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Brain memory working. Optimal control behavior for improved Hopfield-like models

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 to the recognition of every pattern. 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, \(T_{ij}=A_{ij}+\xi_{ij}\). Additionally, we introduce a reasonable controlled variational functional for optimization. This allows us to simulate three potential outcomes when a pattern is submitted to the learning system: (1) if the dynamics converges to an existing stationary point without activating controls, the system has \emph{recognized} an existing 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} without reaching any stationary point, the system is unable to recognize or learn the submitted pattern. An additional feature (4) models the processes of \emph{forgetting and restoring} memory.

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