Mathematics Colloquium: “Act-and-wait” control strategies in the nervous system
Neill Hall 5W
John G. Milton, M.D. Ph.D.
The combination of feedback delay and noise pose particularly challenging problems for the design of strategies to stabilize unstable states. An inherent problem is that of distinguishing those fluctuations that need to be acted upon by the controller from those do not. This is because, by definition, there is a finite probability that an initial deviation away from a set point will be counter--balanced by one towards the set point just by chance. Moreover, too quick a response by a controller to a given deviation can lead to the phenomenon of ``over control'' leading to destabilization, particularly when time delays are appreciable. On the other hand, waiting too long runs the risk that the control may be applied too late to be effective. Thus methods based on continuous feedback control are not only anticipated to be very difficult to implement by the nervous system, but are also unlikely to be effective. One way to overcome these problems is to use an intermittent type open-loop control strategies such as ``act-and-wait'': the feedback is switched on (act) and off (wait) when the dynamical variables cross pre-set thresholds. The mathematical models take the form of stochastic delay differential equations with piecewise constant feedback. Despite their simplicity, these models provide robust control and are remarkably successful in reproducing experimental observations.