
THEMATIC PROGRAMS 

December 18, 2018  
Jacques Bélair, Université
de Montréal Drug administration currently takes the form of sophisticated vectors and techniques designed to reproduce as closely as possible the physiological regimens the medication tries to replace. We discuss a number of examples in which the normal condition is not stationnary, and thus therapeutic interventions must be determined in terms of pulsatile release, for example, or some form of oscillation. We also present a model to describe the time course of plasma concentration
of neuromuscular blocking agents used as anaesthetics during surgery.
The model overcomes the limitations of the classical compartmental models
commonly used in pharmacokinetics by incorporating spatial effects due
to heterogeneity in the circulation. Ross Cressman, Wilfrid Laurier University Reaction diffusion equations were first applied by Fisher (1930) to model the spread of an allele in a spatial version of natural selection. I will consider spatial patterns that emerge for both continuoustime and discretetime diffusion when natural selection is also frequency dependent. It is shown that the existence of these patterns in discrete time depends critically on how diffusion is incorporated into the biological model. This result means that care must be taken when interpreting data from corresponding computer simulations.
Leon Glass, McGill University There are a very large number of different abnormal cardiac
rhythms. Some of these rhythms are dangerous and might lead to imminent
death, whereas others may persist for decades with few adverse effects.
All cardiac arrhythmias are characterized by interesting dynamical properties.
Some of these are obvious to the clinician and are the basis for diagnosis
and therapy, whereas others are sufficiently subtle that they are not
yet appreciated. I describe a range of different experimental and theoretical
models that capture key features of cardiac arrhythmias and discuss
the possibility for better diagnoses based on a better dynamical characterization
of arrhythmia. I also describe the optical mapping of reentrant rhythms,
in which the period is set by the time it takes for the excitation to
travel in a circuitous path with special emphasis on situations in which
there is a paroxysmal onset and offset of abnormal rhythms.
Leon Glass, McGill University Gene networks underly the development and functioning of organisms.
Activities of genes are controlled by transcriptions factors that in
turn result from activities of other genes. A mathematical representation
of genetic networks is introduced that allows one to relate the patterns
of gene activity to the underlying network structure. Gene networks
are represented by differential equations. The dynamics in these equations,
and also the network structure are represented schematically using a
directed graph on an ndimensional hypercube. These methods can be used
to help design in vitro genetic networks that show oscillation and multistability.
They can also be used to determine gene network structure based on the
patterns of activation of genes, such as might be determined using gene
expression chips.
Alf Gerisch, The Fields Institute We describe a method of lines (MOL) technique for the simulation of taxisdiffusionreaction (TDR) systems. These timedependent PDE systems arise when modelling the spatiotemporal evolution of a population of organisms which migrate in direct response to e.g. concentration differences of a diffusible chemical in their surrounding (chemotaxis). Examples include pattern formation and different processes in cancer development. The effect of taxis is modelled by a nonlinear advection term in the TDR system (the taxis term). The MOLODE is obtained by replacing the spatial derivatives in the TDR system by finite volume approximations. These respect the conservation of mass property of the TDR system, and are constructed such that the MOLODE has a nonnegative analytic solution (positivity). The latter property is natural (because densities/concentrations are modelled). The MOLODE is stiff and of large dimension. We develop integration schemes which treat the discretization of taxis and diffusion/reaction differently (splitting). We employ operator (Strang)splitting and/or the approximate matrix factorization technique. The splitting schemes are based on explicit RungeKutta and linearlyimplicit Wmethods. Results on the positivity and the stability of integration schemes are discussed. Numerical experiments with a variety of splitting schemes applied to some semidiscretized TDR systems confirm the broad applicability of the splitting schemes. These methods are more efficient than (suitable) standard ODE solvers in the lower and moderate accuracy range. Altogether, the numerical technique developed is appropriate and efficient for the simulation of TDR systems. John Guckenheimer, Cornell University Multiple time scales are present in all biological systems. These can
be readily incorporated into dynamical models, but A.B. Gumel, University of Manitoba Intermittent administration of immune activators such as interlukin2
(IL2) in combination with highlyactive anti retroviral therapy (HAART)
is considered to be an effective strategy for longterm control of HIV
replication in vivo. This talk focusses on the design and simulation
of a deterministic model that enable the assessment of therapeutic strategies
of HIV. The model, which monitors the temporal dynamics of HIV, uninfected
CD4+ T cells, CD8+ T cells, productivelyinfected and latentlyinfected
CD4+ T cells, shows that current therapeutic strategies (including intermittent
HAART and IL2) are insufficient in eradicating HIV. It, however, suggests
that eradication is feasible if an uninterrupted HAART and IL2 therapy
is combined with an agent (such as a putative antiHIV vaccine) that
