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Control of neural field equations with step-function inputs

Cyprien Tamekue, ShiNung Ching

TL;DR

The paper studies controllability of Amari-type neural field equations under constant/step inputs with the goal of steering the cortical activity profile from $a_0$ to a prescribed $a_1$ within a finite horizon $T$. It develops two complementary approaches: (i) a Banach fixed-point synthesis that yields an implicit constant input under mild regularity, and (ii) a robust generic drift-flow framework that leverages the nonlinear flow $U_t$ and its inverse to construct explicit piecewise-constant and constant-in-time inputs, aided by forward/backward solution representations. A central contribution is the explicit forward nominal-state synthesis and its practical explicit approximations, together with spectral conditions ensuring solvability; these are validated by extensive 1D and 2D numerical experiments showing improved performance over naive linearizations. The results offer a mathematically rigorous toolkit for steering neural-field dynamics, with potential applications in computational neuroscience, psychophysics, and noninvasive neurostimulation, and provide a foundation for predicting visually paradoxical percepts via controlled stimulation regimes.

Abstract

Wilson-Cowan and Amari-type models capture nonlinear neural population dynamics, providing a fundamental framework for modeling how sensory and other exogenous inputs shape activity in neural tissue. We study the controllability properties of Amari-type neural fields subject to piecewise/constant-in-time inputs. The model describes the time evolution of the polarization of neural tissue within a spatial continuum, with synaptic interactions represented by a convolution kernel. We study the synthesis of piecewise/constant-in-time inputs to achieve two-point boundary-type control objectives, namely, steering neural activity from an initial state to a prescribed target state. This approach is particularly relevant for predicting the emergence of paradoxical neural representations, such as discordant visual illusions that occur in response to overt sensory stimuli. We first present a control synthesis based on the Banach fixed-point theorem, which yields an iterative construction of a constant-in-time input under minimal regularity assumptions on the kernel and transfer function; however, it exhibits practical limitations, even in the linear case. To overcome these challenges, we then develop a generic synthesis framework based on the flow of neural dynamics drift, enabling explicit piecewise constant and constant-in-time inputs. Extensive numerical results in one and two spatial dimensions confirm the effectiveness of the proposed syntheses and demonstrate their superior performance compared to inputs derived from naive linearization at the initial or target states when these states are not equilibria of the drift dynamics. By providing a mathematically rigorous framework for controlling Amari-type neural fields, this work advances our understanding of nonlinear neural population control with potential applications in computational neuroscience, psychophysics, and neurostimulation.

Control of neural field equations with step-function inputs

TL;DR

The paper studies controllability of Amari-type neural field equations under constant/step inputs with the goal of steering the cortical activity profile from to a prescribed within a finite horizon . It develops two complementary approaches: (i) a Banach fixed-point synthesis that yields an implicit constant input under mild regularity, and (ii) a robust generic drift-flow framework that leverages the nonlinear flow and its inverse to construct explicit piecewise-constant and constant-in-time inputs, aided by forward/backward solution representations. A central contribution is the explicit forward nominal-state synthesis and its practical explicit approximations, together with spectral conditions ensuring solvability; these are validated by extensive 1D and 2D numerical experiments showing improved performance over naive linearizations. The results offer a mathematically rigorous toolkit for steering neural-field dynamics, with potential applications in computational neuroscience, psychophysics, and noninvasive neurostimulation, and provide a foundation for predicting visually paradoxical percepts via controlled stimulation regimes.

Abstract

Wilson-Cowan and Amari-type models capture nonlinear neural population dynamics, providing a fundamental framework for modeling how sensory and other exogenous inputs shape activity in neural tissue. We study the controllability properties of Amari-type neural fields subject to piecewise/constant-in-time inputs. The model describes the time evolution of the polarization of neural tissue within a spatial continuum, with synaptic interactions represented by a convolution kernel. We study the synthesis of piecewise/constant-in-time inputs to achieve two-point boundary-type control objectives, namely, steering neural activity from an initial state to a prescribed target state. This approach is particularly relevant for predicting the emergence of paradoxical neural representations, such as discordant visual illusions that occur in response to overt sensory stimuli. We first present a control synthesis based on the Banach fixed-point theorem, which yields an iterative construction of a constant-in-time input under minimal regularity assumptions on the kernel and transfer function; however, it exhibits practical limitations, even in the linear case. To overcome these challenges, we then develop a generic synthesis framework based on the flow of neural dynamics drift, enabling explicit piecewise constant and constant-in-time inputs. Extensive numerical results in one and two spatial dimensions confirm the effectiveness of the proposed syntheses and demonstrate their superior performance compared to inputs derived from naive linearization at the initial or target states when these states are not equilibria of the drift dynamics. By providing a mathematically rigorous framework for controlling Amari-type neural fields, this work advances our understanding of nonlinear neural population control with potential applications in computational neuroscience, psychophysics, and neurostimulation.
Paper Structure (23 sections, 18 theorems, 133 equations, 4 figures)

