Abstract
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 tau_{a} and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter epsilon=tau _{a}/tau, 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 epsilonrightarrow0). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. 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/epsilon-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.
Highlights
One of the major challenges in neuroscience is developing our understanding of how noise at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations
We showed how asymptotic methods recently developed to study metastability in other stochastic hybrid systems, such as stochastic ion channels, motor-driven intracellular cargo transport, and gene networks [28,29,30,31,32], can be extended to analyze metastability in stochastic hybrid neural networks, in a regime where the synaptic dynamics is much slower than the spiking dynamics
We first describe a stochastic neural network model that generalizes the neural master equation [17, 18, 20] by incorporating synaptic dynamics. (A more detailed derivation of the model can be found in [27].) Note that there does not currently exist a complete, rigorous derivation of population rate-based models starting from detailed biophysical models of individual neurons, some significant progress has been made in a series of papers by Buice and Chow on generalized activity equations for theta neurons [35,36,37]
Summary
One of the major challenges in neuroscience is developing our understanding of how noise at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Bressloff [18, 19] showed how the path-integral representation of the master equation can be used to investigate large deviations or rare event statistics underlying escape from the basin of attraction of a metastable state, following along analogous lines to previous work on large deviations in chemical master equations [24,25,26] One limitation of both versions of the neural master equation is that they neglect the dynamics of synaptic currents. We recently extended the Buice et al neural master equation by formulating the network population dynamics in terms of a stochastic hybrid system known as a ‘velocity’ jump Markov process [27].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
