Populations of nonlinear oscillators are found in a variety of applications from physics, chemistry, biology, and engineering \mbox{%DIFAUXCMD physics,kuramoto,winfree,Keener2009 The collective behavior of such oscillators varies, and includes synchronization, desynchronization , and clustering. For example, synchronization in beta cells is crucial for efficient insulin secretion \mbox{%DIFAUXCMD beta_synch the beating of the heart is regulated by constant pacing of synchronized cardiac pacemaker cells \mbox{%DIFAUXCMD sn_synch1,sn_synch20pt%DIFAUXCMD , and neural synchrony is essential in visual and odor processing \mbox{%DIFAUXCMD visual,odor}0pt%DIFAUXCMD , } and also in learning and memory recall \mbox{%DIFAUXCMD learn,memory as well. For example, pathological neural synchronization in the thalamus and the subthalamic nucleus (STN) brain region is hypothesized to be one of the causes of motor symptoms for essential and parkinsonian tremor, respectively~\mbox{%DIFAUXCMD thalamus_syncro,STN_syncro to desynchronize an oscillator population. Recently there has also been focus on achieving partial synchrony through clustering instead of complete neural desynchronization \mbox{%DIFAUXCMD tim,Matchen2018,clustered\hspace{0pt}%DIFAUXCMD . One motivation behind such clustering is to rewire neural connections by }  enhancing spike time dependent plasticity which potentiates intra-cluster synaptic connections and depresses inter-cluster connections. This potentially helps in long-term stabilizability of the clusters in the presence of noise .
 The collective behavior of biological oscillators has been recognized as an important problem for several decades, but its control has come into limelight only recently. Much of the focus for control has been on desynchronization of an oscillator population, motivated by the pathological neural synchrony present in essential and parkinsonian tremor. Other applications, such as the beating of the heart 0pt%DIFAUXCMD , and neural synchrony is essential in visual and odor processing \mbox{%DIFAUXCMD visual,odor}0pt%DIFAUXCMD , } and insulin secretion, require synchronization, and recently there has been interest in forming clusters within an oscillator population as well. In this article, we devise a unified control framework to achieve all of these distinct collective behaviors observed in biological oscillators. The control algorithm is based on the partial differential equation governing the evolution of the phase distribution of a population of identical, uncoupled oscillators. Motivated by pathological neural synchrony, we apply our control to desynchronize an \hspace{0pt}%DIFAUXCMD . One motivation behind such clustering is to rewire neural connections by } initially synchronized neural population. Given the proposed importance of enhancing spike time dependent plasticity to stabilize neural clusters and counteract pathological neural synchronization, we formulate the phase difference distribution in terms of the phase distribution, and prove some of its fundamental properties, and in turn apply our control to transform the neural phase distribution to form clusters. Finally, motivated by eliminating cardiac alternans, we apply our control to phase shift a synchronous cardiac pacemaker cell population. To demonstrate the effectiveness of our control for each of these applications, we show that a population of 100 phase oscillators with the applied control mimics the desired phase distribution .
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Populations of nonlinear oscillators are found in a variety of applications from physics, chemistry, biology, and engineering \mbox{%DIFAUXCMD physics,kuramoto,winfree,Keener2009
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such oscillators varies, and includes synchronization, desynchronization , and clustering. For example, synchronization in beta cells is crucial for efficient insulin secretion \mbox{%DIFAUXCMD beta_synch
biological oscillators has been recognized as an important problem for several decades, but its control has come into limelight only recently. Much of the focus for control has been on desynchronization of an oscillator population, motivated by the pathological neural synchrony present in essential and parkinsonian tremor. Other applications, such as
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is regulated by constant pacing of synchronized cardiac pacemaker cells \mbox{%DIFAUXCMD sn_synch1,sn_synch2
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also in learning and memory recall \mbox{%DIFAUXCMD learn,memory
insulin secretion, require synchronization, and recently there has been interest in forming clusters within an oscillator population
R
For example, pathological neural synchronization in the thalamus and the subthalamic nucleus (STN) brain region is hypothesized to be one of the causes of motor symptoms for essential and parkinsonian tremor, respectively~\mbox{%DIFAUXCMD thalamus_syncro,STN_syncro
In this article, we devise a unified control framework to achieve all of these distinct collective behaviors observed in biological oscillators. The control algorithm is based on the partial differential equation governing the evolution of the phase distribution of a population of identical, uncoupled oscillators. Motivated by pathological neural synchrony, we apply our control
D
oscillator population. Recently there has also been focus on achieving partial synchrony through clustering instead of complete neural desynchronization \mbox{%DIFAUXCMD tim,Matchen2018,clustered
A
initially synchronized neural population. Given the proposed importance of
R
which potentiates intra-cluster synaptic connections and depresses inter-cluster connections. This potentially helps in long-term stabilizability of the clusters in the presence of noise
to stabilize neural clusters and counteract pathological neural synchronization, we formulate the phase difference distribution in terms of the phase distribution, and prove some of its fundamental properties, and in turn apply our control to transform the neural phase distribution to form clusters. Finally, motivated by eliminating cardiac alternans, we apply our control to phase shift a synchronous cardiac pacemaker cell population. To demonstrate the effectiveness of our control for each of these applications, we show that a population of 100 phase oscillators with the applied control mimics the desired phase distribution