Conditioned Likelihoods Using Bifurcation Continuation in Inverse   Modeling of Dynamical Systems
  The Morris-Lecar (ML) model has applications to neuroscience and cognition. A simple network consisting of a pair of synaptically coupled ML neurons can exhibit a wide variety of deterministic behaviors including asymmetric amplitude state (AAS), equal amplitude state (EAS), and steady state (SS). In addition, in the presence of noise this network can exhibit mixed-mode oscillations (MMO), which represent the system being stochastically driven between these behaviors. In this paper, we develop a method to specifically estimate the parameters representing the coupling strength (gsyn) and the applied current (Iapp) of two reciprocally coupled and biologically similar neurons. This method employs conditioning the likelihood on cumulative power and mean voltage. Conditioning has the potential to improve the identifiability of the estimation problem. Conditioning likelihoods are typically much simpler to model than the explicit joint distribution, which several studies have shown to be difficult or impossible to determine analytically. We adopt a rejection sampling procedure over a closed defined region determined by bifurcation continuation analyses. This rejection sampling procedure is easily embedded within the proposal distribution of a Bayesian Markov chain Monte Carlo (MCMC) scheme and we evaluate its performance. This is the first report of a Bayesian parameter estimation for two reciprocally coupled Morris-Lecar neurons, and we find a proposal utilizing rejection sampling reduces parameter estimate bias relative to naive sampling. Application to stochastically coupled ML neurons is a future goal. 
Karleigh Cameron, Marissa Saladin
math.DS math.ST stat.TH