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

TL;DR

This work addresses non-gradient, high-dimensional non-equilibrium neural dynamics by introducing a quasi-potential that optimizes for low-speed phase-space regions, enabling a Boltzmann-like steady-state description in the zero-temperature limit. By applying the replica method to a recurrent neural network with asymmetric couplings, the authors derive order parameters for activity and integrated response, and show that the chaos transition is continuous and occurs along the line $g(1+\gamma)=1$, with a peak in the response at the transition. The framework recasts steady-state analysis into an equilibrium-like problem, connects to dynamical mean-field theory, and reveals a fluctuation–response structure encoded in the replica overlaps, thereby providing analytic access to the steady-state landscape of non-gradient high-dimensional dynamics. This quasi-potential approach offers a versatile tool for studying steady states in complex systems—from neural networks to ecological or optimization dynamics—where transient behavior is difficult to capture.

Abstract

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 the path integral approach or dynamical mean-field theory, but the drawback is that 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 states as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running an approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, the distribution of the original steady states can be achieved. The resultant stationary state of the dynamics 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, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.

Figures (9)

• Figure 1: Key idea of our quasi-potential method and phase diagram of the considered RNN model. (a) A three-neuron recurrent neural network is shown in the left panel. Neurons are bidirectionally connected with the strength displayed near the arrows (synaptic directions). Both connections in opposite directions are correlated once $\gamma \neq 0$. Here, we simulate the RNN dynamics [Eq. \ref{['eq: dynamicEquation']}], whose phase space displays two fixed points ($g=1.2,\gamma=0.1$). In the right panel, we show the trajectories flowing to these fixed points, where the color and size of the dots represent the energy of the corresponding position in the phase space, and the arrows represent the direction and the $\ell_2$ norm of the velocity (the length of the arrow). We also show an example of the trivial fixed-point for comparison ($g=0.8,\gamma=0.1$), where nearby trajectories are shown in the light grey color. The cross symbols denote these corresponding fixed points. (b) The phase diagram as a function of gain parameter $g$ and symmetric degree $\gamma$ is divided into two regions: null fixed-point and chaotic phases, separated by an analytic transition line $g(1+\gamma)=1$Brunel-2018. (c) We show the representative behavior of the dynamics [$\phi(x)$] in the two phases, corresponding to two colored points in the phase diagram of (b).
• Figure 2: The average energy (a) and $\ell_2$ norm (b) of network activity versus the inverse temperature $\beta$ in the i.i.d. scenario $\gamma=0$ with $\eta=0$. The orange and the blue curves represent the setting $g=0.8$ and $g=1.2$, respectively. The red dashed line denotes the ground state energy. Two lines of $g=0.8$ and $g=1.2$ are nearly indistinguishable.
• Figure 3: Order parameter profile as a function of gain parameters $g$ and symmetric degree $\gamma$. $g$ is varied from $0.4$ to $1.2$, and $\gamma$ is varied from $-0.2$ to $1.0$ with an interval of $0.1$. The colors of curves correspond to the different $\gamma$ values. The grey vertical plane specifies the location of the critical line $g \qty(\gamma + 1) = 1$Brunel-2018.
• Figure 4: The behavior of correlation function $C$ and response function $R_{\text{int}}$ derived from the DMFT method Zou-2023. We choose $\gamma = 0.0, 0.2, 0.5$, and $g$ varies from $0.5$ to $1.1$ in the simulations. The vertical dashed lines denote the critical points for each $\gamma$ value.
• Figure 5: Fluctuation-response relationship for non-equilibrium steady states. The plotted order parameters follow the saddle-point equation derived in the main text [see Eq. \ref{['sde-rnn']}]. (a) $\sigma_{x\phi}$ vs $q-Q$. (b) $\sigma_{x\phi}$ vs $r-R$. (c) $r-R$ vs $q-Q$.