Nonlinear Hybrid Control of Phase Models for Coupled Oscillators Ali Nabi and Jeff Moehlis Abstract—We present a new approach to the problem of desynchronization of a population of all-to-all coupled oscil- lators. Motivated by Deep Brain Stimulation treatment of Parkinson’s Disease, the objective is to break this synchrony in the fullest measure by means of a control input only applied to one of the oscillators. Specifically, nonlinear hybrid control is proposed as a novel method for robust global asymptotic stabilization of the splay state. The problem setup is presented in a general way, and a simple example is solved that gives an idea of how this method might be applied in practice. I. INTRODUCTION Populations of periodically firing neurons in the brain are often modeled as networks of coupled oscillators, e.g. [1], [2]. Pathological synchrony of these neurons in the motor control region of the brain sometimes results in Parkinson’s disease, for which, Deep Brain Stimulation (DBS) has been proven to be an effective treatment. In DBS, an electrical stimulus is injected into the brain to desynchronize the firing. Recently, the use of phase models has become more common in studies related to controlling neurons [3], [4], [5], [6]. However, these studies have primarily been on a single- neuron level [4], [6], or else, at the population level, they have allowed multiple control inputs to the system [5]. In this study, nonlinear hybrid control [7] is proposed as a new approach to controlling a population of neurons with only a single control input. However, at this early stage, we have made two simplifying assumptions: observability of phases of all neurons and simple additive control. II. MODEL SETUP We consider a phase model for a network of N coupled oscillators, subject to one control input that, without loss of generality is applied to the N th oscillator in the network: ˙ θ i = ω + N  j=1 α ij f (θ j − θ i )+ δ iN u, θ i ∈ [0, 2π), (1) for i =1, 2,...,N . θ i is the phase of oscillator i, ω is the oscillators’ natural frequency, α ij is the coupling strength from oscillator j to oscillator i, f (·) is the 2π-periodic coupling function, δ is the Kronecker delta function, and u is the control input. By convention, neuron i fires when θ i =0. This work was supported by National Science Foundation grant NSF- 0547606 and by the Institute for Collaborative Biotechnologies under grant DAAD19-03-D004 from the U.S. Army Research Office. A. Nabi is with the Department of Mechanical Engineering, Univer- sity of California at Santa Barbara, Santa Barbara, CA 93106, USA nabi@engineering.ucsb.edu J. Moehlis is with Faculty of Mechanical Engineering, Univer- sity of California at Santa Barbara, Santa Barbara, CA 93106, USA moehlis@engineering.ucsb.edu We note that all oscillators are assumed to have identical ω, and that the functional form of the coupling between any pair of neurons is identical, although the strength of such coupling can differ. We can simplify this system by defining ψ i = θ N − θ i for i =1, 2,..,N − 1, to obtain the N − 1 dimensional system ˙ ψ i =Ω i (ψ)+ u where ψ =(ψ 1 ,ψ 2 ,...,ψ N-1 ) T is the vector of phase differences and Ω i (ψ) is the resulting uncontrolled vector field. Our objective here is to not only break the in-phase synchrony between the oscillators, but to break it to the fullest measure and stabilize the splay state, for which the oscillators’ phases are distributed evenly on the unit phase circle, every two neighboring oscillators being 2π N radians apart. That is, we want to stabilize ψ i =(N − i) 2π N . Performing a coordinate transformation to move the splay state to the origin, we get the following ξ system: ˙ ξ i =Ω i (ξ )+ u, i =1, 2,...,N − 1, (2) where ξ =(ξ 1 ,ξ 2 ,...,ξ N-1 ) T and Ω i (ξ ) is the uncontrolled vector field. Now, if we apply the coordinate transformation x 2i-1 = cos(ξ i ) and x 2i = sin(ξ i ), we get the following 2(N − 1) dimensional system: ˙ x 2i-1 = −(Ω i (ξ )+ u)x 2i ˙ x 2i = (Ω i (ξ )+ u)x 2i-1 , (3) with N −1 constraints: x 2 2i-1 +x 2 2i =1, for i =1, 2, ··· ,N − 1. Stabilizing ξ i = 0 in (2) corresponds to stabilizing (x 2i-1 ,x 2i ) = (1, 0) in (3). This is in fact, a problem of stabilization of N − 1 points on a circle. To investigate the control strategy for this system, we consider the following example. We emphasize that the control strategy taken in this example should not be viewed as a definite approach towards controlling such systems, but it is suggestive that there might be a way to control networks of coupled oscillators with only one input. III. EXAMPLE AND CONTROL STRATEGY We consider N = 3, the coupling function f (x) = sin(3x), and symmetric coupling with α ij = α ji . When one does the aforementioned coordinate transformations, one obtains (3) with i = 1, 2, where here, x 1 = cos(ξ 1 ), x 2 = sin(ξ 1 ), x 3 = cos(ξ 2 ), and x 4 = sin(ξ 2 ). The goal is to stabilize the ξ i =0, or equivalently X = (1, 0, 1, 0) T for this system. We will apply a series of different control laws to accomplish our goal, hence the name hybrid control. The control strategy for this example is as follows. We first restrict our attention to one of the oscillators, say ξ 1 . We apply the same approach as in Example 34 of [7] for robust global stabilization of a point on a circle. We make sure that we apply a control that would steer this oscillator 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeB04.5 978-1-4244-7425-7/10/$26.00 ©2010 AACC 922