Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

Abstract

-The elephant walk model originally proposed by Schütz and Trimper to investigate non-Markovian processes led to the investigation of a series of other random-walk models. Of these, the best known is the Alzheimer walk model, because it was the first model shown to have amnestically induced persistence -i.e. superdiffusion caused by loss of memory. Here we study the robustness of the Alzheimer walk by adding a memoryless stochastic perturbation. Surprisingly, the solution of the perturbed model can be formally reduced to the solutions of the unperturbed model. Specifically, we give an exact solution of the perturbed model by finding a surjective mapping to the unperturbed model. Copyright c EPLA, 2012Introduction. -The theoretical foundations of the Brownian motion are well known and were laid more than one hundred years ago (see [1][2][3] and references therein). Since then, the random walk (RW), the simplest realization of the Brownian motion, has been used in the scientific literature as a prototype for modelling applications in various fields. The diffusion of the traditional RW is known as normal diffusion.Superdiffusion [4][5][6][7] is an accelerated diffusion, for which the mean squared displacement grows faster than linearly in time. Theoretical studies of anomalous diffusion are based on generalized Langevin equations [8,9], the fractional Fokker-Planck equation [10,11] and the continuous-time random-walk [12-14] approaches. Superdiffusion is possible only if the necessary and sufficient conditions of the central limit theorem for sums of random variables are not met, otherwise the behavior is always normal diffusion (e.g., Brownian motion). There are two basic mechanisms (not mutually exclusive) by which a random walker can undergo superdiffusion. The first is an infinite second moment for the random-walk step size distribution, as seen in Lévy processes [15][16][17]. The second mechanism is long-range power law correlations, i.e. long-range memory [18,19].

Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

“…For any time t > 0, the concentration of the substances depends on the reaction terms on the righthand side of the equation ( 1). This is known as the Markovian process, but in reality, these concentrations not only depend on the time instance t; rather, they depend on some weighted average concentrations of past time interval, say [t p , t] for t p < t. This is usually termed collectively as the memory effect (e.g., [49,50,51,53,55]). The distribution of the weights is proportional to some power of the length of the time interval, i.e., (t − t p ), following the power law correlation function [48,51].…”

“…Given that a reaction-diffusion process can depend not only on the previous time instance's concentrations but also on all the earlier stages of the concentrations with some weights [48,49,50,51,52,53], this idea is further developed in this paper. A time-fractional reaction-diffusion equation is used to analyze such processes in the AD context, which we call collectively the "memory effect".…”