Chaos 31, 023131 (2021); https://doi.org/10.1063/5.0036508 31, 023131

© 2021 Author(s).

Analysis of input-induced oscillations usingthe isostable coordinate frameworkCite as: Chaos 31, 023131 (2021); https://doi.org/10.1063/5.0036508Submitted: 05 November 2020 . Accepted: 26 January 2021 . Published Online: 18 February 2021

Dan Wilson

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Chaos ARTICLE scitation.org/journal/cha

Analysis of input-induced oscillations using theisostable coordinate framework

Cite as: Chaos 31, 023131 (2021); doi: 10.1063/5.0036508

Submitted: 5 November 2020 · Accepted: 26 January 2021 ·

Published Online: 18 February 2021 View Online Export Citation CrossMark

Dan Wilsona)

AFFILIATIONS

Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA

a)Author to whom correspondence should be addressed: dwilso81@utk.edu

ABSTRACT

Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a sta-ble periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves areinduced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging theisostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcationsthat precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-inducedoscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters nearHopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coor-dinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emergein excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposedtechniques.

Published under license by AIP Publishing. https://doi.org/10.1063/5.0036508

Synchronization, entrainment, and phase-locking of periodic

oscillations are often studied using a weakly perturbed paradigm,

i.e., by first characterizing how small perturbations influence the

oscillation timing and then using this information to predict the

steady state behavior once perturbations are applied. This general

approach, however, cannot be applied for input-induced oscil-

lations, that is, for dynamical systems that are not intrinsically

oscillatory but nonetheless support stable oscillations when cou-

pling or other exogenous inputs are applied. In this work, two

separate reduced order modeling frameworks are proposed that

capture the important dynamical behaviors that can give rise to

these input-induced oscillations. The proposed reduction frame-

works yield reduced order coordinate systems that are compatible

with other analysis techniques that have been previously devel-

oped for intrinsically oscillatory systems. Indeed, in the examples

presented here, analysis of the proposed reduced order models

predicts the emergence of input-induced oscillations as well as the

associated steady state phase-locking relationships.

I. INTRODUCTION

Nonlinear oscillations are widely studied in the physical,chemical, and biological sciences.11,13,20,22,29,34,35,38,40,43,61 In many prac-tical applications involving limit cycle oscillations, because of thecomplexity and high-dimensionality of the associated model equa-tions, model reduction is often an imperative first step in the analy-sis. Phase reduction13,22,34,61 is one commonly used strategy that canbe used to represent the perturbed behavior of an oscillatory systemof the form

x = F(x)+ U(t), (1)

where x ∈ Rn is the system state, F gives the unperturbed dynam-

ics, and U represents an exogenous perturbation, in terms of anasymptotic phase

θ = ω + Z(θ) · U(t), (2)

where the phase θ ∈ [0, 2π) gives a sense of the oscillation tim-ing, ω = 2π/T where T is the unperturbed period of oscillation,

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Z(θ) is the phase response curve that represents the gradient of thephase evaluated on the periodic orbit, and the “dot” denotes the dotproduct. The phase reduced framework (2) has been used as a start-ing point in many applications to both understand and control thebehaviors of weakly coupled,1,4,11,43 and weakly perturbed,31,58,62 limitcycle oscillators.

Oscillator models such as (1) are well-studied in regimes wherethe underlying system has a periodic orbit so that either the trans-formation to (2) (or to one of the many other phase-based reductionframeworks3,8–10,24,47,55,56) can be used. However, comparatively fewframeworks have been developed that are valid in situations wherethe oscillations themselves are induced by either coupling or exter-nal inputs [i.e., the terms that comprise U(t)]. As a concrete example,consider a model for a coupled population of N dynamical elements,

xi = σxi(µi − r2i )− yi(1 + ρ(r2

i − µi))+K

N

∑

j6=i

xj,

yi = σyi(µi − r2i )+ xi(1 + ρ(r2

i − µi)),

(3)

where i = 1, . . . , N, x and y are Cartesian coordinates, r2i = (x2

i

+ y2i ), N = 10 is the number of elements considered, σ = ρ

= 0.05 are model parameters, µi = −0.4 + (i − 1)/30 is a bifur-cation parameter, and K is the coupling strength. These modelequations are a modified version of the radial isochron clock equa-tions from Ref. 61 where a stable limit cycle emerges through a Hopfbifurcation forµi > 0; note here thatµi < 0 for all i. As illustrated inFig. 1, qualitatively different oscillatory patterns emerge for couplingstrengths of various magnitudes. Panels (a)–(e) show the transientbehavior in the first few moments of simulation, and panels (f)–(j)show the limiting behavior of each simulation. If it were the casethat µi > 0 for all i so that each model had a stable limit cycle in theabsence of coupling, the coupled behavior of (3) could be analyzedaccording to the phase reduction

θi = ωi +K

NZi(θi)

∑

j6=i

xj. (4)

For the current example, however, all µi < 0 for all i so that in theabsence of coupling each dynamical element has a globally attractingfixed point at xi = yi = 0. Consequently, it is not possible to write(3) in the form (4) and some other strategy must be used to analyzethe phase-locking behaviors.

The focus of this work is on dynamical systems of the form(1) with limiting behavior that approaches a stable fixed point whenU(t) = 0, but also supports periodic oscillations that can be inducedwhen U(t) 6= 0. Such oscillations will be referred to input-inducedoscillations and cannot be analyzed using the weakly perturbedparadigm under normal circumstances because no limit cycle existsin the absence of input. One often used strategy to handle such sys-tems is to consider the entire coupled system of oscillators as a singlelimit cycle and subsequently analyze the collective oscillation usingphase-based analysis techniques.18,19,21,25,57 These collective oscilla-tions are notoriously difficult to analyze in a reduced order setting,however, as the Floquet exponents associated with the truncatedtransient dynamics are often near-zero.57 Consequently, the magni-tude of inputs that can be considered is often prohibitively small.Other techniques have been developed to consider input-induced

FIG. 1. Panels (a)–(e) illustrate the transient behavior of (3) for coupling strengthsof variousmagnitudes. For each oscillator,µi < 0 so that xi = yi = 0 is a globallyattracting fixed point in the absence of coupling. Panels (f)–(j) show the corre-sponding steady state behavior; each ring shows the coupling-induced periodicorbit of an individual cell with black dots showing a representative snapshot ofthe system state after transient behaviors have decayed. When K = 0 [panels(c) and (h)], each oscillator approaches its globally stable fixed point in the limitas time approaches infinity. For positive coupling values with large enough magni-tude [panels (d), (e), (i), and (j)], synchronized oscillations of varying amplitudesemerge. For moderate amplitude negative coupling strengths [panels (b) and (g)],stable oscillations emerge for some oscillators while others stay near their stablefixed points. For larger amplitude negative coupling strengths [panels (a) and (f)],each dynamical element begins to oscillate at an amplitude governed by µi , withan even spread of the oscillation phases.

oscillations in applications where U(t) is periodic; by augment-ing the state with a time-like variable the resulting input-inducedoscillation can be considered autonomous and phase-amplitudereduction frameworks can be used to analyze the resulting behavior.This technique has been used to study phase-locking in oscillatorsthat are entrained to a specific periodic input10,52,60 but cannot easilyaccommodate changes to the entraining stimulus. Other more gen-eral techniques such as those involving master stability functions32,33

or contraction theory26 could be useful in some applications involv-ing input-induced oscillations, but can be difficult to implement forhigh-dimensional systems.

In this work, the notion of isostable coordinates will be lever-aged in order to capture the salient features of input-induced oscil-lations using a reduced order framework. The isostable coordinateframework, as originally explored in Ref. 28, can be used to representthe dynamical behavior of a general nonlinear dynamical systemon the basis of its slowest decaying Koopman eigenfunctions.5,6 For

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limit cycle oscillators, isostable coordinates can be used in conjunc-tion with the notion of asymptotic phase in order to yield phase-amplitude reductions that are valid far beyond the weakly perturbedregime.50,53 Likewise, these isostable coordinates can be used toaccurately characterize nonlinear behaviors that emerge in systemswith stable fixed point.48 This work will build upon the isostablecoordinate reduced framework to provide reduction strategies thatcan be used to analyze the dynamical behaviors of input-inducedoscillations. The key feature of the isostable coordinate reductionframework that makes it useful for analyzing input-induced oscil-lations is that it allows for high-accuracy, nonlinear, reduced orderrepresentations of the dynamics near a fixed point attractor therebyproviding a framework that is valid beyond the weakly perturbedregime.

