Compact Dynamical Mean-Field Theory of Oscillator Networks

Abstract

We present a compact dynamical mean-field theory (DMFT) for large networks of coupled phase oscillators whose phases live on the circle $S^1$ and interact with both coherent mean-field coupling and quenched randomness. Starting from wrapped Langevin dynamics, we build a path-integral representation that keeps the $2π$-periodicity of the phases explicit. After averaging over the disorder in the thermodynamic limit, this construction reduces to a single-oscillator stochastic equation driven by a deterministic mean field and a self-consistent colored Gaussian noise, whose covariance is fixed by a circular two-time correlator. In the limit of vanishing disorder, the formalism reproduces the Ott--Antonsen reduction and recovers standard Kuramoto and theta-neuron neural-mass equations. The same framework accommodates arbitrary $2π$-periodic coupling functions, including those obtained from infinitesimal phase response curves (iPRCs) of biophysical neuron models. As an example, we show that for adaptive exponential integrate-and-fire neurons, inserting an iPRC-fitted coupling into the compact DMFT yields quantitative predictions for synchronization thresholds, providing a direct route from single-neuron phase response data to network-level mean-field predictions for arbitrary phase-reducible oscillators.

Figures (2)

• Figure 1: iPRC-based parameterization of compact DMFT. (a) Infinitesimal phase response curve $Z(\phi)$ for an AdEx neuron, computed via the adjoint method. The first Fourier harmonic has phase $\alpha \approx -0.98$ rad and synchronizing component $s = -\sin\alpha \approx 0.83$. (b) Mean-channel synchronization: order parameter $R = |Z|$ versus coupling strength $J_0$ for a network of $N = 2000$ phase oscillators with AdEx-derived coupling. Blue circles: network simulations (mean $\pm$ SEM across runs). Black curve: OA prediction \ref{['eq:R_above_threshold']}. Dashed line: theoretical threshold $J_{0,c}^{\rm th} = \Delta/(|h_1| s) \approx 0.121$. Parameters: $\Delta = 0.1$, $D = 0$.
• Figure 2: DMFT correlator closure and finite-size scaling. (a--c) Magnitude $|Q(\tau)|$ from direct network simulations (black curves, mean $\pm$ SEM across probes and disorder realizations) compared with single-rotor DMFT dynamics driven by self-consistent colored noise (shaded bands, mean $\pm$ SEM across independent noise draws) at three subcritical couplings $g/g_c^{\rm eff} \in \{0.70, 0.85, 0.97\}$. Here $g_c^{\rm eff} = \gamma_0/|h|$ is set by the measured incoherent dephasing rate. (d) Finite-size deviation $\Delta_N$ at $g = 0.85\,g_c^{\rm eff}$ scales as $N^{-1/2}$, consistent with central-limit fluctuations around the DMFT limit. Parameters: $D = 0.05$, $\Delta = 0.3$, Lorentzian $p(\omega)$.