We consider applications of path-integral methods to the analysis of a
stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two
stochastic variables, a continuous synaptic variable and a discrete activity variable.
The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical
equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates
depend on the synaptic variables. We assume a separation of time scales between
fast spiking dynamics with time constant τa and slower synaptic dynamics with time
constant τ . This naturally introduces a small positive parameter = τa/τ , which can
be used to develop various asymptotic expansions of the corresponding path-integral
representation of the stochastic dynamics. First, we derive a variational principle for
maximum-likelihood paths of escape from a metastable state (large deviations in the
small noise limit → 0). We then show how the path integral provides an efficient
method for obtaining a diffusion approximation of the hybrid system for small . The
resulting Langevin equation can be used to analyze the effects of fluctuations within
the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects
of intrinsic noise on pattern formation in a spatially structured hybrid network. In
particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a 1/-loop expansion of
the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural
master equation.