An optimization-based equilibrium measure describes non-equilibrium steady state dynamics: application to edge of chaos

Understanding neural dynamics is a central topic in machine learning, non-linear physics and neuroscience. However, the dynamics is non-linear, stochastic and particularly non-gradient, i.e., the driving force can not be written as gradient of a potential. These features make analytic studies very challenging. The common tool is to use path integral approach or dynamical mean-field theory, but the drawback is one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed form solutions in general. From the aspect of associated Fokker-Planck equation, the steady state solution is generally unknown. Here, we treat searching for the steady state as an optimization problem, and construct an approximate potential closely related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running a stochastic gradient dynamics. The resultant stationary state follows exactly the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and different scaling behavior with respect to inverse temperature in both sides of the transition is also revealed. Our method opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.