The organization of this paper is as follows: Sec. II pro-vides necessary background information on the isostable coordinateframework and provides a brief overview of previously developedmodel order reduction strategies that use isostable coordinates.Section III considers a reduced order framework that can be usedto analyze input-induced oscillations for dynamical systems withstable fixed points that are close to a Hopf bifurcation. In this sce-nario, the necessary nonlinear behaviors can be captured with anasymptotic expansion of the dynamics on an isostable coordinatebasis. Section IV considers the recently developed notion of adaptivephase-amplitude reduction51 to yield a reduced order frameworkthat is well-suited for the analysis of input-induced oscillations thatemerge in excitable systems. Multiple examples are presented inSec. V where these strategies are applied to (3) in addition to otherexamples with relevance to both circadian oscillations and neuralphysiology. Concluding remarks are provided in Sec. VI.

II. BACKGROUND: ISOSTABLE COORDINATES AND

ASSOCIATED REDUCED ORDER MODELING

FRAMEWORKS

The isostable coordinate framework will be used to analyze theinput-induced oscillations considered in this work. As a brief back-ground on isostable coordinates, consider the ordinary differentialequation

x = F(x)+ U(t), (5)

where x ∈ Rn denotes the system state, F gives the nominal unper-

turbed dynamics, and U(t) is an external input (e.g., that captures theinfluence of coupling). Supposing that (5) has a stable fixed point atx0, through local linearization one finds

d1x

dt= J1x + O(|1x|2), (6)

where 1x = x − x0 and J is the Jacobian evaluated at x0. The locallinearization (6) alone is only valid in the vicinity of the fixedpoint making it unsuitable for analysis of the coupling-inducedoscillations discussed in Sec. I. The isostable reduced coordinateframework, a strategy that considers the basis of the level sets ofthe slowest decaying Koopman eigenfunctions,5,6 can be leveraged toidentify reduced order models that are substantially more accuratethan (6) for states that are far from the nominal fixed point. To do

so, let wk, vk, and λk correspond to left eigenvectors, right eigenvec-tors, and eigenvalues of J, respectively, ordered so that |Real(λj)| ≤

|Real(λj+1)|. For λ1 (i.e., corresponding to solutions with the slow-est decay), an associated isostable coordinate ψ1(x) can be definedexplicitly for all initial conditions in the basin of attraction of thefixed point by considering the infinite time decay of solutions,59

ψ1(x) = limt→∞

(wT1 (φ(t, x)− x0) exp(−λ1t)), (7)

where φ(t, x) represents the unperturbed flow of (5) and T indicatesthe transpose. Note that (7) is similar to the definition of isostablecoordinates proposed in Ref. 28 that considers Fourier averages ofsystem observables. Equation (7) provides an explicit definition forthe slowest decaying isostable coordinate; faster decaying isostablecoordinates ψ2, . . . ,ψn can also be defined implicitly as level setsof Koopman eigenfunctions with decay rates determined by theirassociated λj.23 Other definitions for ψ1, . . . ,ψn are possible, but foreach λj that is not repeated, an associated isostable coordinateψj canbe defined that is unique up to a constant scaling.

A. Model reduction on the basis of isostable

coordinates

When considering (5) in situations where U(t) = 0, all isostablecoordinates decay exponentially according to ψk = λkψk in theentire basin of attraction of the fixed point. In many applications,the dynamics associated with rapidly decaying coordinates play anegligible role in the underlying system behavior and can be trun-cated by taking them to be zero at all times.7,27,29,41,42,50 Subsequently,the dynamics of (5) on a hypersurface governed by M non-truncatedisostable coordinates can be represented as

ψk = λkψk + Ik(ψ1, . . . ,ψM) · U(t),

k = 1, . . . , M,

x = x0 + G(ψ1, . . . ,ψM),

(8)

where Ik is the gradient of the ψk, G maps the isostable coordinatesto the state, and the dot denotes the dot product. Taylor expansionof Ik and G in powers of the isostable coordinates yields

G(ψ1, . . . ,ψM) ≈

M∑

k=1

[

ψkgk]

+

M∑

j=1

j∑

k=1

[

ψjψkgjk]

+

M∑

i=1

i∑

j=1

j∑

k=1

[

ψiψjψkgijk

]

+ . . . , (9)

In(ψ1, . . . ,ψM) ≈ I0n +

M∑

k=1

[

ψkIkn

]

+

M∑

j=1

j∑

k=1

[

ψjψkIjkn

]

+

M∑

i=1

i∑

j=1

j∑

k=1

[

ψiψjψkIijkn

]

+ . . . . (10)

Here, the terms gk and I0k are the right and left eigenvectors asso-

ciated with the eigenvalue λk of J and appropriately scaled so thatI0k · gk = 1. The remaining terms can be computed using a strategy

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described in Appendix A of Ref. 48 (which itself is based on the moregeneral formulation from Ref. 53 valid for computing high-accuracyrepresentations of the isostable coordinate dynamics for limit cycleattractors).

B. Isostable-based adaptive reduction of models with

periodic orbit attractors

As mentioned previously, standard phase reduction techniquescannot generally be applied to input-induced oscillators becausethere is no stable periodic orbit in the absence of input. Instead, aswill be discussed in Sec. IV, the recently proposed technique of adap-tive phase-amplitude reduction51 is useful to analyze input-inducedoscillations of excitable systems. The adaptive phase-amplitudetechnique is discussed in detail in Ref. 51 and is briefly summarizedhere. To proceed, consider a general dynamical equation

x = F(x, p0)+ U(t), (11)

where p0 ∈ Rm represents a nominal parameter set and x, F, and U

are defined identically to (5). Let Pa be some allowable range of sys-tem parameters. For any p ∈ Pa, it will be assumed that x = F(x, p)has a stable limit cycle x

γp that is continuously differentiable with

respect to both t and p. Note that it is not necessary that p0 ∈ Pa.One can rewrite Eq. (11) as

x = F(x, p)+ Ue(t, p, x), (12)

where F(x, p) captures the nominal dynamics for a given choice of pand

Ue(t, p, x) = U(t)+ F(x, p0)− F(x, p) (13)

is the effective input. Following the derivation from Ref. 51, allowingp to change in time, the phase and isostable dynamics can be found

via the chain rule as dθdt

= ∂θ

∂x· dx

dt+ ∂θ

∂p·

dp

dtand

dψj

dt=

∂ψj

∂x· dx

dt+

∂ψj

∂p·

dp

dt. Upon simplification, the equations of an adaptive reduction

can be written as

θ = ω(p)+ Z(θ , p) · Ue(t, p, x)+ D(θ , p) · p,

ψj = κj(p)ψj + Ij(θ , p) · Ue(t, p, x)+ Qj(θ , p) · p,

j = 1, . . . , M,

p = Gp(p, θ ,ψ1, . . . ,ψM),

(14)

whereψ1, . . . ,ψM are the non-truncated isostable coordinates, ω(p)and κj(p) give the natural frequency and Floquet exponents asa function of p, Z(θ , p) and Ij(θ , p) are the phase and isostableresponse curves that represent the gradients of θ and ψj, respec-tively, evaluated on x

γp , and D(θ , p) and Qj(θ , p) characterize the

influence of changing p on the phase and isostable coordinates.Additionally, model outputs can be approximated as a function ofthe reduced coordinates as

x(t) = xγ

p(t)(θ(t))+

M∑

i=1

gk(θ(t), p(t))ψk(t), (15)

where gk(θ , p) are Floquet eigenfunctions of the periodic orbit xγp (θ).

As part of (14), the function Gp must be designed to actively updatep so that the isostable coordinates ψj remain small. Provided Gp

can be designed so that each ψj(t) = O(ε) uniformly in time with0 < ε � 1 and that the neglected isostable coordinates have suffi-ciently large magnitude Floquet exponents, Eq. (14) is accurate toleading order ε even if U(t) is large. Computation strategies for allnecessary terms of (14) when the underlying model equations areknown are discussed in Ref. 51. Terms of this reduction can alsobe computed using purely data-driven techniques as discussed inRef. 49.

