Wave Dynamics in the Transmission of Neural Signals - Advanced Models of Neural Networks - page 131

Chapter 6
Wave Dynamics in the Transmission of Neural
Signals

Abstract The chapter analyzes wave-type partial differential equations (PDEs) 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 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.

6.1 Outline

As explained in Chap. 1 , spatiotemporal variations of voltage in neurons' membrane
are usually described by partial differential equations (PDEs) of the wave or cable
type [ 1 , 38 , 176 , 185 , 194 ]. Certain models of the voltage dynamics consider also
the propagation of solitary waves through the membrane. Thus, voltage dynamics in
the membrane can be viewed as a distributed parameters system [ 23 , 211 ]. State
estimation in distributed parameter systems and in infinite dimensional systems
described by PDEs is a much more complicated problem than state estimation
in lumped parameter systems [ 46 , 72 , 74 , 174 , 213 , 218 ]. Previous results on state
estimation of wave-type nonlinear dynamics can be found in [ 29 , 81 , 201 , 208 ].