Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside

Clinicians are increasingly aware of the remarkable upsurge of interest in non-linear dynamics, the branch of the sciences widely referred to as chaos theory. Those attempting to evaluate the biomedical relevance of this subject confront a confusing array of terms and concepts, such as non-linearity, fractals, periodic oscillations, bifurcations, and complexity, as well as chaos." Therefore, I hope to provide an introduction to some key aspects of non-linear dynamics and review selected applications to physiology and medicine. Linear systems are well behaved. The magnitude of their responses is proportionate to the strength of the stimuli. Further, linear systems can be fully understood and predicted by dissecting out their components. The subunits of a linear system add up-there are no surprises or anomalous behaviours. By contrast, for non-linear systems proportionality does not hold: small changes can have striking and unanticipated effects. Another complication is that non-linear systems cannot be understood by analysing their components individually. This reductionist strategy fails because the components of a non-linear network interact-ie, they are coupled. Examples include the interaction of pacemaker cells in the heart or neurons in the brain. Their non-linear coupling generates behaviours that defy explanation by traditional (linear) models such as self-sustained, periodic waves (eg, ventricular tachycardia); abrupt changes (eg, sudden onset of a seizure); and, possibly, chaos. One important class of abrupt, non-linear transitions is called a bifurcation.1,4 This term describes situations in which a very small increase or decrease in the value of some factor controlling the system causes it to change abruptly from one type of behaviour to another. A common type of bifurcation is the sudden appearance of regular oscillations that alternate between two values. This dynamic may underlie various alternans patterns in cardiovascular dysfunction. A familiar example is the beat-to-beat alternation in QRS axis and amplitude seen in some cases of cardiac tamponade.5 Many other examples of alternans in perturbed cardiac physiology have been described, including ST-T alternans that may precede ventricular fibrillation 6 and pulsus alternans during heart failure. Although the focus of much recent attention, chaos per se actually consists of only one specific subtype of non-linear dynamics.' Chaos refers to a seemingly random type of variability that can arise from the operation of even the most simple non-linear system. Because the equations that generate such erratic, and apparently unpredictable, behaviour do not contain any random terms-this mechanism is referred to as deterministic chaos.' The colloquial …