III. ANALYSIS OF INPUT-INDUCED OSCILLATIONS

USING ASYMPTOTIC EXPANSIONS OF THE ISOSTABLE

DYNAMICS

For a general model (5), let x0 be a stable fixed point. Largeamplitude input-induced oscillations often emerge when the systemparameters are near a Hopf bifurcation, i.e., with two complex-conjugate eigenvalues of the Jacobian that have real componentsthat are near-zero. In the following analysis, it will be assumedthat Re(λ1), Re(λ2), and the input U(t) are O(ε) terms where0 < ε � 1. Here, input-induced oscillations will be analyzed byconsidering the isostable coordinates associated with these slowlydecaying complex-conjugate isostable coordinates. Previous work48

describes a strategy that can be used to capture the dynamics in thisisostable basis that yields a reduced order set of equations of the form(8). This will be used as a starting point in the derivation below. Themodel will be assumed to have two non-truncated isostable coor-dinates, but it would be straightforward to incorporate additionalisostable coordinates in the analysis to follow. With two isostablecoordinates, the expansions from (9) and (10) can be written as

G(ψ1,ψ2) ≈

A∑

k=1

k∑

j=0

ψk−j1 ψ

j2g(k−j)(j), (16)

In(ψ1,ψ2) ≈ I0n +

A∑

k=1

k∑

j=0

ψk−j1 ψ

j2I(k−j)(j)n , (17)

for n = 1, 2 where, for instance, the notation g(3)(2) is used to denoteg11122 and A is the order in the isostable coordinates to which theexpansion is taken. The resulting model (8) with associated func-tions (16) and (17) represents a straightforward implementation ofthe results from Ref. 48; however, this alone does not provide anydirect insight about the model dynamics and is difficult to workwith on its own. The subsequent manipulations presented belowultimately yield a set of equations that are compatible with othertechniques developed for the analysis of limit cycle oscillators. Asexplained in Ref. 50, ψ1 and ψ2 must be complex conjugate forreal-valued initial conditions of (5). As such, the transformation

ψC = |ψ1|,

θC = arg(ψ1),(18)

can be applied to focus on the dynamics of ψ1. Here, ψC is the mag-nitude of ψ1 and θC is its argument. The subscript here highlightsthe fact that this coordinate transformation (18) is applicable whenψ1 is part of a complex-conjugate pair. Using this transformation,note that Re(ψ1) = ψC cos(θC) and Im(ψ1) = ψC sin(θC). For the

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moment, considering a situation where U(t) = 0, one can take thetime derivatives of (18) using the chain rule [for instance, startingwith ψC =

(

Re(ψ1)Re(ψ1)+ Im(ψ1)Im(ψ1))

/|ψ1| and simplify by

noting that ψ1 = λ1ψ1] to yield

ψC = ζψC,

θC = �,(19)

where ζ = Re(λ1) and� = Im(λ1). The influence of nonzero inputscan be incorporated by considering the gradient of the relationshipsfrom (18) to yield

∂ψC

∂x=

1

|ψ1|

[

Re(ψ1)Re

(

∂ψ1

∂x

)

+ Im(ψ1)Im

(

∂ψ1

∂x

)]

= [cos(θC)Re(IC(ψC, θC))+ sin(θC)Im(IC(ψC, θC))] ,

∂θC

∂x=

cos2(θC)

Re(ψ1)2

[

Re(ψ1)Im

(

∂ψ1

∂x

)

− Im(ψ1)Re

(

∂ψ1

∂x

)]

=1

ψC

[cos(θC)Im(IC(ψC, θC))− sin(θC)Re(IC(ψC, θC))] ,

(20)

where IC(ψC, θC) = I1(ψC exp(iθC),ψC exp(−iθC)) gives the gradi-ent of ψ1 as defined in (17), but takes θC and ψC as its argu-ments. Likewise, GC(ψC, θC) = G(ψC exp(iθC),ψC exp(−iθC)) canbe defined to denote the output as a function of the argumentsθC and ψC. With this information, in the transformed coordinatesystem specified by (18), the dynamics becomes

θC =∂θC

∂x·

dx

dt

=∂θC

∂x· (F(x)+ U(t))

= �+1

ψC

[cos(θC)Im(IC(ψC, θC))

− sin(θC)Re(IC(ψC, θC))] · U(t),

ψC =∂ψC

∂x·

dx

dt

=∂ψC

∂x·

dx

dt

= ζψC + [cos(θC)Re(IC(ψC, θC))

+ sin(θC)Im(IC(ψC, θC))] · U(t).

(21)

Recall that both ζ and U(t) are O(ε) terms so thatψC can be assumedto be an O(1) term. In situations where ψC is an O(ε) term, simplelinearization near the fixed point of (5) is generally sufficient to cap-ture the dynamics; simple linearization cannot, however, be used toanalyze the behavior of input-induced oscillations which drive thestate far from its fixed point.

In the analysis to follow, it will be implicitly assumed thatthe approximations G(ψ1,ψ2) and In(φ1,φ2) [and by extensionGC(ψC, θC) and IC(ψC, θC)] are sufficiently accurate to capture thesalient dynamics. This can, in general, be accomplished by settingan a priori threshold on the allowable values of ψ1 and ψ2 and tak-ing the expansions from Eqs. (16) and (17) to high enough orders

FIG. 2. Isostable coordinates for a single dynamical element from (3) takingµ = −0.3 and σ = ρ = 0.2. The eigenvalues associated with the linearization,λ1,2 = −0.06 ± 1.06i, characterize the decay rates of the isostable coordinates.Isostable coordinates are computed directly using the definition (7) and the cor-responding values of arg(ψ1) are shown in panel (a). The approximation ofG(ψ1,ψ2) is computed to various orders of accuracy in its isostable coordinateexpansion from (16) and used to approximate the level sets of arg(ψ1) as shownin panels (b)–(d). Black lines represent appropriate level sets from panel (a). Ingeneral, as the deviations from the fixed point become larger, the expansions ofG(ψ1,ψ2) and each Ij(ψ1,ψ2) must be taken to higher orders of accuracy toobtain an accurate representation.

of accuracy. For example, the relationship between the order ofthe expansion and its accuracy is illustrated in Fig. 2 for a singledynamical element from (3).

A. Resonance with a periodic input

Let U(t) be a Tp-periodic input. It will be assumed that�− ωp = O(ε) where ωp = 2π/Tp. In this situation, the input isnear the resonant frequency of the complex-conjugate eigenvalues.In a rotating reference frame φC = θC − ωpt, Eq. (21) becomes

φC = �− ωp +1

ψC

[

cos(φC + ωpt)Im(IC(ψC,φC + ωpt))

− sin(φC + ωpt)Re(IC(ψC,φC + ωpt))]

· U(t),

ψC = ζψC +[

cos(φC + ωpt)Re(IC(ψC,φC + ωpt))

+ sin(φC + ωpt)Im(IC(ψC,φC + ωpt))]

· U(t).

(22)

Noticing that (22) is Tp-periodic and is of the general form x =

εF(x, t), formal averaging techniques16,39 can be applied to approx-imate (22) according to

8C = �− ωp + fφ(8C,9C),

9C = ζ9C + fψ (8C,9C),(23)

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where fφ(x, y) = 1Tp

∫ Tp

01y

[

cos(x + ωpt)Im(IC(y, x + ωpt))− sin(x

+ωpt)Re(IC(y, x + ωpt)]

· U(t) dt and fψ (x, y)= 1Tp

∫ Tp

0

[

cos(x + ωpt)

Re(IC(y, x + ωpt))+ sin(x + ωpt)Im(IC(y, x + ωpt))]

· U(t) dt. Asdiscussed in Refs. 16 and 39, fixed points of (23) correspond to peri-odic orbits of the unaveraged equations (22) with the same stabilityproperties.

B. Phase-locking of coupling-induced oscillations

Consider a population of N coupled dynamical systems, eachwith stable fixed points that are close to a Hopf bifurcation [suchas the system from Eq. (3)]. Such a system can be written in theisostable reduced form (21)

θC,i = �i +1

ψC,i

[

cos(θC,i)Im(IC,i(ψC,i, θC,i))

− sin(θC,i)Re(IC,i(ψC,i, θC,i))]

· Ki(θC,1,ψC,1, . . . , θC,N,ψC,N),(24)

ψC,i = ζiψC,i +[

cos(θC,i)Re(IC,i(ψC,i, θC,i))

+ sin(θC,i)Im(IC,i(ψC,i, θC,i))]

· Ki(θC,1,ψC,1, . . . , θC,N,ψC,N).

for i = 1 . . .N where �i, ζi, θC,i, ψC,i, Gi, and IC,i are terms of thereduction that correspond to element i. Here, the coupling relation-ship is considered an input U(t) = Ki(θC,1,ψC,1, . . . , θC,N,ψC,N) andcan most generally be assumed to be a nonlinear function of thereduced coordinates.

