Let
$$n_i(t)= H(u_i(t))$$
where $N\geq i\geq 1$, $H(.)$ is the Heaviside function and
$$ u_i(t) = \sum_{j=1}^N J_{ij} n_j(t) $$
We start with a random $\vec{n}(0)$ and each step of time $t'<t$ pick up a unit $n_k$ uniformly and update its states as $u_k(t')=H(u_k(t'))$.
The question is what is the distribution of $u_i$ at time $t$, for different realisation of $\vec{n}(0)$ given a fix realisation of $J_{ij}$. I assume this is not known generally. Thus, still it would be great if someone can point out the conditions for convergence of $u_i(t)$ to the Gaussian distribution.
Perhaps, an obvious condition is that $J=[J_{ij}]$ matrix is distributed like the iid Gaussian with mean $0$ an standard deviation of $1/\sqrt{N}$, thereafter for $N, t \rightarrow \infty$ leads to $u_i \sim N(0,1)$. Is this correct? Anyone here knows more general statement for matrix $J$? Can we do anything whenever $J_{ij}$ are not generally independent?