Closed-Loop Theory, Programmable Propagation, and Path Optimality in Memory-Modulated Reaction-Diffusion Systems

To address the widespread lack of memory modulation mechanisms, the absence of closed loop design criteria, and insufficient engineering realizability in existing reaction-diffusion spatiotemporal computing systems, this paper proposes the third-generation Memory-Modulated Spatiotemporal Computing Model (MSCM 3.0). A minimally complete coupled partial dif ferential equation (PDE) system consisting of a signal field, a memory field, and an obstacle field is constructed. This work rigorously proves the necessary and sufficient conditions for the existence of propagation wavefronts, the criteria for convergence and non-convergence, the weak optimality of endogenous path selection, and the criteria for history dependence, thereby establishing a complete theoretical closed loop. Numerical experiments are designed to validate phase-diagram partitioning, wavefront propagation dynamics, and path formation mechanisms, and measurable engineering design indicators are provided. A benchmarking analysis is completed against classical reaction-diffusion systems, neural field models, and memristor networks, thereby clarifying the shared properties and core advantages of the pro posed model. By modulating the core control quantity—the effective growth rate µ(m, z)— MSCM 3.0 realizes programmable propagation-path design and completes the transition from a theoretical PDE system to an implementable spatiotemporal computing system, thus pro viding a new theoretical framework and implementation scheme for adaptive path planning, environmental perception, and related spatiotemporal computing tasks.