Balanced-regime convergence under strong coupling

Prove the conjecture for the stochastic network dx^i_t = (f_{p_i}(x^i_t) − ∑_{j=1}^n J^{p_i,p_j}_{ij}(n) b_{p_i,p_j}(x^i_t,x^j_t)) dt + σ_{p_i} dW^i_t with P populations and interaction scaling such that n J^{p,q}_{ij}(n) → ∞ for all (p,q). Let γ(n) be the maximal divergence rate and define g_{p,q} = lim_{n→∞} J^{p,q}_{ij}(n)/γ(n). Define the balanced regime B as the set of measures (μ^1,…,μ^P) for which, for every x_p in Supp(μ^p), ∑_{q=1}^P g_{p,q} ∫ b_{p,q}(x_p,y) μ^q(dy) = 0 for all p. Establish that: (i) if B ≠ ∅, there exist asymptotic solutions confined to B for all times; and (ii) if B is attractive for the ODE system dx_p/dt = −∑_{q=1}^P g_{p,q} ∫ b_{p,q}(x_p,y) μ_q(dy), then for any initial conditions outside B the process collapses onto B on a timescale proportional to 1/γ(n), i.e., for any fixed t > 0, P[X^n_t ∈ B] → 1 as n → ∞.

Background

The paper studies networks of interacting stochastic agents partitioned into P populations with dynamics given by dx^i_t = (f_{p_i}(x^i_t) − ∑{j=1}^n J^{p_i,p_j}{ij}(n) b_{p_i,p_j}(x^i_t,x^j_t)) dt + σ{p_i} dW^i_t. Unlike classical mean-field scaling J ∼ 1/n, the focus is on strong coupling where nJ^{p,q}{ij}(n) diverges as n grows. In this regime, the usual McKean–Vlasov limit does not directly apply.

Motivated by the balance phenomenon in excitatory/inhibitory neural networks introduced by Sompolinsky and van Vreeswijk, the authors define a balanced manifold B of measures for which the net interaction input cancels for all states in the support. They conjecture that, when B is non-empty, the dynamics collapse rapidly onto B and remain confined there, and, furthermore, that if the corresponding deterministic ODE flow is attractive, the collapse occurs from arbitrary initial conditions on a timescale inversely proportional to the divergence rate γ(n). The paper provides numerical evidence and proves partial results in specific one-dimensional settings, but the general conjecture remains open.