  The critical brain hypothesis posits that neural circuits may operate close
to critical points of a phase transition, which has been argued to have
functional benefits for neural computation. Theoretical and computational
studies arguing for or against criticality in neural dynamics have largely
relied on establishing power laws in neural data, while a proper understanding
of critical phenomena requires a renormalization group (RG) analysis. However,
neural activity is typically non-Gaussian, nonlinear, and non-local, rendering
models that capture all of these features difficult to study using standard
statistical physics techniques. We overcome these issues by adapting the
non-perturbative renormalization group (NPRG) to work on network models of
stochastic spiking neurons. Within a ``local potential approximation,'' we are
able to calculate non-universal quantities such as the effective firing rate
nonlinearity of the network, allowing improved quantitative estimates of
network statistics. We also derive the dimensionless flow equation that admits
universal critical points in the renormalization group flow of the model, and
identify two important types of critical points: in networks with an absorbing
state there is a fixed point corresponding to a non-equilibrium phase
transition between sustained activity and extinction of activity, and in
spontaneously active networks there is a physically meaningful \emph{complex
valued} critical point, corresponding to a discontinuous transition between
high and low firing rate states. Our analysis suggests these fixed points are
related to two well-known universality classes, the non-equilibrium directed
percolation class, and the kinetic Ising model with explicitly broken symmetry,
respectively.
