such models are relevant in AI, and in the study of mental life, because memory is an essential ingredient inmany phenomena explored by neuroscience and psychology. We investigate how the nonlinear Fokker-Planck approach to network dynamics is related to the nonequilibrium landscape description of this dynamics analyzed in [36]. We prove that, within general nonequilibrium thermostastical settings, the landscape treatment uncovers deep links between the Liapunov function of the network dynamics, the deterministic phase-space flow of the network, and the properties of the diffusion term in the nonlinear Fokker-Planck equations. These connections, in turn, lead to H-theorems involving free-energy-like functionals related to the generalized entropies. This contribution extends and generalizes our previous studies that focused only on the S-q entropies [35]. We illustrate our present developments by applying them to the celebrated Cohen-Grossberg family of neural network models.