Assume that maxi,j(�j −�i) = O(ε) and let �

= (1/N)∑N

i=1�i. Also define T = 2π/�. As in Sec. III A, towardusing an averaging method of analysis, a rotating reference frameφC,i = θC,i −�t will be used for each oscillator from Eq. (24) to yield

φC,i = �i −�+1

ψC,i

[

cos(φC,i +�t)Im(IC,i(ψC,i,φC,i +�t))

− sin(φC,i +�t)Re(IC,i(ψC,i,φC,i +�t))]

· Ki(φC,1 +�t,ψC,1, . . . ,φC,N +�t,ψC,N),

ψC,i = ζiψC,i +[

cos(φC,1 +�t)Re(IC,i(ψC,i,φC,i +�t))

+ sin(φC,1 +�t)Im(IC,i(ψC,i,φC,i +�t))]

· Ki(φC,1 +�t,ψC,1, . . . ,φC,N +�t,ψC,N),

(25)

for i = 1, . . . , N. Phase-locking can be explored by defining υC,i

≡ φC,i − φC,N and using this to rewrite (25) as

υC,i = �i −�N + hυ,i(φC,1 +�t,ψC,1, . . . ,φC,N +�t,ψC,N),

i = 1, . . . , N − 1,

ψC,j = ζjψC,j + hψ ,j(φC,1 +�t,ψC,1, . . . ,φC,N +�t,ψC,N),

j = 1, . . . , N,

(26)

where each hυ,i and hψ ,j are defined so that (25) is satisfied noting

that υC,i = φC,i − φC,N. As in Sec. III A, Eq. (26) can be written in theform x = εF(x, t) and is periodic in t so that averaging techniques16,39

can be applied to approximate (26) as

ϒC,i = �i −�N + fυ,i(ϒC,1,9C,1, . . . ,ϒC,N−1,9C,N−1,9C,N),

i = 1, . . . , N − 1,

9C,j = ζj9C,j + fψ ,j(ϒC,1,9C,1, . . . ,ϒC,N−1,9C,N−1,9C,N),

j = 1, . . . , N,

(27)

where

fυ,i(x1, y1, . . . , xN−1, yN−1, yN)

=1

T

∫ T

0

hυ,i(x1 +�t, y1, . . . , xN−1,i +�t, yN−1,�t, yN) dt,

fψ ,j(x1, y1, . . . , xN−1, yN−1, yN)

=1

T

∫ T

0

hψ ,j(x1 +�t, y1, . . . , xN−1,i +�t, yN−1,�t, yN) dt.

(28)

Here, each ϒC,i and 9C,i provide close approximations of the cor-responding υC,i and ψC,i, respectively. Additionally, as detailed inRefs. 16 and 39, fixed points of (27) correspond to periodic orbitsof (26) with the same stability allowing (27) to be used to ana-lyze input-induced phase-locking in the reduced model (24) and itscorresponding full model equations.

C. Phase-locking to limit cycle oscillations and other

phase-locking situations

The general structure of (21) makes it compatible with phasereduction techniques for analyzing limit cycle oscillators. For exam-ple, consider two coupled dynamical models, the first of which has astable limit cycle that can be represented in terms of a standard phasereduction (2). Suppose the second dynamical model has a stablefixed point near a Hopf bifurcation so that its reduced order dynam-ics can be represented by (21). In general, the reduced dynamicalequations for this set of coupled equations can be represented by

θ = ω + Z(θ) · K1(θ , θc,ψC),

θC = �+1

ψC

[cos(θC)Im(IC(ψC, θC))− sin(θC)Re(IC(ψC, θC))]

· K2(θ , θc,ψC), (29)

ψC = ζψC + [cos(θC)Re(IC(ψC, θC))+ sin(θC)Im(IC(ψC, θC))]

· K2(θ , θc,ψC),

where K1 and K2 characterize the coupling between the dynami-cal models. Considering the phase difference υ ≡ θ − θC betweenthe two models, provided ω −� = O(ε), one can proceed to ana-lyze the resulting stable and unstable phase-locking relationshipsusing dynamical averaging techniques similar to those employedin Secs. III A and III B. Many other caveats can be consideredstraightforwardly (e.g., phase-locking in the presence of an addi-tional external stimulus, larger systems of coupled oscillators withsome having stable limit cycles and others having stable fixed pointsnear Hopf bifurcations) by making appropriate modifications.

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IV. ANALYSIS FOR EXCITABLE SYSTEMS USING

ADAPTIVE PHASE-AMPLITUDE REDUCTIONS

Section III provided a framework for exploring the emergenceof input-induced oscillations using an asymptotic expansion of thedynamics of fixed point attractors in the basis of isostable coordi-nates. This general strategy is most useful when the salient dynamicscan be captured by expansions of relatively low order (e.g., tenthorder or less). While the actual cutoff will vary on a model-by-modelbasis, the computational effort required to determine the terms of(16) and (17) grows exponentially with the order of the terms con-sidered. Furthermore, when higher order expansions are required,small errors in the computations of the highest order terms cansignificantly degrade the accuracy of the resulting reductions. Asdiscussed in this section, when asymptotic expansions in the basisof isostable coordinates are not sufficient to capture the dynamicsthat give rise to input-induced oscillations, an alternative strategythat leverages the adaptive reduction framework (14) can be moreeffective.

For concreteness in the illustration of this general approach,consider a FitzHugh–Nagumo-based model for an excitable systemoriginally considered in Ref. 37 (cf. Ref. 14)

V = c1V(V − a)(1 − V)− c2Vw + u(t),

w = b(V − dw).(30)

Here, variables V and w can be used to represent a transmembranecell voltage and an associated gating variable, respectively, u(t) isa time dependent input, and parameters are taken to be a = 0.13,b = 0.013, c1 = 0.26, c2 = 0.1, and d = 1. When u = 0, the model(30) has a stable fixed point at V = w = 0 with associated eigen-values λ1 = −0.013 and λ2 = −0.034. For this model, the isostablecoordinate ψ1 is computed according to (7) and shown in panel (a)of Fig. 3, and panel (b) shows the plots of V(t), for example, tra-jectories. Note that because (30) is excitable, small perturbationscan result in large excursions from the fixed point. Consequently,a separatrix exists over which the gradient of the isostable coordi-nates is relatively large. Such gradients are difficult to capture usingasymptotic expansions of the form (9) and (10).

Phase-amplitude reduction techniques cannot generally beapplied to excitable systems, such as the one from (30) because nounderlying limit cycle exists. However, as part of the reduction strat-egy proposed here, a shadow system with similar dynamical equa-tions that does exhibit stable oscillatory behavior will be designed.As shown in the results to follow, by leveraging the phase-amplitudereduction framework from (14), it is possible to understand and ana-lyze the behavior of the nominal (excitable) system (30) in terms ofa reduced order version of the shadow system. As a concrete exam-ple, consider a shadow system that is identical to (30) except for theaddition of a virtual activating function

V = c1V(V − a)(1 − V)− c2Vw + u(t)+ uvirt(w, w0),

w = b(V − dw),(31)

where uvirt(w, w0) = 1 − 11−exp(−80(w−w0))

. The additional term

uvirt(w, w0) is a sigmoid that provides an activating stimulus whenw drops below a threshold determined by w0. With this additionalterm, when u(t) = 0, asymptotically stable limit cycles emerge as

FIG. 3. Panel (a) shows the principal isostable coordinate ψ1 of the excitablesystem (30) along with the stable fixed point (white dot) and two representativetrajectories (white lines). Time courses of these two trajectories are shown in panel(b). Notice the sharp gradient in the isostable coordinate in panel (a) across theseparatrix—such gradients (a general characteristic of excitable systems) makeit difficult to capture the dynamics of the isostable coordinates using asymptoticexpansions of the form (9) and (10). After adding an activating stimulus as in(31), periodic orbits emerge as shown in panel (c) that are parameterized by w0.Panel (d) gives time courses of these periodic orbits with each color representinga specific value of w0. As discussed here, the adaptive phase-amplitude reductionframework can be used to understand the excitable system dynamics in referenceto the collection of periodic orbits shown in panel (c).

shown in panels (c) and (d) of Fig. 3 with periods that are governedby the term w0 in the activating function. In Sec. IV A, a strategythat leverages such a family of periodic orbits will be discussed toprovide a general reduction technique to characterize the behaviorof excitable systems in situations where methods discussed in Sec. IIIare not feasible.

