A new mathematical model for brain memory working. Optimal control behavior for Hopfield networks
Recent works have highlighted the need for a new dynamical paradigm in the modeling of brain function and evolution. Specifically, these models should incorporate non-constant and asymmetric synaptic weights $T_{ij}$ in the neuron-neuron interaction matrix, moving beyond the classical Hopfield framework. Krotov and Hopfield proposed a non-constant yet symmetric model, resulting in a vector field that describes gradient-type dynamics, which includes a Lyapunov-like energy function. Firstly, we will outline the general conditions for generating a Hopfield-like vector field of gradient type, recovering the Krotov-Hopfield condition as a particular case. Secondly, we address the issue of symmetry, which we abandon for two key physiological reasons: (1) actual neural connections have a distinctly directional character (axons and dendrites), and (2) the gradient structure derived from symmetry forces the dynamics towards stationary points, leading for every pattern to a recognition or to a free association, if the equilibrium is rather far from the input. We propose a novel model that incorporates a set of limited but variable controls $|\xi_{ij}|\leq K$, which are used to adjust an initially constant interaction matrix, $T_{ij}=A_{ij}+\xi_{ij}$ according to a controlled variational functional. We simulate three potential outcomes when a pattern is submitted: (1) if the dynamics converges to an existing stationary point without activating controls, the system has \emph{recognized} an incoming pattern; (2) if a new stationary point is reached through control activation, the system has \emph{learned} a new pattern; and (3) if the dynamics \emph{wanders}, the system is unable to recognize or learn the submitted pattern. An additional feature (4) models the processes of \emph{forgetting and restoring} memory. Numerical simulations on a basic neural network model support the theoretical results.