Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods

Abstract : Parameter estimation in multi-dimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem. In neuro-science, the membrane potential evolution in single neurons can be measured at high frequency, but biophysical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris-Lecar model, defined by a non-linear two-dimensional stochastic differential equation. The coordinates are coupled, i.e. the unobserved coordinate is non-autonomous, the model exhibits oscillations to mimick the spiking behavior, which means it is not of gradient-type, and the measurement noise from intra-cellular recordings is typically negligible. Therefore the hidden Markov model framework is degenerate, and available methods break down. The main contributions of this paper are an approach to estimate in this ill-posed situation, and non-asymptotic convergence results for the method. Specifically, we propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parameters maximizing a pseudo-likelihood through a stochastic version of the Expectation- Maximization algorithm. It turns out that even the rate scaling parameter governing the opening and closing of ion channels of the unobserved coordinate can be reasonably estimated. An experimental data set of intracellular recordings of the membrane potential of a spinal motoneuron of a red-eared turtle is analyzed, and the performance is further evaluated in a simulation study.
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Annals Of Applied Statistics, Institute Mathematical Statistics, 2014, 8 (2), pp.674-702. <10.1214/14-AOAS729 >
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Susanne Ditlevsen, Adeline Samson. Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods. Annals Of Applied Statistics, Institute Mathematical Statistics, 2014, 8 (2), pp.674-702. <10.1214/14-AOAS729 >. <hal-00712331v2>

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