Two types of criticality in neural network rate models: mapping to equilibrium phase transitions
Alessia Annibale (KCL London)
*SPECIAL TIME: 14 h*
The seminar will be onsite, and also on zoom (see below).
We consider a simple neural network model, evolving via non-linear coupled stochastic differential equations, where neural couplings are randomly asymmetric Gaussian variables with non-vanishing mean. We analyze the dynamics, averaged over the network ensemble, in the thermodynamic limit, using generating functional analysis. In the absence of noise, the fixed points of the dynamics can be characterized by two order parameters, namely the mean and variance of the neural activity, determined through a set of self-consistency equations, which close when couplings are fully asymmetric. These equations show that, for any degree of coupling asymmetry, two types of criticality emerge, corresponding to ferromagnetic and spin-glass order, respectively. The transition from the disordered phase to either of the ordered phases is consistent with spectral properties of the coupling matrix. Non-fixed point steady states are analysed for fully asymmetric interactions. Such solutions cannot be described by a closed set of equations for the stationary mean and variance, as the latter depends on a non-persistent order parameter, which quantifies time correlations in neural activity. This, in turn, evolves according to a gradient-descent dynamics on a potential, which depends on the variance itself. Our analysis reveals how the variance is dynamically selected by the system: whenever the potential is confining, so to allow a multitude of bounded steady-state solutions, the system selects the steady state corresponding to the separatrix curve. Chaotic motion is the manifestation, at the level of single network instances, of such ensemble-averaged dynamics laying on the curve that separates different realizable steady states. Finally, we show that when either the external signal or the noise exceed a certain threshold, that we calculate explicitly, the system is brought to a phase where only one bounded solution exists and chaos is suppressed.
Meeting ID: 991 7209 3004