A. A general strategy for obtaining reduced order

models of excitable systems using adaptive

phase-amplitude reduction

Consider a general excitable system of the form (5) with a stablefixed point x0. For an excitable system, relatively small perturbationsfrom the fixed points can cause large transient excursions. Here, ashadow system will be considered that is identical to (5) except forthe addition of a virtual activating function, Uvirt, that is designed toelicit an excitation when the state approaches x0, resulting in a largetransient excursion from the fixed point. With the incorporation ofthis activating function, the dynamics become

x = F(x)+ Uvirt(x, p)+ U(t), (32)

where p is the adaptive parameter. Supposing that for a continuousfamily of p ∈ Pa, (32) has a stable limit cycle x

γp , then using the adap-

tive phase-amplitude reduction framework described in Sec. II B bytaking the effective input to be Ue(t, p, x) = U(t)− Uvirt(x, p), onecan obtain a reduced order model of the form (14) provided that

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FIG. 4. For the excitable model (30), when a virtual input is added, the resulting shadow system of Eq. (31) has a continuous family of periodic orbits parameterized by w0,which is taken to be the adaptive parameter in the transformation (14). All functions of the resulting adaptive reduction are computed numerically, with panels (a), (b), and(c) showing the periodic orbits, IV(θ) (the isostable response curve for perturbations to V), and Q(θ), respectively, for various values of w0. The input u(t) from panel (h) isapplied to both the full model (30) and the transformed model that results from applying the adaptive reduction framework. The output, V(t), of the reduced order model ofthe form (14) is nearly identical to the output from the true model simulations (30) with results shown in panel (e). Panels (f) and (g) show the time course of the associatedisostable coordinate and the adaptive parameter w0, respectively, during the application of this u(t). Intuitively, the adaptive parameter w0 is continuously updated so thatthe instantaneous state is always close to the periodic orbit of the shadow system associated with the instantaneous value of w0. Additionally, panel (d) shows a trace of theresulting output (black line) when the input from panel (h) is applied to the transformed equations; this trace is superimposed on a collection of the nominal periodic orbits forvarious choices of w0.

each xγp is continuously differentiable with respect to both t and p

and that some function Gp(p, θ ,ψ) can be identified that keeps theresulting isostable coordinates small. More details including strate-gies for the computation of the necessary terms from the adaptivereduction are presented in Ref. 51.

Figure 4 shows results of this strategy applied to the excitablesystem (30) with the activating function specified by (31). Theamplitude coordinates of each periodic orbit can be representedwith a single isostable coordinate ψ . These periodic orbits, alongwith the corresponding functions IV(θ) (i.e., ∂ψ/∂V), and Q(θ)are computed numerically and shown in panels (a), (b), and (c),respectively, of Fig. 4. The function Gp = −ψQ(θ , p) as proposedin Ref. 51 is used and serves to decrease the magnitude of theisostable coordinates. Intuitively, noting that Q is strictly negative,to counteract an increase (resp., decrease) in the isostable coordi-nate due to an applied input, Gp will increase (resp. decrease) thevalue of w0.

A series of pulse inputs with a pacing period decreasing from360 to 250 time units is shown in panel (h) of Fig. 4 and applied toboth the full and transformed models that use the adaptive phase-amplitude framework. This input is obtained taking u(t) = 0.016when mod(t, 360 − 0.04t) < 25, and u(t) = 0, otherwise. Results forpulses applied at similar intervals are comparable. The output ofthe transformed model is shown in black in panel (d) and superim-posed on the family periodic orbits induced by the virtual activatingfunction described in (31). Outputs from simulations of (30) are

nearly identical to the outputs of the transformed model of the form(14) with outputs determined according to (15).

As a point of emphasis, the model equations (31) are quite sim-ple and are primarily considered for illustrative purposes. Indeed,the transformation to the adaptive reduction yields an increase in thenumber of variables required to represent the dynamics. Neverthe-less, as shown in examples to follow, the general strategy describedhere can be readily applied to other more complicated models wherea resulting reduction in dimensionality can be observed.

B. Analyzing phase-locking of coupling-induced

oscillations using the adaptive phase-amplitude

reduction framework

A strategy similar to the one used in Sec. III can be used to ana-lyze phase-locking in coupling-induced oscillators using the adap-tive phase-amplitude reduction method. For simplicity of the expo-sition to follow, phase-locking in two identical, coupling-inducedoscillators will be considered, but larger populations of heteroge-neous systems could be considered with straightforward modifi-cations to the analysis. To begin, consider two identical, coupled,excitable systems

xi = F(xi)+ Uc(xi, xj), (33)

for i = 1, 2 and j = 3 − i, where Uc gives the form of the couplingbetween both excitable systems, xi ∈ R

n denotes the state of each

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oscillator, and F gives the uncoupled dynamics. When coupling iszero, both oscillators decay to a quiescent state; for strong coupling,however, it is assumed that stable oscillations may emerge. As inSec. IV A, let Uvirt(x, p) ∈ R

n be a state dependent virtual inputadded to each oscillator as described in Sec. IV A designed so thatstable limit cycles x

γp emerge in the absence of coupling for p ∈ Pa

⊂ R. Once again, assuming that each xγp is continuously differen-

tiable with respect to both t and p, each element from Eq. (33) canbe analyzed using the adaptive reduction framework with dynamicsthat follow

θi = ω(pi)+ Z(θi, pi) · Ue + D(θi, pi) · pi,

ψi = κ(pi)ψi + I(θi, pi) · Ue + Q(θi, pi) · pi,

pi = Gp(pi, θi,ψi),

(34)

where i = 1, 2 and Ue(xi, xj, pi) = Uc(xi, xj)− Uvirt(xi, pi). It will beassumed that both Ue and ω(p1)− ω(p2) are O(ε) terms at all times.For simplicity of exposition, it is assumed that only a single isostablecoordinate is necessary to capture the amplitude dynamics of eachperiodic orbit so that ψi corresponds to the isostable coordinate ofthe ith oscillator and the subscripts on I and Q are unnecessary;more isostable coordinates could be included by making appropri-ate modifications to the derivation. As suggested in Ref. 49, pro-vided minθ ,p |Qj(θ , p)| = ν where ε � ν, one can choose Gp(θ , p)= −1

Q(θ ,p)I(θ , p) · Ue ≡ R(θ , p) · Ue. Substituting this choice of Gp into

the second equation from (34) (i.e., the isostable dynamics) resultsin a cancellation that yields

ψi = κ(pi)ψi. (35)

Because all orbits are stable, κ(pi) < 0 allowing the isostable dynam-ics to be neglected from (34) to yield

θi = ω(pi)+ (Z(θi, pi)+ D(θi, pi)R(θi, pi)) · Ue,

pi = R(θi, pi) · Ue.(36)

To continue, note that the coupling and the virtual input are bothfunctions of the system state—which itself can be written as a func-tion of the reduced coordinates in the form (15). Thus, one canwrite the adaptive input felt by the ith system as Ue = Ur

c(θi, pi, θj, pj)

+ Urvirt(θi, pi), where Ur

c and Urvirt take the reduced order coordinates

as their arguments instead of the system state itself. The behaviorof each oscillator will be considered in a rotating reference frameφi = θi − ωt, where ω = (ω(p1)+ ω(p2))/2, so that (36) becomes

φi = ω(pi)− ω + (Z(φi + ωt, pi)+ D(φi + ωt, pi)R(φi + ωt, pi))

·[

Urc(φi + ωt, pi,φj + ωt, pj)+ Ur

virt(φi + ωt, pi)]

,

= ω(pi)− ω + hφ(φ1 + ωt, p1,φ2 + ωt, p2), (37)

pi = R(φi + ωt, pi) ·[

Urc(φi +ωt, pi,φj +ωt, pj)+ Ur

virt(φi +ωt, pi)]

,

= hp(φ1 + ωt, p1,φ2 + ωt, p2),

for i = 1, 2 and j = 3 − i where hφ and hp are shorthand represen-tations of the corresponding terms from (37). Recall the assumptionthat ω(p1)− ω(p2) = O(ε) so that ω(pi)− ω is also an O(ε) term

for i = 1, 2. In this case, (37) is periodic with period T = 2π/ωand is of the general form x = εF(x, t) so that formal averaging

techniques16,39 can be applied. Phase-locking can be considered bydefining φ ≡ φ1 − φ2 so that φ = φ1 − φ2. Subsequent manipula-tion of (37) yields

φ = ω(p1)− ω(p2)+ hφ(φ1 + ωt, p1,φ2 + ωt, p2)

− hφ(φ2 + ωt, p2,φ1 + ωt, p1),

p1 = hp(φ1 + ωt, p1,φ2 + ωt, p2),

p2 = hp(φ2 + ωt, p2,φ1 + ωt, p1).

(38)

Averaging16,39 yields

8 = ω(P1)− ω(P2)+ fφ(8, P1, P2)− fφ(−8, P2, P1),

P1 = fp(8, P1, P2),

P2 = fp(−8, P2, P1),

(39)

where 8, P1, and P2 approximate φ, p1, and p2, fφ(X, Y, Z) =

1

T

∫ T

0hφ(X + ωt, Y,ωt, Z) dt, and fp(X, Y, Z) = 1

T

∫ T

0hp(X + ωt, Y,

ωt, Z) dt. As in previous examples, fixed points of (39) correspondto periodic orbits of (38) with the same stability allowing (39) to beused to investigate stable phase-locking in the reduced system (36)and ultimately in the full model (33).

