Immunology is one of the most challenging fields in biomedicine. In
the last decades its foundations have been revisited and the molecular
interactions at the basis of immune responses are far from being
accurately known at the microscopic level. The synergetics between microbiology
and genetics have been very fruitful in the last three
decades. However, there is a new approach coming from the exact
sciences that is attracting the attention of some immunologists. Thus,
theoretical immunology (in which mathematical and physical tools are
combined to build biologically consistent models) has become an active
and fruitful field of research.
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The immune system is one of the most complicated multiscale systems
imaginable. The adaptive immune system of a vertebrate is a vast army
of cells and molecules that cooperate to seek out, mark, bind to and
destroy invaders. Stochastic modelling is ideally suited to
immunology at many scales. For example,

Cells live in a Brownian world: motion is partly directed and partly
random. The appropriate mathematical tools describing such motion are
stochastic differential equations.

The battle between invading pathogens and the immune
systems, innate and adaptive, involves millions of encounters and is
best described statistically.

The means by which the body selects and
trains its T cells and B cells is probabilistic. T cells
mature in the thymus, undergoing testing and possible elimination
based on their potential future role in identification and elimination
of ``nonself'', while at the same time respecting self cells.

The immunoglobulin gene rearrangement that occurs in the development
of a B cell that generates diversity of the mature antibody repertoire
is believed to involve a random cutting and splicing of gene segments.
Scientists in many fields face a similar challenge: to understand a
large system, driven by many noisy and nonlinear influences,
containing persistent identifiable structures. The challenge is being
met with a twopronged approach. Firstly, using the most powerful
computers available, perform numerical simulations of the full model
on the largest domain and with the highest spatial resolution
possible, for as long a time as possible. Secondly, develop
theoretical methods that efficiently identify the structures of
interest and predict their number, structure, dynamics and
interactions.
Noise, fluctuations or randomness is part of every real phenomenon.
It provides an extra fascinating element for the nonlineardynamicist
to study. Just as differential equations were invented to describe
deterministic dynamics, stochastic differential equations are the tool
needed to describe dynamics that have a random, or stochastic,
component. In practice, solving a nonlinear stochastic differential
equation is like solving an nonlinear ordinary differential equation:
exact analytical solutions are seldom available, but paths can be
generated in a few seconds on a computer and analysed using
tools from pure and applied mathematics. A new area is the study of
stochastic partial differential equations, which describe spatially
extended systems with randomness, such as vortices in a fluctuating
environment. Even though the theory of such equations is still in its
infancy, the recent rapid increase in the computing power is starting
to make it possible to explore their dynamics.
The focus of my research in stochastic partial differential equations
is on the statistics and dynamics of coherent structures that maintain
their identity as they move and are buffeted by local fluctuations. In
their simplest manifestation they are known as kinks and a systematic
study is possible, which will serve as a stepping stone towards the
study of more complicated structures such as vortices.
Kinks
are examples of coherent structures: clearly identifiable
localized features in a noisy, spatiallyextended system that can be
followed as they move about under the influence of fluctuations. We
study kinks in the Ø^{4} stochastic partial differential
equation, where a steadystate mean density is dynamically
maintained: kinks and antikinks are nucleated in pairs, follow
Brownian paths and annihilate on meeting. Current computers can
attain sufficient resolution to perform direct comparisons with
predictions for the steady state, and work at sufficiently low
temperatures to unambiguously locate kinks and identify nucleation
events. In order to perform precise comparison between numerical
and exact results at finite temperature, it is important to use the
exact correlation length and not a lowtemperature approximation.
Numerical convergence of thermodynamic properties, where analytical
results are available, makes it possible to proceed with confidence
to an exploration of the fascinating stochastic dynamics of kinks.
 
In collaboration with Salman Habib (Los Alamos National Laboratory),
largescale simulations of the stochastic PDEs for Ø
^{4}
field theory at finite tempearture are being combined with new
theoretical results.

Grant Lythe and Salman Habib.
CISE
8,3 1015 (2006)

Grant Lythe and Franz Mertens.
Physical Review E 67 027601 (2003)

Grant Lythe and Salman Habib.
p435444 in
proceedings of the IUTAM symposium on Nonlinear Stochastic Dynamics
N. Sri Namachchivaya and Y.K.Lin (Eds.)
(2003)

Salman Habib and Grant Lythe.
Physical Review Letters
84 1070 (2000)

