Dynamical systems theory in physiology

Readers of the current issue will find something unfamiliar , perhaps tantalizing, perhaps unsettling. I am referring to the articles by Cha et al. (see " Ionic mechanisms and Ca 2+ dynamics... " and " Time-dependent changes in membrane excitability... "). The first of these articles is less exotic; it presents computer simulations of a model for bursting electrical activity in pancreatic  cells. The second uses bifurcation diagrams to analyze the behavior of the model. I will argue that this is relevant far beyond  cells—the leading edge of a wedge driving the methods of dynamical systems theory into the heart of biology. Mathematical modeling of cell electrical activity has a long history in physiology, going back to the work of Hodgkin and Huxley (Hodgkin and Huxley, 1952; Chay and Keizer, 1983) for action potential generation and propagation in squid giant axon. The model of Cha et al. (2011a,b) is based on the Hodgkin and Huxley formalism , but augmented with mechanisms for maintaining ionic balance (pumps and exchangers for Ca 2+ , Na + , and K + , and the endoplasmic reticulum). In addition, a nod is given in the direction of metabolism, as  cells are first and foremost metabolic sensors and use ATP-dependent K + (K(ATP)) channels, to transduce the rate of glucose metabolism into intensity of electrical activity. Cha et al. (2011a,b) follow the path blazed by Chay and Keizer (1983). Their model was based on the simple idea, first proposed by Atwater et al. (1980), that bursting results from slow modulation of spiking by calcium. That is, during the active spiking phase of the burst, calcium builds up and turns on calcium-activated K + (K(Ca)) channels until membrane potential falls below the threshold level and spiking terminates. During the ensuing silent phase, calcium would be pumped out of the cell, lowering the spike threshold and allowing the next active phase to begin. Rinzel (1985) formalized this mathematically, recognizing that the key element of Chay–Keizer was a fast spiking system modulated by slow negative feedback. This led to a profusion of models, different biophysically but essentially equivalent mathematically , with alternate proposals for the source of the negative feedback. These included inactivation of the L-type Ca 2+ channel (Chay, 1990), indirect activation of K(ATP) channels by Ca 2+ via its effects on ATP consumption or production (Keizer and Magnus, 1989), and ac-Correspondence to Arthur …