Long-range intercellular Ca²⁺ wave patterns

Modulational instability is utilized to investigate intercellular Ca²⁺ wave propagation in an array of diffusively coupled cells. Cells are supposed to be connected via paracrine signaling, where long-range effects, due to the presence of extracellular messengers, are included. The multiple-scale expansion is used to show that the whole dynamics of Ca²⁺ waves, from the endoplasmic reticulum to the cytosol, can be reduced to a single differential-difference nonlinear equation whose solutions are assumed to be plane waves. Their linear stability analysis is studied, with emphasis on the impact of long-range coupling, via the range parameter s. It is shown that s, as well as the number of interacting cells, importantly modifies the features of modulational instability, as small values of s imply a strong coupling, and increasing its value rather reduces the problem to a first-neighbor one. Our theoretical findings are numerically tested, as the generic equations are fully integrated, leading to the emergence of nonlinear patterns of Ca²⁺ waves. Strong long-range coupling is pictured by extended trains of breather-like structures whose frequency decreases with increasing s. We also show numerically that the number of interacting cells plays on the spatiooral formation of Ca²⁺ patterns, whilst the quasi-perfect intercellular communication depends on the paracrine coupling parameter.