Luis M. A. Bettencourt, Salman Habib and Grant Lythe.
Physical Review D
60 105039 (1999)
Nucleation
Many spatiallyextended systems in physics, chemistry or biology
exhibit two locallystable states or phases. The two
states coexist in the sense that, at any one time, different parts of
the system are in different states. Fluctuations cause transitions of
part of the system from one state to another, called nucleation
events. We consider a string of particles, each coupled to its two neighbours,
subject to noise and to the doublewell potential:
V(φ) = 1/2φ^{2} + 1/4φ^{4}.
The equation of motion for the ith particle is
Click on the image below to see the system in real time (java applet)
Reaction rates controlled by collisions between diffusing particles
depend on the distribution of distances between particles as well as
on the density of particles. In particular, as Noyes stated in 1961
Any rigorous treatment of chemical kinetics in solution must
consider concentration gradients that are established by the existence
of the reaction itself
We study the dynamics of
point particles in one dimension, nucleated at random positions and
times then diffusing until colliding and annihilating with another
particle. Competition between nucleation and annihilation produces a
statistically steady state with a welldefined mean density of
particles and distribution of distances between particles.
The dynamics is as follows:

Particles are nucleated in pairs with initial separation b;

Nucleation occurs at random times and positions with rate Γ

Once born, all particles diffuse independently with diffusivity D;

Particles annihilate on collision.
For unpaired nucleation 1. and 2. are replaced by
4.'
Particles are nucleated at random times and positions with rate Q.


A striking difference between paired and unpaired nucleation
is the scaling of the steady state density of particles,
r
_{0}, with the nucleation rate:
r_{0}
∼ Γ^{1/2} (paired)
versus
r_{0}
∼
Q^{1/3} (unpaired).
We exhibit the crossover between these two cases
in terms of the following dimensionless quantity:
ε = (2Γ/D)^{1/3} b.
Grant Lythe
Physica D
222 159163 (2006)
Salman Habib, Katja Lindenberg, Grant Lythe and
Carmen MolinaParís.
Journal of Chemical Physics
115 73 (2001)
Newton's second law of motion relates force to acceleration.
Consequently, secondorder differential equations are common in
scientific applications, in the guise of ``Langevin'', ``Monte
Carlo'', ``molecular'' or ``dissipative particle'' dynamics, and the
study of methods for secondorder ordinary differential equations is
one of the most mature branches of numerical analysis. Numerical
methods that preserve geometric properties of a the flow of a
differential equation are increasingly useful in deterministic
dynamics when it is important that the numerics reproduce the dynamics
as accurately as possible over long times. In particular, when solving
separable Hamiltonian problems, explicit partitioned RungeKutta (PRK)
methods can be constructed that are symplectic. In stochastic
dynamics, an ensemble of different trajectories originates even from a
single initial condition, corresponding to different realizations of
the stochastic process. The geometric property of the flow defined by
a bundle of neighbouring initial conditions is not relevant. There
is, however, an appropriate longterm quantity: the stationary
density. We construct explicit PRK methods that approximate the
stationary density with highorder accuracy.
Kevin Burrage, Ian Lennane and Grant Lythe
SIAM Journal of Scientific Computing in press (2006)
Exponential timestepping algorithms are efficient for exittime
problems because a boundary test can be performed at the end of
each timestep, giving highorder convergence in numerical
evaluation of mean exit times. Successive time increments are
independent random variables with an exponential distribution.
In the figure on the right, one realization is depicted.
Exponential timesteps
are jumps from one black point to the next. The values of
the process at these times are generated, but not the
corresponding times. 

Kalvis Jansons and G.D. Lythe:
Journal of Statistical Physics 100 1097 (2000)
SIAM Journal of Scientific Computing 24 1809 (2003)
SIAM Journal of Scientific Computing 27 793808 (2005)
Nonlinear dielectrics, such as SrTiO3 (STO), present unique opportunities
to develop practical electrically tunable devices, and also to explore
novel scientific and technological concepts that exploit strong
nonlinearities.
Experimental device


Our theoretical modelling of the coplanar waveguide system uses a
set of coupled partial differential equations in a nonlinear medium with
two boundaries. The nonlinearity depends on the applied bias and is used to
determine the inputoutput curve of the system, which allows us to
calculate the conditions for the appearance of stochastic resonance.


