Microscopic and Macroscopic States in Neural Networks

Abstract

Firing patterns in neurons are thought to be key in the understanding of how the brain works. Trying to model the behaviour of neural networks is the current state-of-the-art in several scientific disciplines, but often these models are analytically intractable due to their complexity and high dimensionality. Recently, the Lorentz Ansatz proposed by Montbrió, Pazó and Roxin, showed that an exact description of macroscopic observables for a neural network is possible under some constraints. This thesis is aimed to re-deriving the ansatz based on the existing work of Montbrió et. al. and analyzing the different states encountered in the model using Bifurcation Theory. We also extend the ansatz and relax some of the constraints. Furthermore, we build and run some simulations of a Quadratic Integrate-and-Fire network to test the theory and its generalization.