Estimation of wave-type dynamics in neurons' membrane with the use of the Derivative-free nonlinear Kalman Filter

The paper analyzes wave-type partial differential equations that describe the transmission of neural signals and proposes filtering for estimating the spatiotemporal variations of voltage in the neurons' membrane. It is shown that in specific neuron models the spatiotemporal variations of the membrane's voltage follow partial differential equations (PDEs) of the wave type while in other models such variations are associated with the propagation of solitary waves in the membrane. To compute the dynamics of the membrane PDE model without knowledge of initial conditions and through the processing of noisy measurements, a new filtering method, under the name Derivative-free nonlinear Kalman Filtering, is proposed. The PDE of the membrane is decomposed into a set of nonlinear ordinary differential equations with respect to time. Next, each one of the local models associated with the ordinary differential equations is transformed into a model of the linear canonical (Brunovsky) form through a change of coordinates (diffeomorphism) which is based on differential flatness theory. This transformation provides an extended model of the nonlinear dynamics of the membrane for which state estimation is possible by applying the standard Kalman Filter recursion. The proposed filtering method is tested through numerical simulation tests.