Intercellular calcium waves in the fire-diffuse-fire framework: Green's function for gap-junctional coupling

Calcium is a crucial component in a plethora of cellular processes involved in cell birth, life, and death. Intercellular calcium waves that can spread through multiple cells provide one form of cellular communication mechanism between various parts of cell tissues. Here we introduce a simple, yet biophysically realistic model for the propagation of intercellular calcium waves based on the fire-diffuse-fire type model for calcium dynamics. Calcium release sites are considered to be discretely distributed along individual linear cells that are connected by gap junctions and a solution of this model can be found in terms of the Green's function for this system. We develop the "sum-over-trips" formalism that takes into account the boundary conditions at gap junctions providing a generalization of the original sum-over-trips approach for constructing the response function for branched neural dendrites. We obtain the exact solution of the Green's function in the Laplace (frequency) domain for an infinite array of cells and show that this Green's function can be well approximated by its truncated version. This allows us to obtain an analytical traveling wave solution for an intercellular calcium wave and analyze the speed of solitary wave propagation as a function of physiologically important system parameters. Periodic and irregular traveling waves can be also sustained by the proposed model.