T. Langlands

Followers

Following

Co-author

Public Views

Interests

Uploads

Papers by T. Langlands

A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping

Mathematical Modelling of Natural Phenomena, 2016

Generalized Continuous Time Random Walks, Master Equations, and Fractional Fokker--Planck Equations

SIAM Journal on Applied Mathematics, 2015

An Introduction to Fractional Diffusion

Complex Physical, Biophysical and Econophysical Systems - Proceedings of the 22nd Canberra International Physics Summer School, 2010

The mathematical description of diffusion has a long history with many different formulations inc... more The mathematical description of diffusion has a long history with many different formulations including phenomenological models based on conservation of mass and constitutive laws; probabilistic models based on random walks and central limit theorems; microscopic stochastic models based on Brownian motion and Langevin equations; and mesoscopic stochastic models based on master equations and Fokker-Planck equations. A fundamental result common to the different approaches is that the mean square displacement of a diffusing particle scales linearly with time. However there have been numerous experimental measurements in which the mean square displacement of diffusing particles scales as a fractional order power law in time. In recent years a great deal of progress has been made in extending the different models for diffusion to incorporate this fractional diffusion. The tools of fractional calculus have proven very useful in these developments, linking together fractional constitutive laws, continuous time random walks, fractional Langevin equations and fractional Brownian motions. These notes provide a tutorial style overview of standard and fractional diffusion processes.

Download

Anomalous subdiffusion with multispecies linear reaction dynamics

Physical review. E, Statistical, nonlinear, and soft matter physics, 2008

We have introduced a set of coupled fractional reaction-diffusion equations to model a multispeci... more We have introduced a set of coupled fractional reaction-diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics. The model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics. The effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms. One consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species. The general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail. We have carried out Monte Carlo simulations corresponding to these special cases and we find excellent...

$Research paper thumbnail of Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations$

Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations

Physical review. E, Statistical, nonlinear, and soft matter physics, 2006

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level u... more We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term. Results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term. We have also developed further extensions of the...

Fractional Cable Models for Spiny Neuronal Dendrites

Physical Review Letters, 2008

Cable equations with fractional order temporal operators are introduced to model electrotonic pro... more Cable equations with fractional order temporal operators are introduced to model electrotonic properties of spiny neuronal dendrites. These equations are derived from Nernst-Planck equations with fractional order operators to model the anomalous subdiffusion that arises from trapping properties of dendritic spines. The fractional cable models predict that postsynaptic potentials propagating along dendrites with larger spine densities can arrive at the soma faster and be sustained at higher levels over longer times. Calibration and validation of the models should provide new insight into the functional implications of altered neuronal spine densities, a hallmark of normal ageing and many neurodegenerative disorders.

Download

Fractional Fokker-Planck Equations for Subdiffusion with Space- and Time-Dependent Forces

Physical Review Letters, 2010

We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-andtime-d... more We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-andtime-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.

Download

$Research paper thumbnail of Turing pattern formation in fractional activator-inhibitor systems$

Turing pattern formation in fractional activator-inhibitor systems

Physical Review E, 2005

Activator-inhibitor systems of reaction-diffusion equations have been used to describe pattern fo... more Activator-inhibitor systems of reaction-diffusion equations have been used to describe pattern formation in numerous applications in biology, chemistry, and physics. The rate of diffusion in these applications is manifest in the single parameter of the diffusion constant, and stationary Turing patterns occur above a critical value of d representing the ratio of the diffusion constants of the inhibitor to the activator. Here we consider activator-inhibitor systems in which the diffusion is anomalous subdiffusion; the diffusion rates are manifest in both a diffusion constant and a diffusion exponent. A consideration of this problem in terms of continuous-time random walks with sources and sinks leads to a reaction-diffusion system with fractional order temporal derivatives operating on the spatial Laplacian. We have carried out an algebraic stability analysis of the homogeneous steady-state solution in fractional activator-inhibitor systems, with Gierer-Meinhardt reaction kinetics and with Brusselator reaction kinetics. For each class of reaction kinetics we identify a Turing instability bifurcation curve in the two-dimensional diffusion parameter space. The critical value of d , for Turing instabilities, decreases monotonically with the anomalous diffusion exponent between unity (standard diffusion) and zero (extreme subdiffusion). We have also carried out numerical simulations of the governing fractional activator-inhibitor equations and we show that the Turing instability precipitates the formation of complex spatiotemporal patterns. If the diffusion of the activator and inhibitor have the same anomalous scaling properties, then the surface profiles of these patterns for values of d slightly above the critical value varies from smooth stationary patterns to increasingly rough and nonstationary patterns as the anomalous diffusion exponent varies from unity towards zero. If the diffusion of the activator is anomalous subdiffusion but the diffusion of the inhibitor is standard diffusion, we find stable stationary Turing patterns for values of d well below the threshold values for pattern formation in standard activator-inhibitor systems.

