ABSTRACT
Biological systems change over time; they change over time in two important and very different ways. First, animals breathe, populations rise and fall and your heart keeps beating. All these changes are regular and continual. In each case the output of the system is changing in a rhythmic manner and past behaviour provides (at least some) indication of what will happen in the future. This change is the continuous fluctuation that makes up the rhythm of life, and varies within some acceptable range. The system is in some sort of balanced equilibrium. Mathematically, the system is said to be stationary, not because the system is not moving, but because the mathematical description of the system and the system parameters are not changing. There is a second type of change in biological systems. This occurs when
the system itself (and by extension, the mathematical description of that system) changes. For millennia the global climate system has been in a quite stationary state — certainly there is seasonal variation and longer climatic cycles , but all this variation is within some natural bounds. Both sorts of change are illustrated in Fig. 1.1. Relative global warm and cool periods occur and fluctuation occurs, but the system is stationary. Then, man dramatically increases the carbon dioxide content in the atmosphere to a level which is unlike anything in natural history and at a speed far faster than nature has ever managed. This man-made change in the state of one variable in the global climate system means that the system is now nonstationary. Its behaviour is entering a regime unlike any that has been seen in the past. Fig. 1.1 illustrates the change in a system, the human cardiovascular sys-
tem, as a patient suffers a heart attack. Similarly, individual neurons within your brain are rapidly and spontaneously pulsing. The neurons (and there are about one thousand billion of them) in your brain are the individual cells which process information and are collectively responsible for consciousness and thought. Despite this massive responsibility, the state of each individual neuron can be fairly well described with a simple set of mathematical equations. The behaviour of each neuron can be qualitatively predicted with reasonable accuracy — by qualitative prediction we mean that we are able to provide a description of the type of predicted behaviour, without predicting
Figure 1.1: Dynamics of systems. The top four plots show the stable dynamic behaviour for the theoretical Ro¨ssler system (dxdt = −y − z, dydt = x + 0.1y, and dzdt = 0.1 + z(x − c)). The horizontal axis is time and in each case the system changes continuously. For a parameter (c) value of 4, the system is periodic. However, as we change the value of the parameter, the behaviour of the system changes. For c = 6 the system is now bi-periodic and for c = 12 the system becomes period three. Of course, for all intermediate values of c the system behaviour changes gradually. But for any fixed value of c, the system will oscillate — changing continuously. Finally, for c = 18, the system behaviour is bounded and not periodic — it exhibits what is mathematically defined as chaos. The lower panel illustrates a real recording of the electrocardiogram of a human and shows a second example of the change in dynamics between (in this case, at least) three states: from a stable (almost periodic) regular heartbeat on the left, to ventricular tachycardia in the middle and ventricular fibrillation on the right. This trace illustrates the progression of onset of a heart attack and in this case the patient (in a coronary care unit) only recovered after medical intervention. Within each window, past behaviour provides (some) guide to the future. But when the system parameter changes, this is not possible because it is not possible to predict how the system parameter changes.