V. EXAMPLES

A. Phase-locking of coupling-induced oscillators near

a Hopf bifurcation

As a preliminary illustration of the analysis from Sec. III, con-sider a collection of N = 10 coupled dynamical elements from (3)with σ = ρ = 0.05 andµi = −0.4 + (i − 1)/30. In this case,µi < 0for all i so that in the absence of coupling, each element from (3) hasa fixed point at xi = yi = 0. These are the same parameters chosenfor the simulations shown in Fig. 1.

For each dynamical element from (3), a reduced order ofthe form (8) is computed numerically. For this choice of param-eters, each fixed point is weakly stable and the linearization has apair of complex-conjugate eigenvalues with Re(λ1) ∈ [−0.02, 0.005]depending on the choice ofµi. As such, there are two isostable coor-dinates required to capture the behavior of each dynamical elementfrom (3) with the associated functions Gi and Ii,n taking the form(16) and (17), respectively, with n = 1, 2 (i.e., one for each isostablecoordinate) and i = 1, . . . , N (i.e., one for each dynamical element).The individual terms of the expansions (16) and (17) are computedusing the strategy described in Appendix A of Ref. 48. The expan-sions are taken to fifth order accuracy in the isostable coordinates.For this example, taking the expansions to higher orders of accuracydoes not substantially alter the results.

Once the reduced order models of each dynamical element areobtained, the transformation from isostable coordinates ψ1,i andψ2,i to the phase-like and amplitude-like coordinates θC,i and ψC,i

using (18) is applied for each i = 1, . . . , N. The resulting system ofequations can be written in the form (24). The averaging strategyfrom Sec. III B is then applied to the resulting isostable coordinatebased transformations and validated against the steady state behav-iors that emerge during full model simulations. Results are presentedin Fig. 5. For coupling strengths K ∈ [−0.109, 0.027], the fixed point

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FIG. 5. After computing the necessary terms of the isostable coordinate transfor-mation, the strategy from Sec. III B is used to predict stable phase-locking in themodel (3) takingN = 10 for various values ofK. Stable fixed points of the resultingaveraged equations (27) correspond to periodic, phase-locked solutions of the fullmodel solutions. This information is also used to predict the magnitudes of eachoscillation, i.e., the difference between the maximum and minimum value of xi insteady state. Magnitude predictions are shown in panel (a) and validated againstthe steady state behavior of the true model simulations of (3) with results shown inpanel (b). Likewise, panel (c) shows the steady state phase differences predictedfrom the averaging paradigm. Note that for regimes where coupling is too smallto induce stable oscillations, it is not possible to compute the steady state phasedifferences. Averaging predictions are compared to the results obtained from thetrue model equations in panel (d).

at xi = yi = 0 is stable and no oscillations emerge. For values out-side this region, stable coupling-induced oscillations emerge and thecorresponding stable fixed point of the averaged Eq. (27) is found(corresponding to a stable phase-locked solution). For stable fixedpoints of (27), the steady state value of the amplitude-like coordinateψSS

C,i is used to approximate the output over a full revolution (i.e.,

by evaluating GC(ψSSC,i, θ) for θ ∈ [0, 2π)). Panel (a) of Fig. 5 shows

the predicted values of max(xSSi )− min(xSS

i ) for each dynamical ele-ment using this approach as a function of the coupling strength.Colors correspond to the bifurcation parameter µi of each oscilla-tor. Panel (b) shows max(xSS

i )− min(xSSi ) in steady state for each

oscillator obtained from full simulations of (3). Panel (c) of Fig. 5shows the phase differences for stable fixed points of (27) that cor-respond to phase-locked solutions of the full model. Likewise, panel(d) shows the phase differences obtained from direct simulations of(3) in steady state.

Considering values of K ∈ [−0.2, −0.109], immediately afterthe fixed point at xi = yi = 0 loses stability for values close toK = −0.109, only a few dynamical elements begin to oscillate (i.e.,those with bifurcation parameter µi closest to the Hopf bifurca-tion). Larger amplitude oscillations emerge in other oscillators asthe coupling strength becomes more negative. These results are con-sistent with those from panels (a) and (b) of Fig. 1. For values ofK < −0.2, phase-locked solutions are no longer stable, but oscilla-tions persist in the isostable transformed equations (24) (results not

shown). Positive values of K result in phase-locked solutions thatrapidly grow in amplitude with increasing K. For smaller values ofK, the agreement is nearly perfect, but the accuracy of the averagingmethod begins to degrade as larger magnitude coupling strengthsare chosen. These errors have two primary explanations, the firstbeing that averaging techniques work best for small inputs,16,39 andthe second being that the isostable coordinates become larger as Kincreases thereby degrading the accuracy of the underlying isostablecoordinate transformation.

Note that for the example presented above, the individual ele-ments of (3) are only two-dimensional and thus the transformationto isostable coordinates does not reduce the order of the model. Nev-ertheless, this example provides an illustration and validation of theanalysis techniques from Sec. III. In the examples to follow, higherdimensional models will be considered so that the transformationto isostable coordinates does indeed provide a reduction in modeldimension.

B. Resonance of a circadian oscillator to periodic

input

Here, a model is considered that has been used to study generegulation dynamics that give rise to circadian oscillations,15

X = v1

Kn1

Kn1 + Zn

− v2

X

K2 + X+ Lc + L(t),

Y = k3X − v4

Y

K4 + Y,

Z = k5Y − v6

D

K6 + Z.

(40)

Here, X, Y, and Z represent the concentration of mRNAof a clock gene, the associated protein, and the protein’snuclear form, respectively, Lc is a constant external light source,and L(t) is a time-varying light input. Model parameters aretaken to be n = 7, v1 = 0.7, v2 = 0.35, v4 = 0.35, v6 = 0.35, K1 = 1,K2 = 1, K4 = 1, K6 = 1, k3 = 0.7, and k5 = 0.7. When Lc = L(t)= 0, a stable periodic orbit with a period of 29.9 h emerges. AsLc increases, the periodic orbit disappears through a Hopf bifur-cation resulting in a weakly unstable fixed point. Taking Lc = 0.02yields a fixed point with associated eigenvalues of the linearizationλ1,2 = −0.0076 ± 0.268i. The remaining eigenvalue λ3 = −0.515and its decay is fast enough relative to the others so that the corre-sponding isostable coordinate can be truncated. As in the previousexample, (40) can be represented in a reduced order framework (8)with the associated functions G and In taking the form (16) and (17),respectively, with n = 1, 2. The individual terms of the expansions(16) and (17) are computed using the strategy described in AppendixA of Ref. 48. Expansions are taken up to eighth order accuracy in theisostable coordinates and the resulting system of equations can bewritten in the form (21) after applying the transformations (18).

For this example, resonant locking with the periodic input

L(t) = Lm

[

1

1 + exp(−5(ts − 18))−

1

1 + exp(−5(ts − 6))

]

, (41)

will be considered. Here, ts = mod(t, 24) so that L(t) is a combi-nation of two sigmoids that result in a periodic light input with a

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periodic 24 h, and Lm sets the magnitude of the variation in light.Note here that� = Im(λ1) = 0.268 which is close to ωp = 0.262 sothat the strategy from Sec. III A can be applied to characterize thesteady state response of (40) to this periodic input.

Using the strategy detailed in Sec. III A, fixed points of theaveraged equations (23) are identified starting with expansions ofthe form (16) and (17) that are valid to various orders of accuracyin the isostable coordinate expansion. The corresponding steadystate value of the amplitude-like coordinate and the phase shiftin the rotating reference frame (ψSS

C and ψSSC , respectively) are

used to approximate the model output over a full 24-h cycle asX(t) = GC(ψ

SSC ,φSS

C + ωpt). Resulting predictions are shown inFig. 6 and compared to the steady state values from full simula-tions of (40). Overall, the accuracy of the predictions increases asthe expansions (16) and (17) are taken to higher orders or accuracy.Additionally, the linear reduction (which takes G and each In to firstand zeroth order accuracy in the isostable coordinate expansion) isnot sufficient to accurately capture the system behavior.

