Abstract
Hodgkin and Huxley (1952) contrived a single neuron model (Hodgkin-Huxley model, or HH model) which was given in the form of non linear differential equations of four variables. This model simulates the variation of the cell membrane potential by an electric circuit composed of condensers, resisters and embedded gates through which the sodium and potassium ion current pass. HH model was treated mainly as a deterministic model for simulations and analysis in the past.
We treat HH model as a non-linear stochastic process which will mimic the behavior of the neuron of as a part of the active neural circuit. Discretization is a natural way to treat stochastic process by digital computer. As a general method of discretization of nonlinear stochastic process, Ozaki (1993) proposed ‘the local linearization method’ and applied it to the Van del Pol equation etc., and verified its goodness in precision. Though HH model is more complicated multi variate system, we prove that the combination of the local linearization method and Kalman filter can be effectively applied to HH model. Next, we augment the state equation by introducing the exogenous current trend input. The trend input can be estimated via smoothing procedure. Even if the model parameters are unknown, both the exogenous current input and the model parameters can be estimated simultaneously by the maximum likelihood method.