Continuous-time random walks on networks with vertex- and time-dependent forcing

Physical Review E, 2013

We have investigated the transport of particles moving as random walks on the vertices of a netwo... more We have investigated the transport of particles moving as random walks on the vertices of a network, subject to vertex- and time-dependent forcing. We have derived the generalized master equations for this transport using continuous time random walks, characterized by jump and waiting time densities, as the underlying stochastic process. The forcing is incorporated through a vertex- and time-dependent bias in the jump densities governing the random walking particles. As a particular case, we consider particle forcing proportional to the concentration of particles on adjacent vertices, analogous to self-chemotactic attraction in a spatial continuum. Our algebraic and numerical studies of this system reveal an interesting pair-aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices. The steady states do not exhibit this pair aggregation if the transport is random on the vertices, i.e., without forcing. The manifestation of pair aggregation on a transport network may thus be a signature of self-chemotactic-like forcing.

Fractional chemotaxis diffusion equations

Physical Review E, 2010

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion... more We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master 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, macro-molecular crowding or other obstacles.

Download

$Research paper thumbnail of Turing pattern formation with fractional diffusion and fractional reactions$

Turing pattern formation with fractional diffusion and fractional reactions

Journal of Physics: Condensed Matter, 2007

We have investigated Turing pattern formation through linear stability analysis and numerical sim... more We have investigated Turing pattern formation through linear stability analysis and numerical simulations in a two-species reaction–diffusion system in which a fractional order temporal derivative operates on both species, and on both the diffusion term and the reaction term. The order of the fractional derivative affects the time onset of patterning in this model system but it does not affect critical parameters for the onset of Turing instabilities and it does not affect the final spatial pattern. These results contrast with earlier studies of Turing pattern formation in fractional reaction–diffusion systems with a fractional order temporal derivative on the diffusion term but not the reaction term. In addition to elucidating differences between these two model systems, our studies provide further evidence that Turing linear instability analysis is an excellent predictor of both the onset of and the nature of pattern formation in fractional nonlinear reaction–diffusion equations.

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

Journal of Mathematical Biology, 2009

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macrosco... more 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.

Download

$Research paper thumbnail of The accuracy and stability of an implicit solution method for the fractional diffusion equation$

The accuracy and stability of an implicit solution method for the fractional diffusion equation

Journal of Computational Physics, 2005

We have investigated the accuracy and stability of an implicit numerical scheme for solving the f... more We have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation. This model equation governs the evolution for the probability density function that describes anomalously diffusing particles. Anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur. The implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement. The accuracy of the scheme is O(Dx 2 ) in the spatial grid size and O(Dt 1 + c ) in the fractional time step, where 0 6 1 À c < 1 is the order of the fractional derivative and c = 1 is standard diffusion. We have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 < c 6 1.

Download

A comparison of methods to estimate the roll torque in thin strip rolling

by Jingyu Shi and T. Langlands

International Journal of Mechanical Sciences, 2001

An energy balance method is used to calculate the roll torque associated with non-circular thin s... more An energy balance method is used to calculate the roll torque associated with non-circular thin strip rolling and the results for the torque are compared with the moments of the forces exerted on the roll by the strip. An example is given in which the results of the two methods are quite di!erent. This probably indicates that the axis of the roll is not at the point about which the moment is taken. The energy balance method is also used to estimate the torque associated with circular arc rolling. The results are compared with those calculated with Hill's formula in one example and, in another example, the comparison is made between the results from the energy method formulated in this paper and a commercial package which also uses both an energy method and Hill's formula.

Download

T. Langlands

Uploads

Papers by T. Langlands

Log In