This paper contains 23 sections, 18 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Synthesis based on the Banach fixed-point theorem
  4. Generic syntheses: Proper generalization of the linear case
  5. Abstract Formulation
  6. Representations and series expansions of the solution
  7. Forward nominal-state synthesis
  8. Generic syntheses in practice
  9. Explicit approximations for the remaining syntheses
  10. On the spectral conditions and beyond
  11. Comments on the generic syntheses and perspectives
  12. How can the syntheses be used to predict visual illusions?
  13. Numerical illustration of the proposed constant-in-time input syntheses
  14. Examples in a one-dimensional spatial domain
  15. Example in two-dimensional spatial domain
  16. ...and 8 more sections

Key Result

Proposition 2.3

Let $p\in[1, \infty]$ and $(a_0, I)\in L^p({\mathbb R}^d)\times L^\infty([0, \infty); L^p({\mathbb R}^d))$. There exists a unique solution $a_I\in C^0([0, \infty); L^p({\mathbb R}^d))$ to eq:NF-intro. Letting $a_I(t):=a_I(\cdot,t)$, one has the following representation Moreover, the following estimate holds

Figures (4)

  • Figure 1: Log-scale plot of the endpoint mismatch $\log_{10} \lvert a(x, T) - a_1(x) \rvert$ for each of the four control strategies, with $a_0$ and $a_1$ defined in \ref{['eq:initial and target state faye']} and final time $T = 0.25$. (a) Forward nominal-state input: $\|a_{fn}(\cdot, T) - a_1(\cdot)\|_2 = 4.4 \times 10^{-4}$, $\|a_{fn}(\cdot, T) - a_1(\cdot)\|_\infty = 3.3 \times 10^{-4}$. (b) Forward final-state input: $\|a_f(\cdot, T) - a_1(\cdot)\|_2 = 9.9 \times 10^{-4}$, $\|a_f(\cdot, T) - a_1(\cdot)\|_\infty = 7.9 \times 10^{-4}$. (c) Backward initial-state input: $\|a_b(\cdot, T) - a_1(\cdot)\|_2 = 1.1 \times 10^{-3}$, $\|a_b(\cdot, T) - a_1(\cdot)\|_\infty = 8.8 \times 10^{-4}$. (d) Backward nominal-state input: $\|a_{bn}(\cdot, T) - a_1(\cdot)\|_2 = 9.4 \times 10^{-4}$, $\|a_{bn}(\cdot, T) - a_1(\cdot)\|_\infty = 7.7 \times 10^{-4}$. The mismatch is sharply localized around $x = 0$, where the target bump is centered and where the derivatives $f'(U_T(a_0))$, $f'(a_1)$, $f'(a_0)$, and $f'(V_T(a_1))$ reach their maximum values, amplifying sensitivity to perturbations. Here $\omega$ and $f$ are given in \ref{['eq:wizard 1D']}.
  • Figure 2: Comparison of $\log_{10}$ endpoint errors and input norms for each control synthesis strategy as a function of the time horizon $T \in \{0.0625,\,0.125,\,0.25,\,0.5,\,1,\,2,\,4,\,8\}$, with $\omega$ and $f$ defined in \ref{['eq:DoG 1D']} and $a_0$, $a_1$ given in \ref{['eq:considered initial and target state']}, satisfying $\|a_1 - a_0\|_{L^2(\Omega)} = 1.49$ and $\|a_1 - a_0\|_{L^\infty(\Omega)} = 0.65$. Each panel shows the $\log_{10}$ of the $L^2(\Omega)$ and $L^\infty(\Omega)$ endpoint errors and input norms for: (a) forward nominal-state input $I_{fn}$, (b) forward final-state input $I_f$, (c) backward initial-state input $I_b$, (d) backward nominal-state input $I_{bn}$, (e) input linearized at the initial state $I_{a_0}$, (f) input linearized at the target state $I_{a_1}$. As $T$ increases, the input norms generally decrease for all syntheses except $I_{a_0}$, which decreases up to $T=4$ and then rises. For small time horizons, the forward and backward nominal-state syntheses yield the lowest endpoint errors, followed by the backward initial-state and forward final-state syntheses. Notably, both linearized inputs $I_{a_0}$ and $I_{a_1}$ given in \ref{['eq:linearized input syntheses']} perform significantly worse than the generic syntheses in terms of both error and input norm.
  • Figure 3: Numerical simulation of the two-dimensional neural field model \ref{['eq:NF-intro']} controlled via the backward nominal-state input $I_{bn}$ over the time horizon $T = 0.5$, using parameters from \ref{['eq:DoG 2D']} and initial/target states from \ref{['eq:initial and target state 2D']}. (a) Initial state $a_0(x_1,x_2)$. (b) Target state $a_1(x_1,x_2)$. (c) Final state $a_{bn}(x_1,x_2,T)$. (d) Input $I_{bn}(x_1,x_2)$ used throughout $[0,T]$. The endpoint errors are $\|a_{bn}(T) - a_1\|_2 = 1.7 \times 10^{-4}$ and $\|a_{bn}(T) - a_1\|_\infty = 8.4 \times 10^{-5}$, while the input norms are $\|I_{bn}\|_2 = 5.05$ and $\|I_{bn}\|_\infty = 1.96$. For reference, the initial discrepancy is $\|a_1 - a_0\|_2 = 2.57$ and $\|a_1 - a_0\|_\infty = 0.99$.
  • Figure :

Theorems & Definitions (44)

  • Remark 1.3
  • Definition 2.1
  • Definition 2.2: Step controllability
  • Proposition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proof 1
  • ...and 34 more