Time Delays in Biology
Leon Glass
Isadore Rosenfeld Chair in
Cardiology, McGill University,
Montreal, Quebec
Complex rhythms are ubiquitous in
physiological systems
(Glass, Nature, 2001)
Might some of this complexity be
due to time delays in control
systems?
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)
JTB 1989
Math Biosci
1988
Simple negative feedback systems
do not show oscillations
• Rate of change = production – decay
Rate of change = production –
decay (Mackey-Glass 1977)
Negative feedback
Mixed feedback
Simple Negative Feedback
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)
Milton et al. JTB 1990
On-Off negative feedback
(notation differs from previous slide)
Time delay equation for blood
cell dynamics
rate of change = production – destruction
Mackey and Glass, 1977
Mixed feedback in pupil
control, Milton et al. JTB
1991
Many control systems have multiple delays in parallel
(Glass, Beuter Laroque, 1988; Glass Malta, 1990)
Negative feedback
In motor control task
(Glass, Laroque, Beuter
1988)
Research Problem
• What are the generic dynamics in multilooped 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
nonlinear oscillators
Fixed delay stimulation of the Poincare
oscillator
Lewis, Bachoo, Glass, Polosa 1987
Glass, Zeng 1990
Conduction System of the Normal
Heart
Interpret this slide
Perturbed Cycle Length Curves
  tc / T0
Chaos in periodically stimulated
heart cells
Predict chaos based on
1D circle maps determined
from resetting experiments.
Resetting depends on phase
of stimulus.
(Guevara, Glass, Shrier 1981)
Rapid Stimulation Leads to a Slower
Rhythm (Overdrive Suppression)
(Kunysz, Glass, Shrier 1995)
Fixed Delay Stimulation
Detect activity
Delay
Stimulate after fixed delay
Fixed Delay Stimulation of Cardiac
Aggregates Leads to Bursting
(Kunysz, Shrier, Glass 1997)
Fixed delay stimulation of superior
laryngeal nerve in cat
Lewis, Bachoo,
Polosa, Glass
1990
Controlling Cardiac Alternans
Hall, Christini, et al. (1997)
Target Unstable Fixed Point
Stimulate to Control Alternans
Dynamics in a Ring of Cardiac Cells
Pacemaker
Nagai, Gonzalez, Shrier, Glass, PRL (2000)
Reentry
Cardiac dynamics in a ring
(Courtemanche, Glass Keener PRL 1993)
Instability leading to oscillation in
action potential duration and
propagation times
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