Multistability, travelling waves and autonomous pacemakers in bistable active medium
with periodic boundary conditions
I.A. Shepelev*, T.E. Vadivasova, D.E Postnov
*Saratov State University, Astrakhanskaya Str. 83, Saratov, Russia
e-mail: igor_sar@li.ru
The bistable active medium with periodic boundary conditions is studied. The elementary cell of
this medium is FitzHugh-Nagumo oscillator, which is in bistable state. FitzHugh-Nagumo
oscillator is used for modeling of some processes in neuron system. The medium is described by
following equations:
¶x
x3
¶2 x
= x- - y+d 2 ,
¶t
3
¶s
(1)
¶y
= e (a x + b - y),
¶t
when x  x( s, t ), y  y ( s, t ) are dimensionless real dynamic variables, s and t are dimensionless
spatial coordinate and time, respectively, α, β are medium parameters, d is coefficient of the
diffusion coupling of the elementary cells. So, control parameter α responsible for the kind of
stable state of oscillator. Parameter β define symmetry of research oscillator: when β =0 system
is symmetric else asymmetrical. The system (1) is studied in a case of bistable cells for fixed
parameters β = 0 and changing control parameters α and d.
Simulation showed that the bistable medium (1) could demonstrate different types of behaviors,
such as: 1) stationary spatial structures without oscillations in the time is observed at small d,
2) travelling waves, 3) autonomous pacemakers, 4) state of rest in someone stable state, 5)
Collapse and disappearance travelling waves (for very big d). Also, researching active medium is
multistable.
Characteristically, the regime of travelling waves, the regime of autonomous pacemakers and
regime of state of rest in someone stable state are fundamentally different and these are
observed at same values d but for different initial conditions. Types of initial conditions for
appropriate regimes were found. Revealed that phase velocity and frequency of switching of
travelling wave are increased at increased value of diffusion coupling. It was found for regime of
autonomous pacemakers that local heterogeneities might significantly influence on properties of
medium and cause transition to regime of travelling waves.
References
1. FitzHugh R. Mathematical models of excitation and propagation in nerve.
Ch. 1 H.P. Schwan, ed. Biological Engineering, McGraw–Hill Book Co., N.Y(1969),
pp. 1–85.
2. F. Muller, L. Schimansky-Geier D.E. Postnov. Interaction of noise supported Ising–
Bloch fronts with Dirichlet boundaries, ECONOM-373; No. 2013 pp. 1-16.
3. D. E. Postnov, F. Müller, R. B. Schuppner, and L. Schimansky-Geier. Dynamical
structures in binary media of potassium-driven neurons, PHYSICAL REVIEW E 80,
031921, 2009, pp. 1-12.