A new mathematical model for brain memory working. Optimal control behavior for Hopfield networks
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911240914206720 |
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| author | Cardin, Franco Lovison, Alberto Maritan, Amos Megighian, Aram |
| author_facet | Cardin, Franco Lovison, Alberto Maritan, Amos Megighian, Aram |
| contents | 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 $|ξ_{ij}|\leq K$, which are used to adjust an initially constant interaction matrix, $T_{ij}=A_{ij}+ξ_{ij}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_14360 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new mathematical model for brain memory working. Optimal control behavior for Hopfield networks Cardin, Franco Lovison, Alberto Maritan, Amos Megighian, Aram Neurons and Cognition 91E40, 92B20, 93C95 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 $|ξ_{ij}|\leq K$, which are used to adjust an initially constant interaction matrix, $T_{ij}=A_{ij}+ξ_{ij}$ 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. |
| title | A new mathematical model for brain memory working. Optimal control behavior for Hopfield networks |
| topic | Neurons and Cognition 91E40, 92B20, 93C95 |
| url | https://arxiv.org/abs/2305.14360 |