A nonlinear Fokker-Planck description of continuous neural network dynamics

Abstract

The nonextensive thermostatistical formalism has been increasingly applied to the description of many complex systems in physics, biology, psychology, economics, and other fields. The q-Maximum Entropy (q-MaxEnt) distributions, which optimize the Sq, power-law entropic functionals, are central to this formalism. We have done previous work regarding computational neural models of associative memory functioning, for mental phenomena such as neurosis, creativity, and the interplay between consciousness and unconsciousness, which suggest that q-MaxEnt distributions may be relevant for the development of neural models for these processes. Power-law behavior has also been experimentally observed in brain functioning. We propose here a nonlinear Fokker-Planck model, associated with the continuous-time evolution equations for interconnected neurons of the Hopfield model. The equation which characterizes the model has stationary solutions of the q-MaxEnt type and is associated with a free energy like quantity that decreases during the time-evolution of the system. This framework elucidates a possible dynamical mechanism which can generate q-MaxEnt distributions in Hopfield memory neural networks. It also provides a theoretical framework that supports the choice of different entropic measures for modelling and simulating complex networks such as the brain, as well as other artificial neural networks.