Abstract
Damping of spikes in an array of coupled oscillators by injection of sinusoidal current is studied both electronically and numerically. The effect is investigated using an array consisting of thirty mean-field coupled FitzHugh–Nagumo-type oscillators. The results are considered as a possible mechanism of the deep brain stimulation used to avoid the symptoms of the Parkinson's disease.
Highlights
Undesirable instabilities in dynamical systems can be avoided by applying conventional proportional feedback techniques [8, 11]
An example is a simple second-order system, where the proportional feedback is given by a linear term with a control coefficient k: x = F (x, y) + k(x∗ − x), y = G(x, y)
Our paper [16] on an adaptive feedback technique for damping neuronal activity deals with a single oscillator only
Summary
Undesirable instabilities in dynamical systems can be avoided by applying conventional proportional feedback techniques [8, 11]. A large number of adaptive control techniques have been developed so far, e.g. the tracking filter method [9, 10, 15], and applied to a variety of dynamical systems (see [1,19] and references therein). A specific example is the stabilization of the unstable upside-down position of a mechanical pendulum by vibrating its pivot up and down at a relatively high frequency [20] This “mechanical” idea has been exploited in a seemingly unexpected field [12], namely, to get insight into the mechanism of the so-called deep brain stimulation (DBS) conventionally used for patients with the Parkinson’s disease, essential tremor [2,3,4], and other brain malfunctions. We consider an array of the mean-field coupled electronic FitzHugh–Nagumo (FHN) oscillators, known in literature as the Bonhoeffer–van der Pol oscillators [14]
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