2018 Impact factor 1.686
Soft Matter and Biological Physics

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

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