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 idealized model of Ca2+ release wit hin living cells. The effect of calcium pumps, which drive Ca2+ back into i nternal stores, is often neglected for mathematical simplicity. Here we sho w how to explicitly analyse such effects by extending the work of Keizer et nl. [Keizer, J. E., G. D. Smith, S. Pence Dawson and J. Pearson (1998). Sa ltatory propagation of Ca2+ waves by Ca2+ sparks. Biophys. J. 75, 595-600.] . For travelling waves, in which release events occur sequentially, we cons truct the speed of waves in terms of the time-scale at which pumps operate. An immediate consequence of this analysis is that the inclusion pf calcium pumps leads to multiple solutions. A Linear stability analysis determines those solution branches in parameter space which are stable. Numerical cont inuation is used to provide explicit examples of the bifurcation diagrams o f the speed of waves as a function of physiologically significant system pa rameters. (C) 2000 Society for Mathematical Biology.