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Mathematical Biology / Biologie math\'matique
(Robert Miura, Organizer)

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JACQUES BELAIR, Universite de Montreal, Montreal, Quebec  H3C 3J7
Delayed regulation in hormonal systems: oscillations and control

The release and maintenance of normal levels of hormones require an intricate balance between numerous regulatory mechanisms. Hormonal profiles sometimes display time course with more or less regular oscillations. We present evidence for the normality of such fluctuations, and then discuss a simplified model of the insulin-glucose regulatory system. This model takes the form of interacting compartments, with time delays incorporated in the interactions between the components. In particular, the presence of a technological delay in an external compartment is seen to strongly influence the time course of the physiologically intact system, which also contains a delay representing the nonlocal action of the secreted hormone. The local stability of the stationary solution is determined, and periodic solutions are seen to emanate from Hopf bifurcations. Degenerate (codimension two) bifurcations are detected, and lead to more complicated oscillations, invariant tori and simultaneous existence of stable periodic solutions. [Joint work with Vincent Lemaire; supported by NSERC and FCAR]

DAVID BRYANT, Universite de Montreal, Montreal, Quebec  H3C 3J7
The combinatorics of quartet phylogenies

Phylogenetics is the study and identification of evolutionary patterns and structures in natural history. In this talk we explore some of the mathematics of these structures. We discuss problems and puzzles generated by the application of discrete mathematics to phylogenetics, and survey recent and classical results.

The basic objects of study are the phylogeny (a tree with leaves labelled by different species or taxa) and its substructures: quartets (four leaf induced subtrees), splits, clusters, and so on. We pay particular attention to the problem of assembling multiple phylogenies for small groups of species into larger complete phylogenies, and demonstrate that the interaction between these simple phylogenetics objects exhibits an intriguing complexity.

SUE ANN CAMPBELL, Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario  N2L 3G1
Multistability in coupled, near-identical neural models

We consider a pair of coupled Fitzhugh-Nagumo neurons. When the neurons are identical, we show how the symmetry of the system leads to the coexistence multiple, stable periodic orbits. As the coupling between the neurons is strengthened, these periodic orbits can undergo various bifurcations, leading to the coexistence of multiple, stable chaotic attractors. Finally, we examine how much of this behaviour persists when the neurons are close to, put no longer exactly, identical.

GERDA DE VRIES, Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta  T6G 2G1
From spikers to bursters via coupling: effects of noise and heterogeneity

I will discuss bursting as an emergent phenomenon. Previous studies have shown that the coupling of two identical spiking cells, incapable of bursting by themselves, can lead to bursting under weak coupling conditions. With stronger coupling, the cells revert to tonic spiking. The addition of noise dramatically increases the coupling range over which bursting is seen. Under intermediate coupling conditions, noise is no longer beneficial. However, bursting can be recovered again by introducing heterogeneity in the model parameters. Implications of these results will be discussed in the context of bursting electrical activity in pancreatic beta cells.

MICHAEL DOEBELI, University of British Columbia, Vancouver, British Columbia  V6T 1Z2
Evolutionary branching and sympatric speciation caused by different types of ecological interactions

Evolutionary branching occurs when frequency-dependent selection splits a phenotypically monomorphic population into two distinct phenotypic clusters. A prerequisite for evolutionary branching is that directional selection drives the population towards a fitness minimum in phenotype space. This paper demonstrates that selection regimes leading to evolutionary branching readily arise from a wide variety of different ecological interactions within and between populations. We use classical ecological models for symmetric and asymmetric competition, for mutualism, and for predator-prey interactions to describe evolving populations with continuously varying characters. For these models, we investigate the ecological and evolutionary conditions that allow for evolutionary branching and establish that branching is a generic and robust phenomenon. Evolutionary branching becomes a model for sympatric speciation when population genetics and mating mechanisms are incorporated into ecological models. In sexual populations with random mating, the continual production of intermediate phenotypes from two incipient branches prevents evolutionary branching. In contrast, when mating is assortative for the ecological characters under study, evolutionary branching is possible in sexual populations and can lead to speciation. Therefore, we also study the evolution of assortative mating as a quantitative character. We show that evolution under branching conditions selects for assortativeness and thus allows sexual populations to depart from fitness minima. We conclude that evolutionary branching offers a basis for understanding adaptive speciation and radiation under a wide range of different ecological conditions.

