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Parameter estimation in multidimensional diffusion models with only onecoordinate observed is highly relevant in many biological applications, but astatistically difficult problem. In neuroscience, the membrane potentialevolution in single neurons can be measured at high frequency, but biophysicalrealistic models have to include the unobserved dynamics of ion channels. Onesuch model is the stochastic Morris-Lecar model, defined by a nonlineartwo-dimensional stochastic differential equation. The coordinates are coupled,that is, the unobserved coordinate is nonautonomous, the model exhibitsoscillations to mimic the spiking behavior, which means it is not ofgradient-type, and the measurement noise from intracellular recordings istypically negligible. Therefore, the hidden Markov model framework isdegenerate, and available methods break down. The main contributions of thispaper are an approach to estimate in this ill-posed situation and nonasymptoticconvergence results for the method. Specifically, we propose a sequential MonteCarlo particle filter algorithm to impute the unobserved coordinate, and thenestimate parameters maximizing a pseudo-likelihood through a stochastic versionof the Expectation-Maximization algorithm. It turns out that even the ratescaling parameter governing the opening and closing of ion channels of theunobserved coordinate can be reasonably estimated. An experimental data set ofintracellular recordings of the membrane potential of a spinal motoneuron of ared-eared turtle is analyzed, and the performance is further evaluated in asimulation study.

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