Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers

Abstract

Phase-amplitude reduction is a widely applied technique in the study of limit cycle oscillators with the ability to represent a complicated and high-dimensional dynamical system in a more analytically tractable coordinate system. Recent work has focused on the use of isostable coordinates, which characterize the transient decay of solutions towards a periodic orbit, and can ultimately be used to increase the accuracy of these reduced models. The breadth of systems to which this phaseamplitude reduction strategy can be applied, however, is still rather limited. In this work, the theory of phase-amplitude reduction using isostable coordinates is further developed to accommodate a broader set of dynamical systems. In the first part, limit cycles of piecewise smooth dynamical systems are considered and strategies are developed to compute the associated reduced equations. In the second part, the notion of isostable coordinates for complex-valued Floquet multipliers is introduced resulting in one phase-like coordinate and one amplitude-like coordinate for each pair of complex conjugate Floquet multipliers. Examples are given with relevance to piecewise smooth representations of excitable cardiomyocytes and the relationship between the reduced coordinate system and the emergence of cardiac alternans is discussed. Also, phase-amplitude reduction is implemented for a chaotic, externally forced pendulum with complex Floquet multipliers and a resulting control strategy for the stabilization of its periodic solution is investigated.

Recent advances in coupled oscillator theory

“…For instance, [21] and [41] use hyperplanes to denote surfaces of constant phase as part of a higher order asymptotic expansion, [23] and [38] use a moving orthonormal coordinate frame in the definition of phase-amplitude coordinates, and [5], [42], and [33] define amplitude coordinates based on Floquet theory. The coordinates based on Floquet theory have been shown to be particularly useful as they result in relatively simple second order accurate phase-amplitude reduced dynamics [40] [39]. This strategy will be used in the following analysis to explain the results from Figure 4.…”

Stabilization of Weakly Unstable Fixed Points as a Common Dynamical Mechanism of High-Frequency Electrical Stimulation

“…Given the large dimension of the models considered in this work, model reduction strategies will be used to asses stability changes resulting from periodic stimulation. Specifically, a phase-amplitude reduced coordinate system will be used 24,25 in order to assess the influence of inputs. To do so, let u(t) = u 0 (t) + Δu(t) where u 0 (t) represents a nominal T-periodic input and Δu(t) is some deviation from this nominal input.…”

“…(16) can be written in a form with an autonomous periodic orbit by redefining the time variable according to (cf., 77 ) Here, θ is the phase of oscillation which gives a sense of the location along a stable periodic solution, ψ j is the j th isostable coordinate which gives a sense of the distance from the periodic orbit in a particular basis, ω = 2π/T is the natural frequency, κ j is the Floquet exponent associated with the j th isostable coordinate, Z(θ) and I(θ) are the phase and isostable response curves, respectively, which give the first order accurate dynamics, and B k (θ) and θ C ( ) , Δu(s) represents the deviation from the nominal input u(s). Information about the derivation of (22) and strategies for the numerical calculation of all necessary terms of the reduction can be found in both 24 and 25 . In the specific situation considered in this work, as a consequence of the periodic orbit in (22) being induced by the periodic forcing, phase and time are related according to θ(s) = ωs.…”

“…framework 24,25 to characterize the shift in Floquet exponents resulting from periodic stimulation. As part of this analysis, a nonlinear reduction framework is necessary because the linear terms of the reduction average to zero.…”

“…In the results to follow, the ability of periodic, charge-balanced electrical stimulation to stabilize a weakly unstable fixed point, thereby engendering changes in the qualitative dynamical behavior is investigated. From a theoretical perspective, using phase-amplitude reduction strategies and the notion of isostable coordinates as a starting point 24,25 , it is found that sinusoidal forcing is energy-optimal for modifying the stability of a fixed point of a general set of ordinary differential equations. Computational simulations and subsequent theoretical analysis provides evidence that desynchronization in a large population of neural oscillators and repolarization block in individual cardiomyocites in response to high-frequency stimulation may be governed by the stabilization of underlying unstable fixed points.…”