June  3, 2020
Jacques Bélair Michael C. Mackey
Ross Cressman John Milton
Leon Glass Steven Ruuth
Alf Gerisch Sven Sigurdsson
John Guckenheimer Jianhong Wu
Abba Gumel Simon X. Yang
John J. Hsieh James Yorke
Eugene M. Izhikevich  
Daniel Kobler  
Carlo R Laing  
Dong Liang  
Xinzhi Liu  
Andre Longtin  

Jacques Bélair, Université de Montréal
Dynamical Modeling of Drug Delivery

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
Spatial Patterns and Reaction-Diffusion Equations

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 continuous-time and discrete-time 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
Talk #1: Dynamics of Cardiac Arrhythmias

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
Talk #2: Dynamics in Genetic Networks

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 n-dimensional 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
Numerical Methods for the Simulation of Taxis-Diffusion-Reaction Systems

We describe a method of lines (MOL) technique for the simulation of taxis-diffusion-reaction (TDR) systems. These time-dependent PDE systems arise when modelling the spatio-temporal 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 MOL-ODE 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 MOL-ODE has a nonnegative analytic solution (positivity). The latter property is natural (because densities/concentrations are modelled).

The MOL-ODE 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 Runge-Kutta and linearly-implicit W-methods. Results on the positivity and the stability of integration schemes are discussed.

Numerical experiments with a variety of splitting schemes applied to some semi-discretized 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 in Neural Systems

Multiple time scales are present in all biological systems. These can be readily incorporated into dynamical models, but
our understanding of dynamical systems with multiple time scales is much poorer than that for systems with a single time scale. This lecture will discuss what we know and what we don't know about the bifurcations of multiple time scale systems, using neural models as examples.

A.B. Gumel, University of Manitoba
A deterministic model for assessing therapeutic strategies of HIV

Intermittent administration of immune activators such as interlukin-2 (IL-2) in combination with highly-active anti- retroviral therapy (HAART) is considered to be an effective strategy for long-term 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, productively-infected and latently-infected CD4+ T cells, shows that current therapeutic strategies (including intermittent HAART and IL-2) are insufficient in eradicating HIV. It, however, suggests that eradication is feasible if an uninterrupted HAART and IL-2 therapy is combined with an agent (such as a putative anti-HIV vaccine) that can increase the proliferation of HIV-specific CD4+ and CD8+ T cells as well as the differentiation of CD8+ T cells into anti-HIV cytotoxic T lympocytes.

John J. Hsieh, University of Toronto
Estimation of HIV Infection Rates and Projection of AIDS Incidence from HIV/AIDS Diagnosis Data

This article develops an EM (expectation-maximisation) 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 non-parametric 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
Computational Challenges in Bursting Dynamics

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
x’ = f(x,y)
y’ = \mu g(x,y)
Slow changes in y cause x’ = f(x,y) to bifurcate from equilibrium to limit cycle attractor and back. We review relevant bifurcation theory and use it to classify bursting dynamics. We also review classical methods of nonlinear analysis, such as averaging, singular perturbations, etc., and discuss challenges and pitfalls when one applies those methods to study bursting dynamics.

Izhikevich (2000) Neural Excitability, Spiking, and Bursting,
International Journal of Bifurcation and Chaos, 10:1171--1266.

Daniel Kobler, TM Bioscience
Combinatorial optimization and genomics: the example of a Toronto company

Many areas of biology benefit from theories and tools developed in mathematics. In this talk, I will make a brief overview of how combinatorial optimization can contribute to problems of genomics. I will focus on real-life problems arising in the genomics industry, and use examples from a small biotechnology company based in Toronto.

Carlo R. Laing, University of Ottawa,
Stabilization of ``bumps'' by noise

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 noise--induced 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
Travelling Waves and Numerical Approximations in A Reaction Advection Diffusion Model with Nonlocal Delayed Effects

In this talk, we consider a single species population problem with two age classes and a fixed maturation period living in a spatial transport field. A Reaction Advection Diffusion Model with time delay and nonlocal effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of a travelling wave front for the special case when the birth function is the one appeared in the well-known Nicholson's blowflies equation. Furthermore, we consider and compare numerical solutions of the travelling wave fronts for the problems with nonlocal time delay effects and local time delay effects.

This is a joint work with J. Wu.

Xinzhi Liu, University of Waterloo
Management of Biological Populations via Impulsive Control

This paper investigates the problem of biological population management. A three-species 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
Challenges in stochastic biodynamics: Ants, reading and neural computation

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 drift-diffusion 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 birth-death stochastic processes (state dependent random walks).

Michael C. Mackey, McGill University
Understanding Periodic Hematological Disease: Insights from Mathematics Translate to the Bedside

There are a number of interesting periodic hematological diseases [Haurie et al. Blood (1998), {\it 92}, 2629-2640] and some are understood through mathematical modeling [M.C. Mackey. ``Mathematical models of hematopoietic cell replication and control", pp. 149-178 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 auto-immune 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
Time delays and noise are intrinsic features of the nervous system

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 neuron-computer 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 ("on-off intermittency") and preprogramming, and rapid computation using populations of noisy, multistable elements. Noisy dynamical systems with retarded variables offer a number of under-appreciated advantages for computation and control that may have many applications.

Steven Ruuth, Simon Fraser University
Convolution-generated motion as a link between cellular automata and continuum pattern dynamics

Many front propagation problems have been modeled using cellular automata. Some of the best-known examples of which are the "excitable media" which simulate the behavior of nerve cells, muscle cells, cardiac function and chemical
reaction. Here, the domain is broken into a finite number of cells each of which can have only a finite number of states (e.g, excited, refractory and resting). This crude discretization leads to several unwanted grid effects, including low order accuracy, unwanted anisotropy and incorrect front speeds. We discuss fast algorithms that give a much improved approximation of the front location and use a convolution step rather than discrete cell averages to avoid the grid effects prevalent in cellular automata methods. Preliminary studies indicate that the convolution-based approach accurately reproduces quantitative aspects of models (and hence has greater potential for quantitative comparisons to physical or biological data) whereas cellular automata are often only capable of reproducing certain qualitative properties.

Sven Sigurdsson, Kjartan Magnusson, Petro Babak, Simon Hubbard, University of Iceland
Discrete particle and continous density models of fish migration

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
Multiple periodic attractors in an excitatory delayed neural networks

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
School of Engineering, University of Guelph
Neural Dynamics and Computation for Visual Information Processing
in Vertebrate Retina

A novel neural network architecture is developed to study some basic aspects in early visual information processing in vertebrate retina, which is based on the neural anatomy and function of retinal neurons in tiger-salamanders and mudpuppies. The architecture comprises neural models of photoreceptors, horizontal cells, bipolar cells, amacrine cells, and ganglion cells. The main response characteristics of the retinal neurons are studied, and the model predictions are compared with the corresponding data. The possible action of $\gamma$-aminobutyric acid (GABA) inhibition from horizontal cells are also examined. The simulation results show that isolated or combined action of GABA_A and GABA_B can generate the dynamics observed in bipolar cells. The simulations removing the asymmetry in the inhibitory pathway produces results consistent with the hypothesis that the directional selectivity results from an asymmetrical inhibition. Model responses for directionally selective ganglion cells to moving spots are found to be in qualitative agreements with data from the turtle when tested with and without amacrine inhibition.

James Yorke, University of Maryland
Determining the Sequence (i.e. The ACGT's) of a Genome

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.

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