On a reaction–diffusion model for calcium dynamics in dendritic spines

As was pointed out by Holcman and Schuss, [D. Holcman and Z. Schuss, Modeling calcium dynamics in dendritic spines, SIAM J. Appl. Math. 65 (3) (2005) 1006–1026], the concentration of calcium ions inside the dendritic spines plays a crucial role in the synaptic plasticity, and in consequence in cognitive processes like learning and memory. The goal of this paper is to study the reaction–diffusion model for calcium dynamics in dendritic spines proposed by Holcman and Schuss. We start from the construction of the model of Holcman and Schuss, taking special care on the hypotheses in order to set a problem with realistic biological assumptions, supported by experimental evidence, but still mathematically solvable. We show that the dynamics of the calcium ions and the proteins interacting with them follows a system of coupled nonlinear reaction–diffusion equations, which is degenerate elliptic if the proteins are considered fixed, and strongly elliptic if they diffuse with a diffusion coefficient d>0d>0. In the first case we prove a priori estimates, global existence, global uniqueness and positivity of solutions, whereas in the second case we prove not only the same features but also that the problem is well-posed. Moreover, we show that there is a “continuous” link between the two problems in the sense that the solutions of the problem with d>0d>0 converge to the solutions of the problem with d=0d=0.