




Mathematical Biology / Biologie math\'matique (Robert Miura, Organizer)
 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
insulinglucose 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, nearidentical neural models

We consider a pair of coupled FitzhughNagumo 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 frequencydependent 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 predatorprey 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 EDELSTEINKESHET, 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). Discretetime 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
`goldenmean
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 onedimensional 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
multidimensional 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 followup
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 wisplike 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 informationtheoretical structure of DNA.
 YUEXIAN 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
backandforth 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 tangolike 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 Ptype 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 integrateandfire 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 betacells 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 predatorprey system with nonmonotonic
functional response

A predatorprey 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 saddlenode 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 socalled ``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 cusptype of codimension 2 (i.e.,
BogdanovTakens 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 postictal
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 gapjunctional 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

Timedependent 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 timedependent kinetics of a drug focus
on purely homogenous mechanisms such as interactions of the drug with
its own metabolic pathway or firstpass 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 timedependent and
nonlinear pharmacokinetics by including the effects of fractal
geometry of the liver and MichaelisMenten 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 wholebody clearance to in
vitroin 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 ndimensional dynamical
systems with biological applications

Criteria are given under which the boundary of an oriented surface does
not consist entirely of trajectories of the C^{1} differential equation
x' = f(x)
in R^{n}. 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 threedimensional competitive
LotkaVolterra 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.

