Identifying the dynamical transitions of a stochastic spiking neural network by topological data analysis

The relationship between the structures of neural networks and their dynamics is yet not well understood, and studies on this topic are still greatly needed. Changes in the parameters of a network can cause its state to shift between different regimes, and in this work, we studied a neural network of stochastic spiking neurons, whose dynamics change when the control parameter varies. More specifically, as showed by some numerical results, a bistable region appears and eventually disappears when the control parameter increases in an interval, and a critical-like transition appears. However, this kind of transition is always difficult to be detected. Here we used a new approach based on topological data analysis (TDA) that uses superlevel persistence to visualize this transition through a "homological bifurcation plot", which shows the changes of the 0th Betti numbers of the KDE (kernel density estimate) of the activities of the network.