Spike patterns as a window into non-injective transient diffusive processes

Complex natural systems at times manifest transitions between disparate diffusive regimes. Efforts to devise measurement techniques capable of identifying the cross-over moments have recently borne fruit, however interpretation of findings remains contentious when the bigger picture is considered. This study generalises the 1D Gierer-Meinhardt reaction – diffusion model to a system that permits transitions between regular diffusive regimes with distinct diffusivities as well as sub-diffusion of a variable order. This is a sufficiently general, yet tractable description for the dynamics of a pattern qualitatively redolent of molecular clusters subject to transient anomalous diffusion mechanisms. The resulting system of equations substantiates the difficulties encountered when attempting to distinguish between various diffusive regimes in experimental settings: a non-monotonic dependence of the pat-tern’s evolution on parameters defining the diffusion mechanism is a common occurrence, as is a non-injective mapping between a given sequence of diffusion regimes and ensuing drift behaviour.