Shalizi on Criticality

Cosma Shalizi is guest blogging at Crooked Timber and begins with a discussion of the mechanics of disordered systems and optimized criticality, based on the work of Osame Kinouchi and Mauro Copelli. (For those of you who missed it, see also Azra’s piece on self-organized criticality and cancer.)

Neurons, like muscle cells, are “excitable”, in that the right stimulus will get them to suddenly expend a lot of energy in a characteristic way — muscle cells twitch, and neurons produce an electrical current called an action potential or spike. Kinouchi and Copelli use a standard sort of model of an excitable medium of such cells, which distinguish between the excited state, a sequence of “refractory” states where the neuron can’t spike again after it’s been excited, and a resting or quiescent state when the right input could get it to fire. (These models have a long history in neurodynamics, the study of heart failure, cellular slime molds, etc.) Normally, in these models the cells are arrayed in some regular grid, and the probability that a resting cell becomes excited goes up as it has more excited neighbors. This is still true in Kinouchi and Copelli’s model, only the arrangement of cells is now a simple random graph. Resting cells also get excited at a steady random rate, representing the physical stimulus.

Kinouchi and Copelli argue that the key quantity in their model is how many cells are stimulated into firing, on average, by a single excited cell. If this “branching ratio” is less than one, an external stimulus will tend to produce a small, short-lived burst of excitation, and there will be no spontaneous activity; the system is sub-critical. If the branching ratio is greater than one, outside stimuli produce very large, saturating waves of excitation, and there’s a lot of self-sustained activity, making it hard to use a super-critical network as a detector. At the critical point, however, where each excited cell produces, on average, exactly one more excited cell, waves of excitation eventually die out, but they tend to be very long-lived, and in fact their distribution follows a power law.