LEAH EDELSTEIN-KESHET, Department of Mathematics, University of British Columbia, Vancouver, British Columbia  V6T 1Z2
Recent progress in biomedical modelling

In this talk, I will survey a number of ongoing projects which involve mathematical modelling of biomedical topics. Foremost among these will be a description of work on the pathology of Alzheimer's Disease. I will describe how a combination of analysis and simulations is being used to understand particular aspects of this disease, namely the inflammatory cascade that results in neuronal stress and death. I will also mention other ongoing work at the subcellular, signal transduction level.

RODERICK EDWARDS, Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia  V8W 3P4
Structure and behaviour in gene networks

Tremendous progress has been made in mapping genetic structures in humans and other organisms. This wealth of information will necessitate new analytic tools to deduce function from structure in gene regulatory networks. One way to begin tackling this problem is to investigate simple idealized switching networks that capture the various possible logical structures of real gene networks (or other networks characterized by strong switching). Discrete-time Boolean networks constitute one such attempt, but the assumption of a discrete clock and synchronous updating is not biological and may lead to special behaviours. Instead, we investigate differential equation models that still involve binary functions of binary inputs. We are able to demonstrate the existence, stability and periods of periodic orbits corresponding to cycles in given network structures. The possible patterns of periodic behaviour are remarkably rich in networks with as few as 4 interacting genes and very complex limit cycles are possible. This allows plenty of room for complex but orderly processes of gene expression and regulation. Furthermore, since complex network interactions can produce aperiodic behaviour, it may be important to understand why some network structures lead to chaos, while others lead to periodic or steady state behaviour. We will illustrate how chaos can be revealed in these networks by demonstrating the existence of a `golden-mean map' in a particular network of 4 genes.

HERB FREEDMAN, University of Alberta, Edmonton, Alberta
Competition models of cancer treatment

Cancer treatment by chemotherapy and immunotherapy is modelled by considering the normal and cancer cells as competing for the same resources and the treatment as a predator on both of them. Three modes of treatment in each case is considered, Criteria are established for the cancer to be stabilized at an acceptably low level, either in equilibrium or oscillating using bifurcation techniques.

LEON GLASS, McGill University
Dynamics of reentrant tachycardia

Reentrant tachycardias are abnormal cardiac arrhythmias in which the period is set by the time it takes for the excitation to travel in a circuitous path. I describe a very simple model of reentrant arrhythmias in which the path is modeled by a one-dimensional ring of cells. This model is used to investigate various aspects of cardiac arrhythmias including: the stability of the circulation as the path length of the reentrant circuit is decreased, the effects of single and multiple pulses delivered during the tachycardia, control of instabilities during the reentrant rhythm by adjusting the timing of stimuli delivered during the course of the tachycardia, paroxysmal rhythms in which the arrhythmia undergoes sudden onset and offset. In all cases, mathematical models can be developed suitable for comparison with experimental and clinical data.

JOHN HSIEH, University of Toronto, Toronto, Ontario  M5S 1A8
Statistical inference of stochastic models in life sciences

This talk presents statistical procedures of estimation and hypothesis testing for stochastic models used in biomedical sciences, with special emphasis on the modern counting processes martingale methods for asymptotic inference. The stochastic processes considered include processes with independent increments (which includes uni- and multi-dimensional nonhomogeneous Poisson processes), Markov processes, renewal processes, diffusion processes, point and marked point processes, martingales and processes with stationary independent increments (which include homogeneous Poisson process, symmetric random walk, Brownian motion and Brownian bridge). Statistical inference procedures depend on sampling plan and on how data are collected. Both parametric and nonparametric procedures will be presented using likelihood methods and martingale approaches to account for different sampling plans and types of censoring and truncation on data. The connection between likelihood and martingale methods will be delineated. Examples of applications include chain binomial models and SIR models in epidemiology, branching process models in population genetics, life table analysis in medical demography, spatial and cluster models in ecology, survival analysis for medical follow-up studies and random walk models for cell kinetics and for neuron firing.

LILA KARI, Western Department of Computer Science, University of Western Ontario, London, Ontario  N6A 5B7
Life like a theorem

How do cells and nature ``compute''? They read and ``rewrite'' DNA all the time, by processes that modify sequences at the DNA or RNA level. We study the computational capabilities of cellular organisms with the aim of understanding their computational processes and of harnessing their computational power for our desired purposes. Together with Laura Landweber we developed a formal model for the homologous recombinations that take place during gene rearrangement in ciliates (unicellular protozoans named for their wisp-like cover of cilia). We prove that our model has universal computational power which indicates that in principle, some unicellular organisms may have the capacity to perform any computation carried out by an electronic computer. We show also preliminary results on the information-theoretical structure of DNA.

