A bang-bang optimal control for a nonlinear system modeling the Gate Control Theory of Pain

TL;DR

The paper analyzes a nonlinear ODE model of Gate Control Theory–based pain pathways and studies an optimal control problem where the short-frequency input $x_s$ is used to minimize a combined state-energy and pain cost. It establishes well-posedness and long-time behavior of the state, and shows monotone dependence of the pain-related T-cell potential on inputs and cognitive control. Using the Pontryagin Maximum Principle, the authors prove the optimal control is of bang-bang type with a single switching time and provide explicit switching conditions, offering theoretical guarantees and practical guidance for patient-specific neurostimulation strategies. Overall, the work connects rigorous control-theoretic analysis with a biologically motivated model to yield a tractable, implementable control law for pain modulation.

Abstract

We consider a nonlinear system of coupled ordinary differential equations (representing the excitatory, inhibitory, and T-cell potentials) based on the Gate Control Theory of Pain, initially proposed by R. Melzack and P.D. Wall in 1965, and later mathematically modeled by N.F. Britton and S.M. Skevington in 1988.

Figures (10)

• Figure 1: Basic design of the pain gate
• Figure 2: Qualitative representation of the potentials ${\rm V}_{{\rm I}},{\rm V}_{{\rm E}}$ y ${\rm V}_{{\rm T}}$
• Figure 3: Phase plain for the potentials ${\rm V}_{{\rm I}}$ and ${\rm V}_{{\rm T}}$
• Figure 4: Monotone dependence of ${\rm V}_{{\rm T}}$ with respect to $x_{s}$
• Figure 5: Monotone dependence of ${\rm V}_{{\rm T}}$ with respect to $\alpha_{c{\rm I}}$