posted on 2005-08-01, 14:11authored byYulia Timofeeva, S. Coombes
The De Young Keizer model for intracellular calcium oscillations is
based around a detailed description of the dynamics for inositol trisphosphate
(IP3) receptors. Systematic reductions of the kinetic schemes for IP3 dynamics
have proved especially fruitful in understanding the transition from excitable to
oscillatory behaviour. With the inclusion of diffusive transport of calcium ions
the model also supports wave propagation. The analysis of waves, even in reduced
models, is typically only possible with the use of numerical bifurcation techniques.
In this paper we review the travelling wave properties of the biophysical De Young
Keizer model and show that much of its behaviour can be reproduced by a much
simpler Fire-Diffuse-Fire (FDF) type model. The FDF model includes both a refractory
process and an IP3 dependent threshold. Parameters of the FDF model
are constrained using a comprehensive numerical bifurcation analysis of solitary
pulses and periodic waves in the De Young Keizer model. The linear stability
of numerically constructed solution branches is calculated using pseudospectral
techniques. The combination of numerical bifurcation and stability analysis also
allows us to highlight the mechanisms that give rise to propagation failure. Moreover,
a kinematic theory of wave propagation, based around numerically computed
dispersion curves is used to predict waves which connect periodic orbits. Direct
numerical simulations of the De Young Keizer model confirm this prediction. Corresponding
travelling wave solutions of the FDF model are obtained analytically
and are shown to be in good qualitative agreement with those of the De Young
Keizer model. Moreover, the FDF model may be naturally extended to include
the discrete nature of calcium stores within a cell, without the loss of analytical
tractability. By considering calcium stores as idealised point sources we are able
to explicitly construct solutions of the FDF model that correspond to saltatory
periodic travelling waves.