In
mathematics and
mathematical biology, the Mackey–Glass equations, named after
Michael Mackey and
Leon Glass, refer to a family of
delay differential equations whose behaviour manages to mimic both healthy and pathological behaviour in certain biological contexts, controlled by the equation's parameters.[1] Originally, they were used to model the variation in the relative quantity of mature
cells in the blood. The equations are defined as:[1][2]
(Eq. 1)
and
(Eq. 2)
where represents the density of cells over time, and are parameters of the equations.
There exist an enormous number of
physiological systems that involve or rely on the periodic behaviour of certain subcomponents of the
system.[4] For example, many
homeostatic processes rely on
negative feedback to control the concentration of substances in the blood;
breathing, for instance, is promoted by the detection, by the brain, of high CO2 concentration in the blood.[5] One way to model such systems mathematically is with the following simple
ordinary differential equation:
where is the rate at which a "substance" is produced, and controls how the current level of the substance discourages the continuation of its production. The solutions of this equation can be found via an
integrating factor, and have the form:
However, the above model assumes that variations in the substance concentration is detected immediately, which often not the case in physiological systems. In order to ease this problem,
Mackey, M.C. & Glass, L. (1977) proposed changing the production rate to a function of the concentration at an earlier point in time, in hope that this would better reflect the fact that there is a significant delay before the
bone marrow produces and releases mature cells in the blood, after detecting low cell concentration in the blood.[6] By taking the production rate as being:
we obtain Equations (1) and (2), respectively. The values used by
Mackey, M.C. & Glass, L. (1977) were , and , with initial condition . The value of is not relevant for the purpose of analyzing the dynamics of Equation (2), since the
change of variable reduces the equation to:
This is why, in this context, plots often place in the -axis.
Dynamical behaviour
It is of interest to study the behaviour of the equation solutions when is varied, since it represents the time taken by the physiological system to react to the concentration variation of a substance. An increase in this delay can be caused by a
pathology, which in turn can result in chaotic solutions for the Mackey–Glass equations, especially Equation (2). When , we obtain a very regular periodic solution, which can be seen as characterizing "healthy" behaviour; on the other hand, when the solution gets much more erratic.
^Specht, H.; Fruhmann, G. (1972). "Incidence of periodic breathing in 2000 subjects without pulmonary or neurological disease". Bulletin de physio-pathologie respiratoire. 8 (5): 1075–1083.
PMID4657862.
^Rubin, R.; Strayer, D.S.; Rubin, E. (2008). Rubin's pathology: clinicopathologic foundations of medicine. Lippincott Williams & Wilkins.