Centre for Nonlinear
Studies, Institute of Cybernetics
at Tallinn University of Technology.
Title: On mathematical modelling of waves in nerve fibres.
Abstract: Beside celebrated action potential models for explaining
the nerve pulse propagation, the recent attention is turned also to
modelling of mechanical waves in fibre walls i.e., in cylindrical
biomembranes. The Heimburg-Jackson model (2005) with a special type of
nonlinearities has opened a wide avenue of studies of deformation waves in
biomembranes. Here we present the analysis of the improved
Heimburg-Jackson model which corresponds to the original (well-posed)
Boussinesq equation including the higher order mixed derivatives. In this
case the double dispersion model describes correctly the velocity limits
and accounts for inertia of lipids as microstructural constituents of
biomembranes. Such a governing equation of longitudinal waves along the
biomembrane admits the solitary wave solutions with the width controlled
by the additional dispersion term. Using the ideas from the theory of
rods, the transverse displacement is determined as the derivative (c.f.
experiments by Tasaki, 1988). The analysis includes the description of all the solution
types to such a governing equation, the demonstration of the emergence of
trains of solitary waves from a general initial condition and the study of
their interactions. An attempt is made to unite the physical phenomena of
the action potential, the deformation wave in the surrounding biomembrane
and the pressure wave in the axoplasm into one robust mathematical model.
The coupling mechanism is proposed to be realized by additional forces.