New math enables precise control of neural populations with step inputs

Researchers Cyprien Tamekue and ShiNung Ching have published a rigorous framework for controlling Amari-type neural fields—mathematical models that describe how populations of neurons evolve over a spatial continuum via nonlinear dynamics and synaptic convolution kernels. Their work, appearing in the SIAM Journal on Control and Optimization (arXiv preprint 2510.22022), tackles a fundamental challenge: using step-function (piecewise-constant-in-time) inputs to drive neural activity from an initial state to a prescribed target state. This is critical for understanding phenomena like paradoxical visual illusions and for designing neurostimulation protocols.

The authors first present a control synthesis based on the Banach fixed-point theorem, which iteratively constructs a constant input under minimal assumptions on the kernel and transfer function. However, this approach has practical limitations even for linear cases. To overcome these, they developed a generic synthesis leveraging the flow of neural drift dynamics, yielding explicit piecewise-constant or constant inputs. Extensive 1D and 2D numerical experiments confirm that their method significantly outperforms naive linearization strategies when initial or target states are not equilibria of the drift. This advance offers a mathematically rigorous pathway for steering nonlinear neural populations, with direct implications for computational neuroscience, psychophysics, and next-generation neurostimulation technologies.