Phase resetting for a network of oscillators via phase response curve approach

Abstract

The problem of phase regulation for a population of oscillating systems is considered. The proposed control strategy is based on a phase response curve (PRC) model of an oscillator (the first-order reduced model obtained for linearized system and inputs with infinitesimal amplitude). It is proven that the control provides phase resetting for the original nonlinear system. Next, the problem of phase resetting for a network of oscillators is considered when applying a common control input. Performance of the obtained solutions is demonstrated via computer simulation for three different models of circadian/neural oscillators.

On robustness of phase resetting to cell division under entrainment

“…Details of the standard procedure for a phase model derivation for an oscillator can be found in [5], [6], [16] and they are briefly summarized in the Appendix.…”

“…and γ s (t + θ s,0 ω −1 s ) is a trajectory of the s th cell in (2) for u = 0 and Λ s,k,ns = I ns for all k ≥ 0 with initial conditions in Γ s with the initial phase θ s,0 ∈ [0, 2π], Q s (t) is the infinitesimal PRC derived in the Appendix. This model is constructed around the base trajectory γ s (t + θ s,0 ω −1 s ) under the assumption that the perturbed trajectory with u = 0 and Λ s,k,ns = I ns stays close to that one [16]. Since such a closeness assumption is rather restrictive and may be invalid on a sufficiently long time interval (the excited trajectory can belong to a small vicinity of Γ s for sufficiently small ε and W , but moving away from γ s (t+θ s,0 ω −1 s ) due to a phase shift induced by external inputs), then it is better to recalculate the phase of base trajectory γ s (t + θ s,0 ω −1 s ) after a period T, for example (that is the idea of Poincaré phase map approach [5]).…”

“…Phase synchronization is frequently observed in networks of oscillators, like a colony of the smallest free-living eukaryotes [7], the mammalian circadian pacemaker neural network [8], [9] or networks of neural oscillators [10], [3], [11], to mention a few. Controlled phase resetting has been studied in [12], [13], [14], [15] and for a population of oscillators in [16].…”

“…If the entraining input is a series of pulses, then a Poincaré phase map based on PRC can be calculated to predict the phase behavior [5]. Such a reduced phase model has been used in [14], [16] for pulse amplitude and timing calculation for a controlled phase resetting.…”

Output global oscillatory synchronisation of heterogeneous systems

“…Over the last decade, the synchronization of complex dynamical systems and/or network of systems has attracted a great deal of attention from multidisciplinary research communities due to their pervasive presence in nature, technology and human society [Blekhman (1988); Pikovsky et al (2003); Strogatz (2003); Osipov et al (2007)]. Among potential application domains of synchronization, it is worth to mention the smooth operations of microgrid ; Schiffer et al (2014)], secure communication [Martínez-Guerra et al (2016) ;Fradkov & Markov (1997)], deployment of mobile sensor networks [Wang et al (2012)], formation control [Ren & Beard (2008)], chaos synchronization [Rodriguez et al (2009)], genetic oscillators [Efimov (2015)], etc.…”