The De Young Keizer model for intracellular calcium release 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 these reduced models support wave
propagation. The analysis of waves, even in reduced models, is typically
only possible with the use of numerical bifurcation techniques. Here 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 simpler
Fire-Diffuse-Fire (FDF) type model. The FDF model includes both a
refractory process and an IP3 dependent threshold. Moreover, the FDF model
may be naturally extended to include the discrete nature of calcium stores
within a cell, without 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
travelling waves.
Finally we introduce a computationally inexpensive model of calcium release
based upon a stochastic generalization of the Fire-Diffuse-Fire threshold
model. This model incorporates a notion of release probability via the
introduction of threshold noise. Apart from belonging to the Directed
Percolation universality class this model is shown to generate a form of
array enhanced coherence resonance whereby all calcium stores release
periodically and simultaneously.
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