YUE-XIAN LI, Department of Mathematics, University of British Columbia Vancouver, British Columbia  V6T 1Z2
Tango waves in a bidomain model of fertilization calcium waves

At fertilization, an oocyte becomes an excitable medium capable of generating travelling waves in the level of cytosolic Ca2+. Several Ca2+ wave patterns have been observed including travelling fronts and pulses as well as concentric and spiral waves that are known to exist in other excitable media. We here report the discovery of a new wave phenomenon in the numerical study of a model of fertilization Ca2+ waves. It is characterized by waves that propagate in a back-and-forth pattern. The reversal in the direction of these waves resembles the reversal in the direction of movement typical in a tango dance. Thus we call them tango waves. They are generated by the injection of a large dose of Ca2+ into a medium that is nearly excitable. When the medium becomes excitable, travelling pulses are generated as the tango waves move forward. The advance of tango waves in the medium is associated with the spread of bistability. The study shows that tango waves facilitate the dispersion of localized Ca2+. Key features of the Ca2+ excitable medium that make tango waves possible are revealed. These include the short range of Ca2+ diffusion, the bidomain nature of the system, and the ability of the Ca2+ content in the store to cause transitions between single and bistabilities. Since these features are characteristic to oocytes of many species, this study predicts that tango waves can occur in real oocytes. This is supported by the observation of tango-like waves in nemertean worms and ascidian eggs.

ANDRE LONGTIN, Universite d'Ottawa, Department of Physics, Ottawa, Ontario  K1N 6N5
Dynamical gain control in electrosensory systems

Weakly electric fish are increasingly studied to shed light on key issues of neural coding. They generate a weak oscillatory electric field around their body. Amplitude modulations of this field, caused by environmental stimuli such as food or communication signals from other fish, are read by cutaneous P-type electroreceptors. The firing activity of these receptors is relayed to pyramidal cells. These cells are thought to perform, among other tasks, gain control. Such control maintains their output firing rate within certain bounds, so that the fish can properly discriminate a wide range of stimulus intensities; it may also highlight novel stimuli. Such control is achieved via delayed excitatory and inhibitory feedback pathways from higher brain structures to the pyramidal cells.

We present a mathematical model of the pyramidal cell firing activity in the presence of such feedback. It is based on recently obtained data from intracellular measurements. A simpler integrate-and-fire type model is also derived, which embodies the main features of the full model. Particular attention is paid to the stochastic nature of the synaptic input in each case. With a ``static feedback'', a form often assumed in studies of neural gain control, this noise is shown to produce a novel ``divisive inhibition'' regime for perithreshold stimuli. The analysis of the full model with dynamical feedback further reveals bifurcations to oscillatory activity involving the pyramidal cells and higher brain structures.

MICHAEL MACKEY, Center for Nonlinear Dynamics and Department of Physiology, McGill University, Montreal, Quebec
Periodic hematological diseases: insight into the pathology from mathematical modeling

The periodic hematological diseases are those in which one or more of the circulating cells types (white cells, red blood cells, platelets) oscillate spontaneously with periods ranging from days to weeks. This talk will examine the clinical and laboratory data for several of these diseases (cyclical neutropenia, cyclical thrombocytopenia, and periodic chronic myelogenous leukemia) and the insight obtained from mathematical models for these conditions.

ROBERT MIURA, Department of Mathematics, University of British Columbia, Vancouver, British Columbia  V6T 1Z2
Resonances in excitable cells

The electrical potential activity across the membranes of neurons, cardiac cells, and pancreatic beta-cells reflects their excitability. These temporal patterns include action potentials, bursts of such spikes, and voltage oscillations that are subthreshold to spike initiation. In general, the genesis of this activity is attributable either to oscillations endogenous to the cell or to exogenous stimuli. Our mathematical and experimental studies have examined the possible roles that ``resonance'' has in neuronal activity patterns and that ``stochastic resonance'' (SR) has in neuronal function. In this talk, we define ``resonance'' and ``stochastic resonance'', describe mathematical considerations that motivated some of the experimental protocols, and discuss some results from the modelling studies. The concept of resonance, characterized by an elevated response of a system to stimuli at specific frequencies, was introduced in electrical and mechanical systems. Stochastic resonance is similar, except the presence of noise accentuates the resonant responses of the system. SR has been demonstrated experimentally in different biological systems. As yet, there are no demonstrations of SR in the central nervous system. Mathematical models, however, predict a contribution of SR to neuronal excitability.

