Synchronization of Coupled Equations of Hodgkin-Huxley Type
Abstract
We study a class of differential equations modelling the electrical
activity in biological systems. This class includes the
Hodgkin-Huxley equations for the nerve impulse, as well as models for
other excitable tissue, like muscle fibers, pacemakers and
pancreatic cells.
We show that when two of these
equations are coupled their solutions always synchronize.
Synchronization takes place regardless of the initial condition if the
coupling is strong enough, and even for two
equations with different parameter values, coupled asymetrically.
We find a bounded region in phase space that attracts the flow globally
and thus contains all points with recurrent behaviour. The size of the
region can be calculated from the parameters in the equations. Thus we
show
that the system is uniformly dissipative. We obtain explicit bounds for
this
region in terms of the parameters as a tool for establishing
synchronization. These estimates are also obtained for the uncoupled
equations.