Using the root locus method to analyze pathological oscillations in neurological diseases

In recent years the authors have developed what appears to be a very successful phenomenological model for analyzing the role of deep brain stimulation (DBS) in alleviating the symptoms of Parkinson´s disease. In this paper, we extend the scope of the model by using it to predict the generation of new frequencies from networks tuned to a specific frequency, or indeed not self-oscillatory at all.We have discussed two principal cases: firstly where the constituent systems are coupled in an excitatory-excitatory fashion, which we designate by “+/+”; and secondly where the constituent systems are coupled in an excitatory-inhibitory fashion, which we designate “+/-”. The model predicts that from a basic system tuned to tremor frequency we can generate an unlimited range of frequencies. We illustrate in particular, starting from systems which are initially non-oscillatory, that when the coupling coefficient exceeds a certain value, the system begins to oscillate at an amplitude which increases with the coupling strength. Another very interesting feature, which has been shown by colleagues of ours to arise through the coupling of complicated networks based on the physiology of the basal ganglia, can be illustrated by the root locus method which shows that increasing and decreasing frequencies of oscillation, existing simultaneously, have the property that their geometric mean remains substantially constant as the coupling strength is varied. We feel that with the present approach, we have provided another tool for understanding the existence and interaction of pathological oscillations which underlie, not only Parkinson´s disease, but other conditions such as Tourette´s syndrome, depression and epilepsy.