Predicting the firing behaviour of neural network through the bifurcation analysis of derivative mean-field model

The mean-field model is an important method for understanding the complex dynamics of the nervous system at different spatial levels and for simulating and theoretically analysing the collective dynamic behaviour of large neural populations. In the work, we construct an improved mean-field system of neural networks coupled with quadratic integrate-and-fire neurons and examine the discharge patterns for networks by analysing such a model, which is a three-dimensional smooth differential system. Bifurcation analysis of the mean-field model is conducted from both theoretical and simulation perspectives, we obtain the bifurcation conditions of some co-dimension-two bifurcations and divide the parameter space into different regimes by simulating two parameters bifurcation diagrams. We find a close correspondence, though with some variance, between the mean-field model and neural network when comparing the firing patterns of the two models in various parameter regimes. In summary, the obtained mean-field description builds the bridge between the parameters of neurons or networks and that of a mean-field system to ensure we can compare them and understand the connections between them. Specifically, the mean-field model can reflect the dynamics of neural networks from a macroscopic perspective, and its bifurcation can predict the behaviour of neural networks to a certain extent and understand the mechanisms behind them, such as bursting dynamics of neural networks.