Time Delays in Biology
Leon Glass
Isadore Rosenfeld Chair in
Cardiology, McGill University,
Montreal, Quebec
Why time delays?
• Biological systems have intrinsic delays.
• Natural or artificial control systems
necessarily have delays from the sensing
of a variable and the initiation of
appropriate response.
• Mathematics of systems with time delays
are intriguing and pose basic
mathematical challenges.
Sources of delays
• Diffusion or transport of substances
(cellular metabolites, hormones, oxygen
and carbon dioxide in blood, control
molecules)
• Conduction time of nerves
• Intrinsic times for synthesis, growth, and
reproduction
Effects of delays
• Delays can lead to change of stability of
dynamics. This makes it of interest to
theoretical biology and also for practical
attempts to control dynamics.
Adding time delays to negative feedback
systems can lead to oscillations
Activation – Inhibition Systems
Glass and Kauffman, 1972
Shymko and Glass, 1974
Artificial delays in physiological
control systems
• motor control – Beuter, Laroque, Glass
• pupil diameter – Longtin, Milton
• feedback to pacemakers - Kunysz, Shrier, Glass
• reentrant propagation through the AV node and control of alternans – Billette, Hall, Glass, Christini
• delay equations from pulses propagating in a ring (Courtemanch, Glass, Keener)
Milton et al. JTB 1990
Control of pupil diameter via
a hybrid device that turns on a
light if pupil diameter is bigger
than a threshold following a time
delay (Longtin, Milton 1989)
Simple Negative Feedback
Time delay equation for blood
cell dynamics
rate of change = production – destruction
Mackey and Glass, 1977
Many control systems have multiple delays in parallel
(Glass, Beuter Laroque, 1988; Glass Malta, 1990)
Research Problem
• What are the generic dynamics in multi-
looped feedback systems? Is it possible
that some of the complex dynamics seen
in physiological control can be explained
by dynamics in such systems with multiple
loops?
Fixed delay stimulation of the Poincare
oscillator
Lewis, Bachoo, Glass, Polosa 1987
Glass, Zeng 1990
Chaos in periodically stimulated
heart cells
(Guevara, Glass, Shrier 1981)
Predict chaos based on
1D circle maps determined
from resetting experiments.
Resetting depends on phase
of stimulus.
Main ideas
• Time delays arise in many different ways
in physiological systems
• Experiments and theory have studied the
effects of artificially introduced time delays
in physiological systems
• Time delays in feedback can lead to
oscillations as well as complex rhythms
• Thanks to many collaborators