Optimal phase control of biological oscillators using augmented phase reduction

Abstract

We develop a novel optimal control algorithm to change the phase of an oscillator using a minimum energy input, which also minimizes the oscillator's transversal distance to the uncontrolled periodic orbit. Our algorithm uses a two-dimensional reduction technique based on both isochrons and isostables. We develop a novel method to eliminate cardiac alternans by connecting our control algorithm with the underlying physiological problem. We also describe how the devised algorithm can be used for spike timing control which can potentially help with motor symptoms of essential and parkinsonian tremor, and aid in treating jet lag. To demonstrate the advantages of this algorithm, we compare it with a previously proposed optimal control algorithm based on standard phase reduction for the Hopf bifurcation normal form, and models for cardiac pacemaker cells, thalamic neurons, and circadian gene regulation cycle in the suprachiasmatic nucleus. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier is close to unity; in such cases, the previously proposed control algorithm fails.

Phase distribution control of a population of oscillators

“…To demonstrate our control in the presence of noise, we use (42) to transform an initial uniform phase distribution into a desired bi-modal distribution (33). We take the noise strength √ 2D = 0.03 in equations (38) and (37).…”

“…In [42], we developed a novel strategy to suppress alternans by changing the phase of the pacemaker cells. The control strategy was based on a single oscillator model to change the phase of a single cell.…”

“…Such a control strategy could eliminate the need to excite the tissue at multiple sites. The amount of phase change required to eliminate alternans depends on the discrete APD dynamics [42]. Here we advance the phase by π/4 as an example.…”

“…Such asynchrony can lead to several physiological disorders [52,26], thus motivating researchers to try to develop ways to remove it. In [42], we developed a strategy to eliminate this asynchrony by changing the phase of a single SCN neuron by using a light stimulus as the control input, since light is known to affect the circadian rhythm [10]. This would change the phase of the circadian rhythm so that it gets aligned with the external cycle after the end of the controlled oscillation.…”

“…It reduces the dimensionality of a dynamical system with a periodic orbit to a single phase variable, and captures the oscillator's phase change due to an external perturbation through the phase response curve (PRC). This can make the analysis of high dimensional systems more tractable, and their control [40,62,71,60,38,42] experimentally implementable; see e.g., [57,46,55,71].…”