Neurological disorders, such as epilepsy and Parkinson’s disease, often evidence themselves through pathological neuronal behaviour, including seizures induced by hypersynchronous activity in certain regions of the brain and muscular tremors. While neurologists have the technology to implant electrodes into the brain in order to deliver a stimulus, finding the locations which most significantly influence brain activity in a desired way and the type of stimulus is still largely unknown.
From a mathematical perspective, the brain can be represented as an immensely large-scale complex network that describes the interconnection of neurons (nodes). The mathematical model then describes changes in the state of the brain “system” as changes in the membrane potential of neurons in the network. The objective of neurological implants is then to direct or guide the system state from an initial configuration to a desired final configuration by the application of an external input. In most cases our goal is to direct the system, or brain, away from known pathological behaviour – preventing epileptic seizures before they occur or reducing the symptoms of tremors due to Parkinson’s disease.
Finding the locations where controls should be placed within a network addresses the problem of finding the control structure necessary to influence the system, whereas designing the time-varying input provides the specific stimulus used to modify the state. Selecting locations that are insensitive to perturbations in the network structure and inputs that are robust to parameter values allows us to find mechanisms for robustly controlling networks.