FIG. 6. Entrainment of the circadian oscillator (40) to a resonant input. The aver-aging techniques from Sec. III A are applied to the isostable reduced equationswhich are taken to various orders of accuracy in the isostable coordinate expan-sions of (16) and (17). For the linear averaged system, G is taken to first orderaccuracy in the isostable coordinates and each In is taken to zeroth order accu-racy—such a reduction can be represented with a standard, linear state-spaceformulation. The output predicted from the averaging strategy is compared to thesteady state outputs from the full model simulations. Resulting traces of the vari-ableX are shown in panel (a) taking Lm = 0.0025. Panel (b) shows the associatedlight input. Likewise, traces of the steady state values of X are shown in panel(c) using Lm = 0.0040 with the associated light input in panel (d). In panel (e)[resp., (f)], the averaging framework is used to predict the difference between themaximum and minimum value of X (resp., tpeak, the time at which X reaches itsmaximum value) after all transients have decayed. Fourth and sixth order accu-racy expansions do not predict stable locking for Lm values greater than 0.0025and 0.0041, respectively. When taking the expansions of (16) and (17) to linearaccuracy, the averaging predictions are only valid when Lm is particularly small.The accuracy of these predictions increases as higher order terms are added tothese expansions.

In the context of circadian physiology, models similar to (40)have been used previously to study the mechanisms that give riseto rhythmic oscillations in the suprachiasmatic nucleus (SCN), aregion of the mammalian brain comprised of approximately 20 000coupled neurons that acts as the master circadian pacemaker.30,36

While coupled neurons in the SCN exhibit robust collective oscil-lations, the majority of the individual oscillators are not intrinsicallyrhythmic.2,46 Rather, these collective oscillations are induced by acombination of intracellular coupling and environmental input (e.g.,a 24-h light–dark cycle). The reduction methods presented herecould be used to explain collective behaviors observed in relevantcomputational circadian models that consider both rhythmic andarrhythmic cells.15,17,44 Of specific interest would be the observationthat cellular heterogeneity can speed reentrainment after a suddenadvance or delay in the 24-h light–dark cycle, thereby hasteningrecovery from jet-lag.15,17

C. Phase-locking and input response in an excitable

neural model

Here, phase-locking in a model with N ≥ 1 excitableWang–Buzsaki model neurons45 will be considered, with the addi-tion of an adaptation current,12 and synaptic coupling,13

CVj = −gNam3∞hj(Vj − ENa)− gKn4

j (Vj − EK)

− gL(Vj − EL)− iw,j − isyn,j + u(t),

hj = γ[

αh(Vj)(1 − hj)− βh(Vj)hj

]

,

nj = γ[

αn(Vj)(1 − nj)− βn(Vj)nj

]

,

wj = a(1.5/(1 + exp((b − Vj)/k))− wj),

sj = αs(1 − sj)/(1 + exp(−(Vj − Vt)/σt))− βssj,

(42)

for j = 1, . . . , N. Here, Vj is the transmembrane voltage, hj and nj

are gating variables for neuron j. The variable wj is used to determinethe adaptation current iw,j = gwwj(Vj − EK)with associated conduc-tance gw = 2mS/cm2, sj is a synaptic variable used to determine the

synaptic current isyn,i = gsyn

∑Nj=1 sj(Vi − Esyn), and u(t) is an exter-

nally applied current that is common to each neuron. The synapticcoupling strength is given by gsyn ≥ 0 and the reversal potential ofthe neurotransmitter, Esyn, is taken to be 0 mV so that the cou-pling is excitatory. Full model equations and associated parameterscan be found in the Appendix. In the absence of coupling, (42) isexcitable with a stable fixed point representing a quiescent state.As explained in Sec. IV, excitable systems are difficult to analyzeusing an asymptotic expansion of the isostable coordinate dynamicsabout the fixed point. Instead, (42) will be analyzed by applying theadaptive phase-amplitude reduction framework to each individualneuron.

To proceed, following the strategy set forth in Sec. IV A,

a virtual activating function Uvirt,j =[

ib,j 0 0 0 0]T

will beadded to the dynamics for each neuron to define an associatedshadow system, where ib,j is a baseline current applied to neuron j.This current will be taken as the adaptive parameter in the adap-tive phase-amplitude reduction. For each five-dimensional neuronfrom (42) when taking a constant ib > 0.8µA/cm2, eliminating the

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FIG. 7. Individual neurons from (42) can be represented using the adaptivephase-amplitude reduction strategy suggested in Sec. IV yielding a two-dimen-sional equation of the form (43). Note that the individual neurons from (42) donot have a periodic orbit in the absence of coupling precluding the use of other,commonly used phase reduction techniques for this application. The adaptiveparameter ib (in units of µA/cm2) represents a constant transmembrane cur-rent—panel (a) shows how the natural frequency changes with ib. Panels (b)–(d)show other terms of the reduced order equations as a function of both θ and iB.This reduced order model is validated using a random series of inputs generatedas described in the text. Panel (e) shows the representative voltage traces forboth the full model from Eq. (42) (blue line) and the reduced model from Eq. (43)(black line) in response to the pulses from panel (f) (in units of µA/cm2). Panel(g) shows the time course of the adaptive parameter in the reduced order model.Over a 5000ms simulation, the reduced order model accurately predicts 99% of allaction potentials that occur in the full model, defined as the transmembrane volt-age crossing 0mV with a positive slope. The reduced order model occasionallypredicts an action potential that is not realized by the full model, seen, for exam-ple, at approximately 40 ms in panel (e). Such errors only occur on approximately13% of the spikes predicted by the reduced order model.

synaptic coupling by taking gsyn = 0mS/cm2, and taking u(t) = 0,a stable limit cycle emerges. Here, each neuron will be studied interms of an associated reduced order equation. This setup ulti-mately allows for the application of the strategy from Sec. IV B.As shown in panel (a) of Fig. 7, the natural frequency increases asib grows. The amplitude dynamics of each of these periodic orbitscan be characterized by a single isostable coordinate, which will bedenoted by ψ . Voltage traces of stable periodic orbits correspond-ing to various values of ib as well as traces of IV(θ) (i.e., ∂ψ/∂V)and Q(θ) are shown in panels (b), (c), and (d), respectively. Toimplement the adaptive phase-amplitude reduction framework, let

Ue =[

ue 0 0 0 0]T

, where ue ≡ isyn + u(t)− ib. Noting thatQ(θ , ib) is sufficiently bounded away from 0 for all ib and θ and that

there is only one isostable coordinate, the simplification Gp(θ , ib)= −IV(θ , ib)/Q(θ , ib)ue can be used to eliminate the isostabledynamics yielding a reduced order equation for each neuron of theform (36)

θ = ω(ib)+

(

ZV(θ , ib)−D(θ , ib)IV(θ , ib)

Q(θ , ib)

)

ue,

ib = −Iv(θ , ib)

Q(θ , ib)ue,

(43)

where ZV(θ , ib) is the phase response curve in response to voltageperturbations (i.e., ∂θ/∂V). The reduced order model output of x(t)can be taken from (15), assuming that the isostable coordinate isequal to zero at all times, i.e., using x(t) = x

γ

ib(t)(θ).

The resulting reduced order model is validated for a singleneuron (taking N = 1) without synaptic coupling (i.e., taking gsyn

= 0 µS/cm2) using a series of randomly occurring pulses with ran-dom magnitudes. Results are shown in panels (e)–(g) of Fig. 7.During the simulation, the time between pulses is drawn from aninverse Gaussian distribution with the probability distribution func-tion given by ρ(t) =

√

λg/(2π t3) exp(−λg(t − µg)2/2µ2

g t), where

µg = 7 and λg = 30 are the mean and shape parameter. This dis-tribution is used so that larger times between pulses occur withgreater frequency than they would, for instance, when using a Gaus-sian distribution. Each pulse lasts 1 ms with a magnitude that ischosen randomly from a uniform distribution ranging from 0 to40µA/cm2. Both the full order, single neuron model (42) andthe two-dimensional reduction of the form (43) are simulated for5000 ms with a pulse train that is randomly generated as describedabove. Panel (e) of Fig. 7 shows representative voltage outputs inresponse to the representative pulse train in panel (f). Panel (g)shows how the adaptive parameter changes over time. In general, thereduced order model accurately captures the spiking dynamics, butdoes not perfectly replicate the subthreshold transmembrane voltagedynamics.

Finally, phase-locking for N = 2 neurons of the form (42) isconsidered taking u(t) = 0 and gsyn 6= 0. In the absence of cou-pling, and for small coupling strengths, the state approaches a stablefixed point in the limit as time approaches infinity. When the cou-pling strength is above 0.076µS/cm2, the synaptic coupling is strongenough to sustain periodic oscillations in the full model. Whenconsidering each neuron’s dynamics using the two-dimensionalreduction (43), synaptic coupling-induced oscillations also emergefor coupling strengths at or above 0.072µS/cm2, which is close tothe critical point for the full model. Noticing that the reduced orderequations of the form (43) are of the general form (36), by assum-ing that both ω(ib,1)− ω(ib,2) and ue are each O(ε) terms, one canapply the averaging framework from Sec. IV B to yield the aver-aged equations (39). When gsyn > 0.072µS/cm2, (39) has one stablefixed point and one unstable fixed point where the neurons are inantiphase and synchronized configurations, respectively. Panel (a)of Fig. 8 shows the phase difference dynamics of the averaged equa-tions evaluated using values of ib associated with the stable antiphasesolution associated with the corresponding value of gsyn. Stabilityof the phase-locked solutions can only be evaluated by consideringthe eigenvalues of the Jacobian of (39); nevertheless, these curvesgive a sense of the overall stability of the phase-locked solutions.

