Interacting systems of infinite spiking neurons with weights beyond uniform summability

We consider an infinite system of spiking neurons with a drift and both excitatory and inhibitory connections. We study conditions for non-explosiveness and the uniqueness of the invariant measure. In particular, we examine conditions that allow this infinite interacting system to go beyond the usual interactions of uniformly summable weights. As a result, we extend the Galves-L"ocherbach model beyond the restrictive uniform summability of the model.