can increase the proliferation of HIVspecific CD4+ and CD8+ T cells
as well as the differentiation of CD8+ T cells into antiHIV cytotoxic
T lympocytes. John J. Hsieh, University of Toronto This article develops an EM (expectationmaximisation) algorithm and related computationally intensive procedures for iterative estimation of the HIV infection rates and prediction of the future course of AIDS incidence in Canada, U.S. and Australia.. First, likelihood functions are constructed by assuming nonhomogeneous Poisson and planar Poisson point processes for infections and incidences. Smooth maximum likelihood estimates of the HIV infection rates are then obtained by applying the EM algorithm coupled with a smoothing step at each iteration. Accuracy of the estimates of the infection rates can then be checked by comparing the estimates of the expected AIDS incidence, calculated from the Volterra integral equation of the first kind using the known incubation distribution as the kernel. Furthermore, by extrapolating the estimates of the HIV infection rates so obtained into the future, the Volterra integral equation can be used to project the future course of the AIDS incidence. We have employed various parametric and nonparametric distributions for the kernel. The parametric distributions include Weibull and gamma distributions and several new distribution functions and the nonparametric distributions include linear and cubic spline functions. These are so chosen as to take into account the fact that the HIV screening test was available in these countries only since 1985 and the drug treatment made available to the HIV positive patients only after 1987. Our method has produced results that fit the observed AIDS incidence better than those produced by other existing methods based on data from U.S., Canada and Australia. Eugene M. Izhikevich, The Neurosciences
Institute, La Jolla, CA A neuron is said to have bursting dynamics when its activity alternates
between a quiescent state (equilibrium) and spiking state (limit cycle).
Most models of bursting have singularly perturbed form REFERENCE: Daniel Kobler, TM Bioscience Spatially localized regions of active neurons (``bumps'') have been proposed as a mechanism for working memory, the head direction system, and feature selectivity in the visual system. Stationary bumps are ordinarily stable, but including spike frequency adaptation in the neural dynamics causes a stationary bump to become unstable to a moving bump through a supercritical pitchfork bifurcation in bump speed. Adding spatiotemporal noise to the network supporting the bump can cause the average speed of the bump to decrease to almost zero, reversing the effect of the adaptation and ``restabilizing'' the bump. This restabilizing can be understood by examining the effects of noise on the normal form of the pitchfork bifurcation where the variable involved in the bifurcation is bump speed. This noiseinduced stabilization is a novel example in which moderate amounts of noise have a beneficial effect on a system, specifically, stabilizing a patiotemporal pattern. Determining which aspects of our model system (integral rather than diffusive coupling, a slow variable, traveling structures that appear through a pitchfork bifurcation in speed) are necessary for this type of behavior remains an open problem. Dong Liang, York University Xinzhi Liu, University of Waterloo This paper investigates the problem of biological population management. A threespecies population growth model is considered. With an impulsive control scheme, we establish criteria for keeping all the species from going extinct. We introduce the concept of stabilizing some positive point that is not necessarily the equilibrium of the system. Andre Longtin, University of Ottawa Stochasticity arises in many different contexts and many different forms in biological systems. This talk will present recent novel approaches to modeling noise, from the single cell level to the collective level in high level neural information processing and in insect societies. Stochastic firing in neurons has often been modeled using the point process formalism, in which the intensity of the process may be time dependent or a stochastic process itself. However, this approach is limited when nonlinear dynamical effects related to excitability are of interest, which in turn leads to driftdiffusion process formulations. We describe such a dynamical model for electroreceptor firing activity, in which a fast and a slow stochastic process drive the deterministic dynamics. This is shown to be needed for explaining the first and second order firing statistics, as well as the regularity of spike train over different counting windows (Fano factor). We then present a recent model for the motion of eyes and of attention during reading. The attention is made stochastic by distributing it over words. This model is more parsimonious than existing ones, and is shown to nevertheless fit fixation time data. Finally, we discuss stochastic task allocation in ants societies. Individual ants sample their environment and make decisions at random times to switch task. The inherent stochasticity enables a novel formulation of this problem in terms of iterated function systems as well as birthdeath stochastic processes (state dependent random walks). Michael C. Mackey, McGill University There are a number of interesting periodic hematological diseases [Haurie
et al. Blood (1998), {\it 92}, 26292640] and some are understood through
mathematical modeling [M.C. Mackey. ``Mathematical models of hematopoietic
cell replication and control", pp. 149178 in {\bf The Art of Mathematical
Modeling: Case Studies in Ecology, Physiology and Biofluids} (H.G. Othmer,
F.R. Adler, M.A. Lewis, and J.C. Dallon eds.) Prentice Hall (1997)].
