  In the study of randomly connected neural network dynamics there is a phase
transition from a `simple' state with few equilibria to a `complex' state
characterised by the number of equilibria growing exponentially with the neuron
population. Such phase transitions are often used to describe pathological
brain state transitions observed in neurological diseases such as epilepsy. In
this paper we investigate how more realistic heterogeneous network structures
affect these phase transitions using techniques from random matrix theory.
Specifically, we parameterise the network structure according to Dale's Law and
use the Kac-Rice formalism to compute the change in the number of equilibria
when a phase transition occurs. We also examine the condition where the network
is not balanced between excitation and inhibition causing outliers to appear in
the eigenspectrum. This enables us to compute the effects of different
heterogeneous network connectivities on brain state transitions, which can
provide new insights into pathological brain dynamics.
