B. I. Henry

Fractional reaction–diffusion

We derive a fractional reaction–diffusion equation from a continuous-time random walk model with temporal memory and sources. The equation provides a general model for reaction–diffusion phenomena with anomalous diffusion such as occurs in spatially inhomogeneous environments. As a first investigation of this equation we consider the special case of single species fractional reaction–diffusion in one dimension and show that the fractional diffusion does not by itself precipitate a Turing instability.

Existence of Turing Instabilities in a Two-Species Fractional Reaction-Diffusion System

We introduce a two-species fractional reaction-diffusion system to model activator- inhibitor dynamics with anomalous diffusion such as occurs in spatially inhomogeneous media. Con- ditions are derived for Turing-instability induced pattern formation in these fractional activator- inhibitor systems whereby the homogeneous steady state solution is stable in the absence of diffusion but becomes unstable over a range of wavenumbers when fractional diffusion is present. The condi- tions are applied to a variant of the Gierer-Meinhardt reaction kinetics which has been generalized to incorporate anomalous diffusion in one or both of the activator and inhibitor variables. The anoma- lous diffusion extends the range of diffusion coefficients over which Turing patterns can occur. An intriguing possibility suggested by this analysis, which can arise when the diffusion of the activator is anomalous but the diffusion of the inhibitor is regular, is that Turing instabilities can exist even when the diffusion coefficient of the activator exceeds that of the inhibitor.

Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.

Fractional Fokker-Planck equations for subdiffusion with space-and time-dependent forces

We derive a fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.

Fractional chemotaxis diffusion equations

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.

Fractional Cable Equation Models for Anomalous Electrodiffusion in Nerve Cells: Finite Domain Solutions

In recent work we introduced fractional Nernst–Planck equations and related fractional cable equations to model electrodiffusion of ions in nerve cells with anomalous subdiffusion along and across the nerve cells. This work was motivated by many computational and experimental studies showing that anomalous diffusion is ubiquitous in biological systems with binding, crowding, or trapping. For example, recent experiments have shown that anomalous subdiffusion occurs along the axial direction in spiny dendrites due to trapping by the spines. We modeled the subdiffusion in two ways leading to two fractional cable equations and presented fundamental solutions on infinite and semi-infinite domains. Here we present solutions on finite domains for mixed Robin boundary conditions. The finite domain solutions model passive electrotonic properties of spiny dendritic branch segments with ends that are voltage clamped, sealed, or killed. The behavior of the finite domain solutions is similar for both fractional cable ...

Continuous Time Random Walks with Reactions Forcing and Trapping

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Deterministic KPZ model for stromatolite laminae

The deterministic variant of the Kardar–Parisi–Zhang equation for the evolution of a growing interface is used to model patterning produced by successive laminations in certain stromatolites. Algebraic solutions of the model together with numerical simulations are employed to fit model parameters consistent with digital recordings of individual stromatolite laminae. Numerical values for model parameters, related to lateral growth, vertical growth and surface diffusion, provide a set of indices which may prove useful for classifying different stromatolites.

Turing Patterns from Dynamics of Early HIV Infection

We have developed a mathematical model for in-host virus dynamics that includes spatial chemotaxis and diffusion across a two-dimensional surface representing the vaginal or rectal epithelium at primary HIV infection. A linear stability analysis of the steady state solutions identified conditions for Turing instability pattern formation. We have solved the model equations numerically using parameter values obtained from previous experimental results for HIV infections. Simulations of the model for this surface show hot spots of infection. Understanding this localization is an important step in the ability to correctly model early HIV infection. These spatial variations also have implications for the development and effectiveness of microbicides against HIV.

A Fractional Order Recovery SIR Model from a Stochastic Process

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack–McKendrick age-structured SIR model, and it reduces to the Hethcote–Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.