The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of Ca2+ release

The fire-diffuse-fire model provides an idealised model of Ca2+ release within living cells. The effect of calcium pumps, which drive Ca2+ back into internal stores, is often neglected for mathematical simplicity. Here we show how to explicitly analyse such effects by extending the work of Keizer et al. (J. E. Keizer, G. D. Smith, S. Ponce Dawson and J. Pearson, 1998, Saltatory propagation of Ca2+ waves by Ca2+ sparks, Biophysical Journal, 75, 595-600). For travelling waves, in which release events occur sequentially, we construct the speed of waves in terms of the time-scale at which pumps operate. An immediate consequence of this analysis is that the inclusion of calcium pumps leads to multiple solutions. A linear stability analysis determines those solution branches in parameter space which are stable. Numerical continuation is used to provide explicit examples of the bifurcation diagrams of the speed of waves as a function of physiologically significant system parameters.