Balanced-regime confinement and collapse conjecture for strongly coupled networks

Let $X^n=(x^n_{i_1},\cdots,x^n_{i_p})$ be a vector composed of one representative neuron in each of the $P$ populations, and let

$$\mathcal{B}=\{(\mu^1,\cdots,\mu^P)\in\mathcal{M}^1(\mathbb{R}^d)^P:\ \forall x_p\in\operatorname{Supp}(\mu^p),\ \sum_{q=1}^P g_{p,q}\int_{\mathbb{R}^d}b_{p,q}(x_p,y)\mu^q(dy)=0\}$$

be the balanced regime. The associated deterministic dynamics are

$$\frac{dx_p}{dt}=\sum_{q=1}^{P}g_{p,q}\int_{\mathbb{R}^d}b_{p,q}(x_p,y)\mu_q(dy).$$

Balanced-regime confinement and collapse conjecture. If the set $\mathcal{B}$ is nonempty, there exist asymptotic solutions confined to the balanced regime for all times. Moreover, if $\mathcal{B}$ is attractive for the deterministic dynamical system, in the sense that solutions with initial conditions $x_p^0$ near the support of $\mu_p$ converge to that support, then even initial conditions outside $\mathcal{B}$ lead to collapse onto $\mathcal{B}$ at a timescale inversely proportional to $\gamma(n)$; that is, for any $t>0$, $\mathbb{P}[X^n_t\in\mathcal{B}]\to1$ as $n\to\infty$. This conjecture predicts whether divergent interactions produce confinement to, or rapid attraction toward, the zero-net-input manifold; its resolution depends on establishing asymptotic dynamics and stability for the strongly coupled network.