Application of Describing Function Analysis to a Model of Deep Brain Stimulation

Abstract

Deep brain stimulation effectively alleviates motor symptoms of medically refractory Parkinson's disease, and also relieves many other treatment-resistant movement and affective disorders. Despite its relative success as a treatment option, the basis of its efficacy remains elusive. In Parkinson's disease, increased functional connectivity and oscillatory activity occur within the basal ganglia as a result of dopamine loss. A correlative relationship between pathological oscillatory activity and the motor symptoms of the disease, in particular bradykinesia, rigidity, and tremor, has been established. Suppression of the oscillations by either dopamine replacement or DBS also correlates with an improvement in motor symptoms. DBS parameters are currently chosen empirically using a "trial and error" approach, which can be time-consuming and costly. The work presented here amalgamates concepts from theories of neural network modeling with nonlinear control engineering to describe and analyze a model of synchronous neural activity and applied stimulation. A theoretical expression for the optimum stimulation parameters necessary to suppress oscillations is derived. The effect of changing stimulation parameters (amplitude and pulse duration) on induced oscillations is studied in the model. Increasing either stimulation pulse duration or amplitude enhanced the level of suppression. The predicted parameters were found to agree well with clinical measurements reported in the literature for individual patients. It is anticipated that the simplified model described may facilitate the development of protocols to aid optimum stimulation parameter choice on a patient by patient basis.

Analysis of Oscillatory Neural Activity in Series Network Models of Parkinson's Disease During Deep Brain Stimulation

“…Dopaminergic medication has been shown to decrease the synchronous beta activity and improve the motor symptoms of the disease [18], [19]. A recent study by Beudel et al showed correlation between deep brain stimulation (DBS)-induced suppression in the power of oscillations in the low gamma frequency range (31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45) and the reduction in the amplitude of the resting tremor recorded in Parkinsonian patients [20]. DBS also acts to suppress the pathological oscillations in the beta frequency band, with a parallel improvement in the motor symptoms of Parkinson's disease observed [15], [21], although a direct causal link, as well as an exact mode of action, remains yet to be established.…”

“…In [40], a second-order macroscopic mathematical model used to represent oscillatory activity of a synchronous group of neurons was presented. The model was tuned to specifically capture key features of the physiological system representing a closed loop within the cortico-basal ganglia-thalamocortical system and to generate the pathological oscillations that occur here in Parkinson's disease.…”

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

“…3, and follow the circle to the right as g is increased. A simple application of the root locus calibration equation (16) at either crossing point of the imaginary axis, s = ±jb, gives the critical value of gl = g 2…”

“…In [16], [17], a computationaL model of Parkinsonian pathoLogical oscillatory activity and its suppression with the application of high frequency stimuLation is presented. The model is a macroscopic neural-mass type model and aims to capture the key features of a synchronized group of neurons in a mathematically tractable manner.…”

“…In this study, the model presented in [16] is used as the basis with which to explore the oscillatory activity in self-oscillating and non-seLf-oscillating coupLed Loops.…”