On the control of recurrent neural networks using constant inputs
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912606690738176 |
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| author | Tamekue, Cyprien Chen, Ruiqi Ching, ShiNung |
| author_facet | Tamekue, Cyprien Chen, Ruiqi Ching, ShiNung |
| contents | This paper investigates the controllability of a broad class of recurrent neural networks widely used in theoretical neuroscience, including models of large-scale human brain dynamics. Motivated by emerging applications in non-invasive neurostimulation such as transcranial direct current stimulation (tDCS), we study the control synthesis of these networks using constant and piecewise constant inputs. The neural model considered is a continuous-time Hopfield-type system with nonlinear activation functions and arbitrary input matrices representing inter-regional brain interactions. Our main contribution is the formulation and solution of a control synthesis problem for such nonlinear systems using specific solution representations. These representations yield explicit algebraic conditions for synthesizing constant and piecewise constant controls that solve a two-point boundary value problem in state space up to higher-order corrections with respect to the time horizon. In particular, the input is constructed to satisfy a tractable small-time algebraic relation involving the Jacobian of the nonlinear drift, ensuring that the synthesis reduces to verifying conditions on the system matrices. For canonical input matrices that directly actuate $k$ nodes, this implies that the reachable set (with constant inputs) of a given initial state is an affine subspace whose dimension equals the input rank and whose basis can be computed efficiently using a thin QR factorization. Numerical simulations illustrate the theoretical results and demonstrate the effectiveness of the proposed synthesis in guiding the design of brain stimulation protocols for therapeutic and cognitive applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the control of recurrent neural networks using constant inputs Tamekue, Cyprien Chen, Ruiqi Ching, ShiNung Optimization and Control Dynamical Systems Neurons and Cognition This paper investigates the controllability of a broad class of recurrent neural networks widely used in theoretical neuroscience, including models of large-scale human brain dynamics. Motivated by emerging applications in non-invasive neurostimulation such as transcranial direct current stimulation (tDCS), we study the control synthesis of these networks using constant and piecewise constant inputs. The neural model considered is a continuous-time Hopfield-type system with nonlinear activation functions and arbitrary input matrices representing inter-regional brain interactions. Our main contribution is the formulation and solution of a control synthesis problem for such nonlinear systems using specific solution representations. These representations yield explicit algebraic conditions for synthesizing constant and piecewise constant controls that solve a two-point boundary value problem in state space up to higher-order corrections with respect to the time horizon. In particular, the input is constructed to satisfy a tractable small-time algebraic relation involving the Jacobian of the nonlinear drift, ensuring that the synthesis reduces to verifying conditions on the system matrices. For canonical input matrices that directly actuate $k$ nodes, this implies that the reachable set (with constant inputs) of a given initial state is an affine subspace whose dimension equals the input rank and whose basis can be computed efficiently using a thin QR factorization. Numerical simulations illustrate the theoretical results and demonstrate the effectiveness of the proposed synthesis in guiding the design of brain stimulation protocols for therapeutic and cognitive applications. |
| title | On the control of recurrent neural networks using constant inputs |
| topic | Optimization and Control Dynamical Systems Neurons and Cognition |
| url | https://arxiv.org/abs/2410.17199 |