From SIAM News , Volume 40 , Number 2 , March 2007

Spatiotemporally organized waves are among the most ubiquitous activity patterns in the brain. Recent advances in electrophysiology, imaging, and other technologies allow experimentalists to observe these activity patterns simultaneously in the responses of many individual neurons and in the ensemble response of the neural circuit, both in vivo (the whole animal/subject) and in vitro (e.g., cultured cells, brain slices); see Figure 1. By making explicit the process by which wave-like activity emerges from neuronal circuitry, applied mathematics can contribute substantially to our understanding of the dynamics of the brain. Propagating waves clearly tell us something about the intrinsic circuitry of the brain—for example, its preferred directionality and excitability. Any active medium that is locally connected is able, and indeed expected, to generate wave-like behavior. Recognizing that nature generally tries to exploit the intrinsic dynamics of systems, several investigators have proposed theories as to how wave-like activity might be useful in the brain. Wave-like patterns are evident in both normal and pathological activity throughout the nervous system. The clearest examples of organized spatiotemporal activity for normal function are central pattern generators, which coordinate activity in muscles involved in such motor functions as swimming, feeding, and even breathing. Recordings from the spinal cord of the eel-like lamprey, for instance, show a clear pattern of oscillatory waves that produce the necessary patterns for swimming. Similar waves are observed in the leech and in the crayfish. Other possible roles for waves in normal brain function include establishing the spatial map between the retina and brain during development, scanning sensory inputs for novel features, and encoding target location during motor control. Waves are also common in many pathological brain states, such as epilepsy. Focal epilepsy, in particular, is characterized by waves of activity emanating from a local cortical malformation resulting either from injury or from a developmental disorder. Epileptiform activity waves can also be seen in many animal models of epilepsy, both in vivo and in vitro. In most cases, waves are induced experimentally by increasing the excitability of neural tissue with drugs or other manipulations. Neural waves can be roughly divided into three classes: (i) stationary waves; (ii) active waves; and (iii) oscillation-generated, or phase waves. Stationary waves are spatial patterns that do not propagate—zero-velocity waves. Although their properties are extremely interesting, we focus here on the latter two classes. Active waves emerge from spatial networks when each local point exhibits the dynamics of an excitable system. Waves of this type—which are responsible for spiral waves in chemical systems, for the movement of flame fronts, and for many other traveling phenomena—are the most familiar to applied mathematicians. In the brain, the best known examples include epileptiform activity waves emanating from diseased or damaged tissue or from experimental brain tissue treated with drugs. In certain tissue slices it is also possible to induce spiral waves and other twodimensional phenomena. Phase waves emerge when the local circuitry is intrinsically oscillatory and local heterogeneities, anisotropy, or dynamic instabilities induce phase differences that vary over space. Waves observed in the slug brain, in the visual cortex of the turtle, in the motor cortex of monkeys, and in the central pattern generators of many swimming animals are likely examples of phase waves. Neurophysiology and Waves