A number of these diseases are most certainly due to a Hopf bifurcation
in the dynamics of peripheral control, triggered by alteration of cellular
death rates, e.g. periodic autoimmune anemia and cyclical thrombocytopenia.
Others, like cyclical neutropenia, are due to a bifurcation in stem
cell dynamics arising from elevated levels of apoptosis. This talk will
give an overview of the status of mathematical modeling of these diseases,
and the role that this modeling is playing in shaping treatment strategies.
For papers related to these topics, go to John Milton, The University of Chicago Despite these apparent obstacles, the human nervous system is able to maintain control of processes that occur on time scales shorter than the delay, to respond quickly in an ever changing environment, and to perform complex computational tasks, such as imaging processing, with surprising speed. These issues are addressed in the context of experiments involving neuroncomputer circuits, stick balancing at the fingertip, and the golf swing. Appropriate mathematical and computer models are formulated in terms of stochastic delay differential equations. A number of novel mechanisms for control, memory storage, and computation are identified: rapid responses can be achieved by making use of an underlying multistability, fast control with delayed feedback by using parametric noise ("onoff intermittency") and preprogramming, and rapid computation using populations of noisy, multistable elements. Noisy dynamical systems with retarded variables offer a number of underappreciated advantages for computation and control that may have many applications. Steven Ruuth, Simon Fraser University Many front propagation problems have been modeled using cellular automata.
Some of the bestknown examples of which are the "excitable media"
which simulate the behavior of nerve cells, muscle cells, cardiac function
and chemical Sven Sigurdsson, Kjartan Magnusson,
Petro Babak, Simon Hubbard, University of Iceland We present a discrete stochastic particle model of fish migration that is under development, and contrast it with a continous density model, implemented numerically by a finite element method on a triangular grid. The underlying aim is to aid in the assessment of what type of migration models might be included into a more general prediction model of changes in fish stock size, as well as testing various assumptions on what external factors may influence fish migration. In this respect the finite element model forms a bridge between the particle models on one hand and compartmental models on the other. These are the models presently used to predict changes in stocksize and in these models migration between compartments is simply described in terms of transition matrices. The focus of our presentation is on compuatitonal similarities and differences between the discret and continous models and on what insight we may gain from contrasting one with the other. Jianhong Wu, York University We present some recent work on the structure of the global attractor
for a simple excitatory network of two identical neurons with delayed
recurrent loops. We address the issue of the coexistence of multiple
limit cycle attractors or periodic solutions which are unstable but
with large domains of attraction, and we discuss some potential applications
to dynamic memory storage and encoding. Simon X. Yang, Advanced Robotics and Intelligent
Systems (ARIS) Lab James Yorke, University of Maryland Scientists have created draft versions of the human genome, as well
as for many species with smaller, simpler genomes, including several
varieties of bacteria, a worm, a plant, and a fly. Efforts are underway
to find the sequence of mouse, rat, zebrafish, mosquito and many others.
The mathematical and computational problems are significant, and the
efforts of our University of Maryland group to make this procedure more
efficient and effective will be described. 
