  Parameter estimation in multidimensional diffusion models with only one
coordinate observed is highly relevant in many biological applications, but a
statistically difficult problem. In neuroscience, 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 nonlinear
two-dimensional stochastic differential equation. The coordinates are coupled,
that is, the unobserved coordinate is nonautonomous, the model exhibits
oscillations to mimic the spiking behavior, which means it is not of
gradient-type, and the measurement noise from intracellular 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 nonasymptotic
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.
