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

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

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 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

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

0/ct T

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.

Rapid Stimulation Leads to a Slower

Rhythm (Overdrive Suppression)

(Kunysz, Glass, Shrier 1995)

Fixed Delay Stimulation

Delay Detect activity

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

Pacemaker

Nagai, Gonzalez, Shrier, Glass, PRL (2000)

Dynamics in a Ring of Cardiac Cells

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