Optimal phase control of biological oscillators using augmented phase reduction
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.
KeywordsOptimal control Phase reduction Alternans Circadian rhythms Spike timing control
This work was supported by National Science Foundation Grants Nos. NSF-1363243 and NSF-1635542. We thank Dan Wilson for helpful discussions on numerical computation of the augmented phase reduction.
Compliance with ethical standards
Conflict of Interest
The authors declare that they have no conflict of interest.
- Efimov D, Sacré P, Sepulchre R (2009) Controlling the phase of an oscillator: a phase response curve approach. In: Proceedings of the 48h IEEE conference on decision and control (CDC) held jointly with 2009 28th Chinese control conference, pp 7692–7697. https://doi.org/10.1109/CDC.2009.5400901
- Guevara M, Ward G, Shrier A, Glass L (1984) Electrical alternans and period doubling bifurcations. IEEE Comp Cardiol 562:167–170Google Scholar
- Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, CambridgeGoogle Scholar
- Kühn A, Tsui A, Aziz T, Ray N, Brücke C, Kupsch A, Schneider GH, Brown P (2009) Pathological synchronisation in the subthalamic nucleus of patients with Parkinson’s disease relates to both bradykinesia and rigidity. Exp Neurol 215(2):380–387. https://doi.org/10.1016/j.expneurol.2008.11.008 CrossRefPubMedGoogle Scholar
- Malkin I (1949) Methods of Poincare and Liapunov in the theory of nonlinear oscillations. Gostexizdat, MoscowGoogle Scholar
- Monga B, Wilson D, Matchen T, Moehlis J (2018) Phase reduction and phase-based optimal control for biological systems: a tutorial (Under Review)Google Scholar
- Netoff T, Schwemmer M, Lewis T (2012) Experimentally estimating phase response curves of neurons: theoretical and practical issues. In: Schultheiss N, Prinz A, Butera R (eds) Phase response curves in neuroscience. Springer, New York, pp 95–129. https://doi.org/10.1007/978-1-4614-0739-35 CrossRefGoogle Scholar
- Rubin J, Terman D (2004) High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. J Comput Neurosci 16(3):211–235. https://doi.org/10.1023/B:JCNS.0000025686.47117.67 CrossRefPubMedGoogle Scholar
- Shirasaka S, Kurebayashi W, Nakao H (2017) Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems. Chaos 27(023):119Google Scholar
- Wever R (1985) Use of light to treat jet lag: differential effects of normal and bright artificial light on human circadian rhythms. Ann N Y Acad Sci 453(1):282–304. https://doi.org/10.1111/j.1749-6632.1985.tb11818.x CrossRefPubMedGoogle Scholar
- Zhang J, Wen J, Julius A (2012) Optimal circadian rhythm control with light input for rapid entrainment and improved vigilance. In: Proceedings of the 51st IEEE conference on decision and control (CDC). IEEE, pp 3007–3012. https://doi.org/10.1109/CDC.2012.6426226