1. RHYTHMS IN HUMAN NATURE



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     Glass and Mackay illustrate simple perturbations in an experiment that analyzes

“The effects of a brief electrical shock delivered to an aggregate of spontaneously beating cells derived from the ventricles of an embryonic chick heart. In response to a brief electrical stimulus, there is a resetting of the phase of subsequent action potentials, but the original cycle time is reestablished within several beats… Poincare called such oscillations stable limit cycles… If, on the other hand, a small perturbation induces a change in the dynamics so that the original dynamics are not reestablished, then the steady state or limit cycle is unstable…. Any value of a parameter at which the number and/or stability of steady states and cycles change is called a bifurcation point and the system is said to undergo a bifurcation… Since small changes in parameters lead to qualitatively different dynamics at bifurcation points, systems are not structurally stable at bifurcation points.” 3

     Arthur Winfree, a pioneering biophysicist and chronobiologist, makes the case that small perturbations to rhythms, when precisely timed, can have big effects. In his studies on cardiac arrhythmias, especially ventricular fibrillation, he observed, “a small local depolarization can trigger transition to fibrillation…If the stimulus is too big or too small or wrongly timed, it won’t work. There is an atrial vulnerable phase and, later, a ventricular vulnerable phase.” 4













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