On the control of recurrent neural networks using constant inputs

Area of Science:

• Computational Neuroscience
• Control Theory
• Systems Neuroscience

Background:

• Recurrent neural networks are crucial for modeling large-scale brain dynamics.
• Non-invasive neurostimulation, like transcranial direct current stimulation (tDCS), shows promise for therapeutic and cognitive applications.
• Controlling these complex nonlinear systems remains a significant challenge.

Purpose of the Study:

• To investigate the control synthesis of Hopfield-type recurrent neural networks used in theoretical neuroscience.
• To develop explicit algebraic conditions for synthesizing constant and piecewise constant controls for these nonlinear neural systems.
• To demonstrate the application of these control methods in designing brain stimulation protocols.

Main Methods:

• Formulation and solution of a control synthesis problem for continuous-time Hopfield-type neural networks.
• Utilizing specific solution representations to derive explicit algebraic conditions for control synthesis.
• Employing small-time algebraic relations involving the Jacobian of the nonlinear drift for tractable input construction.
• Analyzing the reachable set of initial states using constant inputs, characterized as an affine subspace.

Main Results:

• Explicit algebraic conditions were derived for synthesizing constant and piecewise constant controls.
• The control synthesis was shown to reduce to verifying conditions on system matrices.
• For canonical input matrices, the reachable set was identified as an affine subspace with efficiently computable basis.
• Numerical simulations validated the theoretical results and the effectiveness of the proposed synthesis.

Conclusions:

• The study provides a novel framework for the control synthesis of nonlinear recurrent neural networks relevant to brain dynamics.
• The derived conditions offer a pathway for designing effective non-invasive neurostimulation protocols.
• The findings have implications for advancing therapeutic and cognitive applications of brain stimulation.