Decision making times in mean-field dynamic Ising model
Attempts in understanding of distributional asymptotics for the exit time in describing the vanishing noise asymptotics for noisy heteroclinic networks are an important topic in many papers. Such systems occur naturally in the context of neural dynamics and sequential decision making. Exit times (also for diffusion models) have been used in psychology to describe reaction times in decision tasks and it seems to natural to ask if the limiting behavior of exit times is reproduced in statistical mechanics models of neural computation. Based on this question, here, one of the simplest models of this kind is studied, namely, the dynamic mean field ferromagnetic model, also known as the Curie-Weiss model, in the low temperature regime with two minima of free energy. The author starts the evolution of the system at the completely disordered state with zero magnetization (the number of plus spins equals the number of minus spins). They stopped the dynamics as soon as magnetization enters a neighborhood of one of the stable equilibrium values and interpreted that event as a decision made by the system between the two choices. As it turns out, the number \(N\) of spin variables (representing individual neurons in the neural computation context) goes to infinity, the exit time behaves as \(\ln N\) and the correction to the main term converges to an affine transformation of \(\ln |G|^{-1}\) (where \(G\) is a standard Gaussian random variable), thus reproducing the above result for the diffusion in the neighborhood of an unstable equilibrium. This paper is organized as follows. After presenting preliminaries in the introduction, the author studies the model giving all necessary requirements (Section 2). Section 3 contains the proof of this model which is based on the theory of Markov processes, martingales and their convergence.
scientific article; zbMATH DE number 6063534
scientific article