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FIG. 8. Phase-locking results taking u(t) = 0 and gsyn large enough so thatstable oscillatory solutions emerge in the neural model (42). The associatedaveraged equations (39) are used to predict phase-locking solutions and theirstability. For gsyn large enough, there is one stable antiphase solution (with 8 =

π ) and an additional unstable synchronized solution (with 8 = 0). For variousvalues of gsyn, Panel (a) shows d8/dt = ω(ib,1)− ω(ib,2)+ fφ(8, ib,1, ib,2)−

fφ(−8, ib,2, ib,1) from (39) with values of ib identical to those associated with thestable antiphase solution. Panel (b) shows the period of the stable phase-lockedsolutions predicted from the averaged equations (39) (black line) as well as theactual period of the stable phase-locked solutions for the full equations (42)(dashed line). Panel (c) shows voltage traces of the stable antiphase solutionstaking gsyn = 0.1µS/cm2 for the full and reduced order equations (dashed andsolid lines, respectively), and panel (d) shows the associated effective input forthe reduced order model. Outputs of different neurons in panels (c) and (d) areplotted with different colors.

In panel (b), the value of ω(ib) for stable solutions of (39) is usedto predict the period once the stable antiphase locking is reached.These predictions match the actual phase-locked solution periodobtained with direct simulations of the full model equations (42).Taking gsyn = 0.1µS/cm2, voltage traces of each neuron once thestable antiphase solution is reached are shown in panel (c) for thereduced and full order models (solid and dashed lines, respectively).Panel (d) shows the associated effective input ue(t) for both neuronsfrom the reduced order model. While the effective input is relativelylarge, the averaging method still accurately predicts the steady statebehavior.

VI. CONCLUSION

In this work, reduced order modeling and analysis strategies aredeveloped to characterize phase-locking relationships that emergein coupled populations of input-induced oscillators. In comparisonto more commonly used phase reduction techniques that are validfor systems that have stable oscillations in the absence of couplingor other exogenous inputs, dynamical systems with input-inducedoscillations have stable fixed points in the absence of input—as such,

standard reduction techniques that employ the weakly perturbedparadigm cannot be used.

As part of this work, two general reduced order modelingframeworks for analysis of input-induced oscillations are pro-posed and validated. The first is well-suited to systems with stablefixed points that are close to a Hopf bifurcation. Here, dynami-cal behaviors are considered on the basis of the slowest decayingKoopman eigenfunctions with a transformation to isostable coor-dinates. Asymptotic expansion in the isostable coordinate basisyields high-accuracy, reduced order models that accurately predictcoupling-induced bifurcations that precipitate stable oscillations inaddition to the associated steady state phase-locking relationships.By contrast, for excitable systems, asymptotic expansions are gen-erally insufficient to accurately capture the isostable dynamics inthe domain of interest. For these systems, a strategy based on therecently proposed adaptive phase-amplitude reduction framework51

can be used. By designing an appropriate set of adaptive parametersand the corresponding virtual activating function, an accurate setof reduced order equations can be found that captures the essentialbehavior of an excitable system. Both methods can be further ana-lyzed with formal averaging techniques,16,39 and theoretical resultsare validated with multiple example systems.

While the proposed reduction and analysis strategies work wellfor the models considered in this work, there are many opportunitiesfor improvement. First, when considering input-induced oscilla-tions in weakly unstable fixed points (for instance, with dynamicsnear a Hopf bifurcation), it is required for the asymptotic expansionsof the isostable reduced dynamics (17) and the output equations (16)to be accurate to at least O(ε) in the allowable range of isostablecoordinates (where ε bounds the magnitude of the allowable input).In many applications, including those considered in this work, thisrequires that the expansions of the functions G and I be taken to fifthorder accuracy or higher in the isostable coordinate expansion. Tak-ing the terms of the expansion to such high orders of accuracy canbecome difficult for high-dimensional systems as the computationaleffort increases exponentially with the dimension of the associatedfull order model. Alternatively, it would be useful to be able to inferthe necessary terms of G and each In in a data-driven manner, i.e.,by considering the input–output relationships and estimating therequired functions accordingly. Preliminary results48,54 have shownpromise for accomplishing this task by considering the steady stateoutputs in response to sinusoidal input applied at various frequen-cies. Future work will consider deep learning techniques and otherstatistical approaches for inferring these reduced order models in adata-driven manner.

A more thorough investigation of the adaptive phase-amplitude reduction framework in the context of excitable systemswould also be warranted in future studies. The efficacy of thisreduction strategy proposed in Sec. IV hinges on the choice of anappropriate virtual activating function and a set of adaptive param-eters that yield accurate reduced order equations. In the excitablemodels considered in this work, activation functions were chosen inan ad hoc manner to yield inputs that elicit excitations as the stateof the excitable system approaches its fixed point. While such vir-tual activating functions worked well for the systems (30) and (42),it would be useful to have a systematic understanding of how acti-vation functions can be designed for general systems. Additionally,

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a more thorough investigation on the design of Gp from (14) (i.e.,the function that actively sets the adaptive parameters in order tokeep the isostable coordinate small) would be of interest and will beconsidered in future work.

While the primary focus of this work is on phase-locking ofinput-induced oscillations, the approaches considered in this workare quite general and could be used to study other applications suchas entrainment, synchronization, and oscillation timing. Further-more, these reduction frameworks yield phase-like and amplitudecoordinates that make them compatible with analysis techniquesthat have been developed applications that involve limit cycle oscil-lators. The proposed approaches represent a valuable and generalframework for understanding the behaviors of input-induced oscil-lations for which standard phase-amplitude reduction approachesare not applicable.

ACKNOWLEDGMENTS

This material is based on the work supported by the NationalScience Foundation (NSF) under Grant No. CMMI-1933583.

APPENDIX: THE WANG–BUZSAKI MODEL

The equations and parameters for a coupled population of NWang–Buzsaki model45 from (42) with an adaptation current12 anda synaptic variable13 are given below,

CVj = −gNam3∞hj(Vj − ENa)− gKn4

j (Vj − EK)

− gL(Vj − EL)− iw,j − isyn,j,

hj = γ[

αh(Vj)(1 − hj)− βh(Vj)hj

]

,

nj = γ[

αn(Vj)(1 − nj)− βn(Vj)nj

]

,

wj = a(1.5/(1 + exp((b − Vj)/k))− wj),

sj = αs(1 − sj)/(1 + exp(−(Vj − Vt)/σt))− βssj,

(A1)

for j = 1, . . . , N. Here, Vj represents the transmembrane voltage,gating variables are denoted by hj and nj, wj is used to set the adap-tation current, and sj is used to determine the synaptic current. Themembrane capacitance, C, is taken to be 1µF/cm2, with additionalequations

m∞ = αm(V)/(αm(V)+ βm(V)),

βn(V) = 0.125 exp(−(V + 44)/80),

αn(V) = −0.01(V + 34)/(exp(−0.1(V + 34))− 1),

βh(V) = 1/(exp(−0.1(V + 28))+ 1),

αh(V) = 0.07 exp(−(V + 58)/20),

βm(V) = 4 exp(−(V + 60)/18),

αm(V) = −0.1(V + 35)/(exp(−0.1(V + 35))− 1),

governing various ionic currents. Reversal potentials and conduc-tances are as follows:

ENa = 55 mV,

EK = −90 mV,

EL = −65 mV,

gNa = 35 mS/cm2,

gK = 9 mS/cm2,

gL = 0.1 mS/cm2.

Here, γ = 5. The adaptation current is given by

iw,j = gwwj(Vj − EK),

with associated parameters

a = 0.02 ms−1,

b = −5 mV,

k = 0.5 mV,

gw = 2 mS/cm2.

The synaptic current for the neuron i is given by

isyn,i = gsyn

N∑

j=1

sj(Vi − Esyn),

where Esyn = 0 mV so that coupling is excitatory. The couplingstrength gsyn ≥ 0 is variable. Parameters associated with the synapticdynamics are

αs = 3,

Vt = −30 mV,

σt = 0.8 mV,

βs = 0.1.

DATA AVAILABILITY

The data that support the findings of this study are availablewithin the article.

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Chaos 31, 023131 (2021); doi: 10.1063/5.0036508 31, 023131-15

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