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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 bewritten as gradient of a potential. These features make analytic studies verychallenging. The common tool is to use path integral approach or dynamicalmean-field theory, but the drawback is one has to solve theintegro-differential or dynamical mean-field equations, which iscomputationally expensive and has no closed form solutions in general. From theaspect of associated Fokker-Planck equation, the steady state solution isgenerally unknown. Here, we treat searching for the steady state as anoptimization problem, and construct an approximate potential closely related tothe speed of the dynamics, and find that searching for the ground state of thispotential is equivalent to running a stochastic gradient dynamics. Theresultant stationary state follows exactly the canonical Boltzmann measure.Within this framework, the quenched disorder intrinsic in the neural networkscan be averaged out by applying the replica method. Our theory reproduces thewell-known result of edge-of-chaos, and further the order parameterscharacterizing the continuous transition are derived, and different scalingbehavior with respect to inverse temperature in both sides of the transition isalso revealed. Our method opens the door to analytically study the steady statelandscape of the deterministic or stochastic high dimensional dynamics.

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