# zbMATH — the first resource for mathematics

Wave bifurcation and propagation failure in a model of Ca$$^{2+}$$ release. (English) Zbl 1037.92014
Summary: The De Young Keizer model [see G. W. De Young and J. Keizer, Proc. Natl. Acad. Sci. USA 89, 9895–9899 (1992); J. Theor. Biol. 166, 431–442 (1994)] for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP$$_3$$) receptors. Systematic reductions of the kinetic schemes for IP$$_3$$ 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.
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 IP$$_{3}$$ 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.

##### MSC:
 92C37 Cell biology 92C05 Biophysics 90C40 Markov and semi-Markov decision processes 37N25 Dynamical systems in biology
Full Text: