Reaction-subdiffusion front propagation in a comblike model of spiny dendrites

Fractional differentiation method: Application to the trapping reactions in the comb-like structures with relaxation

The theory of subdiffusion and reactions occurring in confined domains such as comb structures has widespread applications in various fields of science. We study trapping diffusion-controlled reactions taking place in the backbone of a two-dimensional comb-like structure. A brief survey of previously published works on diffusion transport, both without and with inertial effects, is given. To take into account inertial effects, in this paper, we used the known Compte-Metzler analysis to choose an appropriate fractional diffusive Cattaneo system. By means of the chosen suitable fractional diffusive Cattaneo system, we derive the diffusion equation comprising a 1/2-order fractional time derivative as a damping term. With the help of the fractional differentiation method, we analytically evaluate the trapping reaction rate coefficient for short and long time regimes. Numerical calculations performed for the reaction rate coefficient showed a very good agreement with the obtained analytical expressions.

Random Walks on Comb-like Structures under Stochastic Resetting

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.

Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of the comb model, which allows diffusion along the backbone when the particles are not in the branches. We also consider memory effects, which may be handled with different fractional derivative operators involving singular and non-singular kernels. We find exact solutions for the particle distributions in this model that display normal and anomalous diffusion regimes when the mean-squared displacement is determined. As an application, we use this model to fit the anisotropic diffusion of water along and across the axons in the optic nerve using magnetic resonance imaging. The results for the observed diffusion times (8 to 30 milliseconds) show an anomalous diffusion both along and across the fibers.

A Random Walk Approach to Transport in Tissues and Complex Media: From Microscale Descriptions to Macroscale Models

The biological processes necessary for the development and continued survival of any organism are often strongly influenced by the transport properties of various biologically active species. The transport phenomena involved vary over multiple temporal and spatial scales, from organism-level behaviors such as the search for food, to systemic processes such as the transport of oxygen from the lungs to distant organs, down to microscopic phenomena such as the stochastic movement of proteins in a cell. Each of these processes is influenced by many interrelated factors. Identifying which factors are the most important, and how they interact to determine the overall result is a problem of great importance and interest. Experimental observations are often fit to relatively simple models, but in reality the observations are the output of complicated functions of the physicochemical, topological, and geometrical properties of a given system. Herein we use multistate continuous-time random walks and generalized master equations to model transport processes involving spatial jumps, immobilization at defined sites, and stochastic internal state changes. The underlying spatial models, which are framed as graphs, may have different classes of nodes, and walkers may have internal states that are governed by a Markov process. A general form of the solutions, using Fourier-Laplace transforms and asymptotic analysis, is developed for several spatially infinite regular lattices in one and two spatial dimensions, and the theory is developed for the analysis of transport and internal state changes on general graphs. The goal in each case is to shed light on how experimentally observable macroscale transport coefficients can be explained in terms of microscale properties of the underlying processes. This work is motivated by problems arising in transport in biological tissues, but the results are applicable to a broad class of problems that arise in other applications.

Diffusion–Advection Equations on a Comb: Resetting and Random Search

This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical simulations in terms of coupled Langevin equations for the comb structure. The subordination approach is one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting.

Fractional Diffusion to a Cantor Set in 2D

A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set Fμ is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink.

Reaction and ultraslow diffusion on comb structures

A two-dimensional (2D) comb model is proposed to characterize reaction-ultraslow diffusion of tracers both in backbones (x direction) and side branches (y direction) of the comblike structure with two memory kernels. The memory kernels include Dirac delta, power-law, and logarithmic and inverse Mittag-Leffler (ML) functions, which can also be considered as the structural functions in the time structural derivative. Based on the comb model, ultraslow diffusion on a fractal comb structure is also investigated by considering spatial fractal geometry of the backbone volume. The mean squared displacement (MSD) and the corresponding concentration of the tracers, i.e., the solution of the comb model, are derived for reactive and conservative tracers. For a fractal structure of backbones, the derived MSDs and corresponding solutions depend on the backbone's fractal dimension. The proposed 2D comb model with different kernel functions is feasible to describe ultraslow diffusion in both the backbone and side branches of the comblike structure.

Comb Model: Non-Markovian versus Markovian

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional m + n comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the 1 + 1 comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schrödinger equation.

Richardson diffusion in neurons

The dynamics of an initial wave packet affected by random noise is considered in the framework of a comb model. The model is relevant to a diffusion problem in neurons where the transport of ions can be accelerated by an external random field due to synapse fluctuations. In the present specific case, it acts as boundary conditions, which lead to a reaction transport equation with multiplicative noise. The temporal behavior of the mean squared displacement is estimated analytically, and it is shown that the spreading of the initial wave packet corresponds to Richardson diffusion.

Reaction–diffusion with stochastic decay rates

Microscopic physical and chemical fluctuations in a reaction–diffusion system lead to anomalous chemical kinetics and transport on the mesoscopic scale. Emergent non-Markovian effects lead to power-law reaction times and localization of reacting species.

Mesoscopic description of random walks on combs

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study continuous time random walks on combs and present a generic method to obtain their transport properties. The random walk along the branches may be biased, and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone. We analyze the overall diffusion properties along the backbone and find normal diffusion, anomalous diffusion, and stochastic localization (diffusion failure), respectively, depending on the characteristics of the continuous time random walk along the branches, and compare our analytical results with stochastic simulations.

Hitting and trapping times on branched structures

In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time H(i,f) between two arbitrary nodes i and j. We then use this formula to obtain the set of hitting times {H(i,f)} for combs and their expectation values, namely, the mean first-passage time, where the average is performed over the initial node while the final node f is given, and the global mean first-passage time, where the average is performed over both the initial and the final node. Finally, we discuss applications in the context of reaction-diffusion problems.

Fractional diffusion on a fractal grid comb

A grid comb model is a generalization of the well known comb model, and it consists of N backbones. For N=1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1/2. We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.

Slow encounters of particle pairs in branched structures

On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent; i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold, and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comblike structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we provide evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one-particle problem.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.