Roberto Camassa, Alp T. Findikoglu and Grant Lythe
SIAM Journal of Applied Mathematics 66 128 (2005)
Alp Findikoglu, Roberto Camassa, Grant Lythe and Q.X. Jia
Applied Physics Letters
80 3391 (2002)
Alp Findikoglu et al
Integrated Ferroelectrics
22 259268
Differential equations have long been used to model the dynamics of
physical systems. With the availability of computers, the tendency to
focus only on analysis of linear equations is being replaced by a
methodology that profits from a judicious mixture of
numerical generation of paths, bifurcation theory and asymptotic
analysis. However, when random perturbations (i.e. noise)
play an important role, this new spirit is not so widespread. One
reason is that the mathematical tools appropriate for describing
stochastic paths are not sufficiently wellknown. It is partly as a
result of this that there is a widespread misconception that noise
acts only to smear out deterministic dynamics.
Solving a stochastic differential equation
(SDE) is akin to solving an ordinary differential equation: exact
analytical solutions are seldom available, but paths can be
generated in a matter of seconds on a computer. Analysis is based on
Ito calculus, that permits computation of experimentally accessible
quantities inaccessible to traditional methods.
I perform numerics and analysis of nonlinear stochastic differential
equations, stochastic partial differential equations and differential
delay equations.
Kalvis Jansons and Grant Lythe
Journal of Statistical Physics
90 227251 (1998)
(.ps file)
Stochastic partial differential equations describe continuum systems
with noise. The noise can be white in space and time, described
solely by an amplitude, or can have space or time scales of its own.
The numerical and analytical tools for solving SPDEs are
generalisations of the powerful stochastic analysis used for ODEs with
noise.
Because they focus on one realisation at a time, stochastic DEs are
natural tools when noise is an active part of the dynamics, not merely
an agent that rattles the system around in a predetermined potential.
In a simple example of a nonequilibrium spatially extended system,
microscopic white noise produces a characteristic macroscopic domain
size that is a function the rate of change of the parameter and the
amplitude of the noise.
Grant Lythe
Physical Review E 53
R42714274 (1996)
Esteban Moro and Grant Lythe
Physical Review E
59 R13031306 (1999)
Salman Habib and Grant Lythe.
Computer Physics Communications
142 29 (2001)
Semiconductor lasers have a wide range of applications because they
are of relatively small size, they can be massively produced at low
cost, and they are easy to operate. Despite their successful
technology, semiconductor lasers are very sensitive to any external
perturbation. A small amount of optical feedback resulting from the
reflection from an optical disk or from the end of an optical fiber is
sufficient to generate pulsating instabilities. Systematic
experimental studies of semiconductor lasers, in particular time
series analysis, is not possible because the timescale of the
intensity pulsations is typically in the picosecond regime. Most of
the progress in understanding these bifurcations comes from extensive
numerical studies of simple models and their comparison to the
experimentally obtained Fourier spectra.
A. Gavrielides, V. Kovanis, P.M. Farangis,
T. Erneux and G. Lythe
Quantum and semiclassical optics
9 785 (1997)
J.C. Celet, D. Dangoisse, P. Glorieux,
G. Lythe and T. Erneux
Physical Review Letters
81 975978 (1998)
G.D. Lythe and T. Erneux, A. Gavrielides and V. Kovanis
Physical Review A 55 44434448, (1997)
Small amounts of
noise can dramatically change and simplify nonequilibrium dynamics.
Examples include slowfast dynamics, when long quiescent phases are
occassionally interrupted by bursts of activity; and spatially
extended systems undergoing patternforming transitions such as
annealing and the formation of convective patterns. Macroscopic
length and timescales are found to emerge from microscopic
fluctuations.
In noisecontrolled dynamics, length and time
scales depend on the product of a slow inverse time and the
logarithm of the noise level. Examples are described in ordinary
and partial differential equations modelling laser and fluid systems.
The simplest example of the disproportionate
and simplifying effect of noise is a dynamic
bifurcation. In the corresponding situation for a
spatially extended system, microscopic
noise produces a characteristic macroscopic domain size.
Grant Lythe and MRE Proctor
Physical Review E 47 31223127 (1993)
Grant Lythe
Nuovo Cimento D17 855861 (1995)

Grant Lythe and MRE Proctor
Physica D 133 362 (1999)
α = L^{2}μlogε/4π^{2}

Classical Stokes' drift is the small timeaveraged drift
velocity of suspended nondiffusing particles in a fluid due to the
presence of a wave. We consider the effect of adding diffusion to the
motion of the particles, and show in particular that a nonzero
timeaveraged drift velocity exists in general even when the classical
Stokes' drift is zero. Our results are obtained from a general
procedure for calculating ensembleaveraged Lagrangian mean velocities
for motion that is close to Brownian, and are verified by numerical
simulations in the case of sinusoidal forcing.


Kalvis Jansons and Grant Lythe
Physical Review Letters
81 31363139
Many animals manage to detect a weak signal against a noisy
background. The key to understanding this may lie in the concept of
stochastic resonance, a mechanism by which noise actually assists,
rather than hinders, detection of signals. Stochastic resonance has
been demonstrated in a wide variety of mathematical models, from
simple bistable systems to fullscale models of auditory systems and
of neural cell networks. The challenge is to establish whether it is
indeed used by living animals. An example that is being actively
investigated at Leeds is found in moths, which have tympanic organs
sensitive to ultrasound that enable them to detect the echolocation
calls of bat predators and take evasive action. Detection involves the
identification of a periodic train of bat echolocation calls against a
background of biotic and abiotic noise. Resolving the puzzle of how
the twocelled moth organ is capable of performing this detection will
require integrating experiment and theory from beginning to end,
including stochastic modelling and statistical analysis of time
series. A systems approach is needed to understand the throughput
information from the sensory cells to the behavioural consequences,
with input from neurophysiology, signal processing, behavioural
ecology and evolutionary modelling.