SHIGUI RUAN, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia  B3H 3J5
Global Analysis in a predator-prey system with nonmonotonic functional response

A predator-prey system with nonmonotonic functional response is considered. A global qualitative analysis of the model is carried out. The bifurcation analysis of the model depending on all parameters indicates that it exhibits numerous kinds of bifurcation phenomena including the saddle-node bifurcation, the supercritical and subcritical Hopf bifurcations, and the homoclinic bifurcation. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop, or exhibits the so-called ``paradox of enrichment'' phenomenon. Moreover, a limit cycle cannot coexist with a homoclinic loop for all parameters. In the generic case, the model has the bifurcation of cusp-type of codimension 2 (i.e., Bogdanov-Takens bifurcation) but has a multiple focus of multiplicity at least two.

FRANCES SKINNER, Toronto Western Research Institute, University Health Network, Departments of Medicine and Physiology, Institute of Biomaterials and Biomedical Engineering, University of Toronto
Cellular mechanisms of neuronal network behaviour

Extensive inhibitory networks of interneurons play critical roles in generating the synchronous rhythmic output of principal cells in the brain. This suggests that they may provide the precise temporal structure necessary for ensembles of neurons to perform specific functions. Theoretical work has demonstrated how synchrony could occur in mutually inhibitory networks. In addition, gap junctions (or electrotonic coupling) exist between interneurons, but their role is far from clear. The combination of inhibitory and gap junctional coupling in neuronal networks has not been extensively examined.

Slow brain rhythms are associated with deep sleep, some post-ictal epileptic events, and the onset of some seizure states. Experimentally, it has been shown that a combination of gap junctions and inhibitory coupling is needed for slow rhythmic output to occur in the hippocampal cortex. By using a minimal mathematical model of an interneuronal network, we describe cellular based mechanisms which require the coupling of intrinsic and synaptic currents to produce stable, bursting oscillations as observed experimentally. From this work, we postulate that a novel role of gap-junctional coupling in inhibitory networks may lie in the generation and stabilization of slow bursting behaviour.

JACK TUSZYNSKI, Faculty of Pharmacy and Pharmaceutical Sciences, University of Alberta, Edmonton, Alberta  T6G 2J1
Nonlinearity and fractality in the mathematical modelling of pharmacokinetic processes

Time-dependent pharmacokinetic phenomena are observed for a wide variety of drugs, including cardiovascular drugs such as mibefradil, diltiazem, verapamil, lidocaine, etc. Conventional approaches to explaining nonlinearities and time-dependent kinetics of a drug focus on purely homogenous mechanisms such as interactions of the drug with its own metabolic pathway or first-pass elimination processes. Less emphasized is the role played by the heterogeneity of the relevant metabolic and transport processes. Unfortunately, traditional compartmentalized heterogenous models with a finite number of compartments and linear couplings result in purely exponential or multiexponential kinetics, and cannot capture some anomalous kinetics, even empirically, without a great number of adjustable parameters. We have developed models of time-dependent and nonlinear pharmacokinetics by including the effects of fractal geometry of the liver and Michaelis-Menten nonliearities due to the action of enzymes. In particular, the inclusion of fractal geometry as defining the exploration space for the drug molecules being distributed, absorbed and eliminated in the various organs of the body could lead to a markedly improved predictability of elimination curves for drugs in animal and human subjects. We have carried out analyses of the laboratory data for the elimination of mibefradil in the livers of four different dogs. Our comparison of fractal kinetics results with those of standard homogenous models indicates a substantially better fit using fractal approach, especially for a lower dosage concentrations. In addition, work on a spectrum of techniques, from absorption models to whole-body clearance to in vitro-in vivo correlation (IVIVC), is underway aimed at predicting the pharmacokinetic parameters earlier in the development process. Our comparisons studies of three models for IVIVC (well stirred, parallel tube and distributed tube) indicate that a distributed tube model offers improved correlation for high clearance drugs.

ACKNOWLEDGMENTS. Funding for this project was provided by NSERC, MRC and AHFMR.

PAULINE VAN DEN DRIESSCHE, University of Victoria, Victoria, British Columbia  V8W 3P4
The number of limit cycle solutions to n-dimensional dynamical systems with biological applications

Criteria are given under which the boundary of an oriented surface does not consist entirely of trajectories of the C¹ differential equation x' = f(x) in Rⁿ. In the special case of an annulus, the criteria are used to deduce sufficient conditions for the equation to have at most one cycle. For three-dimensional competitive Lotka-Volterra systems, it is shown that no limit cycle may occur in two equivalence classes; thus proving a previous conjecture. The nonexistence of periodic solutions is established for some disease transmission models. These results lead to complete global analyses of the biological systems.