  Bifurcation theory is a powerful tool for studying how the dynamics of a
neural network model depends on its underlying neurophysiological parameters.
However, bifurcation theory has been developed mostly for smooth dynamical
systems and for continuous-time non-smooth models, which prevents us from
understanding the changes of dynamics in some widely used classes of artificial
neural network models. This article is an attempt to fill this gap, through the
introduction of algorithms that perform a semi-analytical bifurcation analysis
of a spin-glass-like neural network model with binary firing rates and
discrete-time evolution. Our approach is based on a numerical brute-force
search of the stationary and oscillatory solutions of the spin-glass model,
from which we derive analytical expressions of its bifurcation structure by
means of the state-to-state transition probability matrix. The algorithms
determine how the network parameters affect the degree of multistability, the
emergence and the period of the neural oscillations, and the formation of
symmetry-breaking in the neural populations. While this technique can be
applied to networks with arbitrary (generally asymmetric) connectivity
matrices, in particular we introduce a highly efficient algorithm for the
bifurcation analysis of sparse networks. We also provide some examples of the
obtained bifurcation diagrams and a Python implementation of the algorithms.
