page_title: Mackey-Glass equations
section_title: Introduction
hierarchical_section_title: Mackey-Glass equations / Introduction
caption_attribution_description: English: Mackey-glass timeseries for a pathological case.
context_page_description: 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. Originally, they were used to model the variation in the relative quantity of mature cells in the blood. The equations are defined as:
  and
  where represents the density of cells over time, and are parameters of the equations.
  Equation, in particular, is notable in dynamical systems since it can result in chaotic attractors with various dimensions.
context_section_description: There exist an enormous number of physiological systems that involve or rely on the periodic behaviour of certain subcomponents of the system. 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. 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:
  where  is any initial condition for the initial value problem.
  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. 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.
caption: Also generated from the Mackey-Glass equations, but now could be seen as pathological blood cell density variation. Here, 
    
      
        
          τ
          =
          22
        
      
      {\displaystyle \tau =22}
    
  .
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