Determine the coefficient c in the effective noise intensity for strong coupling

Determine the constant c appearing in the effective noise intensity approximation for strongly coupled, globally diffusive networks of stochastic bistable elements governed by dx_i = [f(x_i) + K (X − x_i)] dt + alpha dW_i(t), with f(x) = −x(x − r)(x − 1) and X = (1/N) ∑_{j=1}^N x_j. Specifically, in the strong-coupling and large-N regime, the authors propose that the effective noise intensity for x_i is approximately alpha/sqrt(N) * (1 + c/K); establish the value of c by a systematic reduction (e.g., via an appropriate white-noise approximation of the Ornstein–Uhlenbeck displacements y_i = x_i − X) that justifies this form.

Background

In the strong coupling regime, the authors argue that synchronization among nodes reduces the effective noise intensity affecting each element. By linearizing around the mean field X and analyzing the Ornstein–Uhlenbeck dynamics of displacements y_i = x_i − X, they estimate Var[y_i] ≈ alpha^2/(2K) and propose an effective noise intensity for x_i that combines the mean-field noise alpha/sqrt(N) with a correction due to the displacements.

They posit an approximate form for the effective noise intensity as alpha/sqrt(N) * (1 + c/K), where c is a constant, consistent with the observed K^{-1} approach of the mean escape time to its asymptote. However, they explicitly state that they were unable to determine c systematically, noting that this would require an appropriate white-noise approximation of the Ornstein–Uhlenbeck process driving the displacements.