Eur. Phys. J. E 3, 205-219
Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models
V.N. Belykh1 - I.V. Belykh2 - M. Colding-Jørgensen3 - E. Mosekilde4
1 Advanced School of General and Applied Physics,
Nizhny Novgorod University, 23 Gagarin Ave., Nizhny Novgorod,
603600 Russia
2 Radiophysical Department, Nizhny Novgorod State
University, 23 Gagarin Ave., Nizhny Novgorod 603600, Russia
3 Scientific Computing, Novo Nordisk A/S, 2880 Bagsværd,
Denmark
4 Center for Chaos and Turbulence Studies, Department of
Physics, The Technical University of Denmark, 2800 Lyngby, Denmark
ellen@chaos.fys.dtu.dk
Received 24 June 1999 and Received in final form 17 February 2000
Abstract
We present a qualitative analysis of a generic model
structure that can simulate the bursting and spiking dynamics of
many biological cells. Four different scenarios for the emergence
of bursting are described. In this connection a number of theorems
are stated concerning the relation between the phase portraits of
the fast subsystem and the global behavior of the full model. It is
emphasized that the onset of bursting involves the formation of a
homoclinic orbit that travels along the route of the bursting
oscillations and, hence, cannot be explained in terms of
bifurcations in the fast subsystem. In one of the scenarios, the
bursting oscillations arise in a homoclinic bifurcation in which
the one-dimensional (1D) stable manifold of a saddle point becomes
attracting to its whole 2D unstable manifold. This type of
homoclinic bifurcation, and the complex behavior that it can
produce, have not previously been examined in detail. We derive a
2D flow-defined map for this situation and show how the map
transforms a disk-shaped cross-section of the flow into an annulus.
Preliminary investigations of the stable dynamics of this map show
that it produces an interesting cascade of alternating pitchfork
and boundary collision bifurcations.
PACS
05.45.-a Nonlinear dynamics and nonlinear dynamical systems
Copyright EDP Sciences, Società Italiana di Fisica, Springer-Verlag
