Comment on "The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory" [Chaos 30, 073139 (2020)]
CComment on “The Winfree model with non-infinitesimal phase-responsecurve: Ott-Antonsen theory” [Chaos , 073139 (2020)] Diego Pazó and Rafael Gallego Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander,Spain Departamento de Matemáticas, Universidad de Oviedo, Campus de Viesques, 33203 Gijón,Spain
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIPPublishing. This article appeared in Chaos , 018101 (2021) and may be found here In a recent paper [Chaos , 073139 (2020)] we analyzed an extension of the Winfree model with nonlinear inter-actions. The nonlinear coupling function Q was mistakenly identified with the non-infinitesimal phase-response curve(PRC). Here, we asses to what extent Q and the actual PRC differ in practice. By means of numerical simulations, wecompute the PRCs corresponding to the Q functions previously considered. The results confirm a qualitative similaritybetween the PRC and the coupling function Q in all cases.In Ref. we studied this generalization of the Winfreemodel of globally coupled phase oscillators:˙ θ i = ω i + Q ( θ i , A ) , (1a) A = ε N N ∑ j = P ( θ j ) . (1b)Here, A is proportional to the sum over the pulses emitted bythe N oscillators of the population. In contrast to the originalmodel , function Q in Eq. (1a) has a nonlinear dependenceon the mean field A . The motivation for this is the fact thatnonlinearity is an unavoidable consequence of applying phasereduction beyond the first order to oscillator ensembles . Notethat a Taylor expansion of Q to n th order in A yields up to ( n + ) -body phase interactions, similarly to Ref. .We mistakenly called Q ‘non-infinitesimal phase-responsecurve’ in Ref. . Properly speaking, function Q is a non-linear‘coupling function’ . The aim of this comment is to clar-ify to what extent the coupling function Q determines theactual phase-response curve (PRC). The PRC quantifies thephase shift gained by an oscillator in response to an externalstimulus . There is no analytic relation between Q and thePRC beyond the small ε limit; in that case Q ( θ , A ) (cid:39) ˜ Q ( θ ) A ,where ˜ Q turns out to be so-called infinitesimal PRC (iPRC). Inconsequence, we rely here on numerical simulations to com-pute the PRC empirically.The family of functions Q ( θ , A ) considered in was: Q ( θ , A ) = f ( A )( − cos θ ) − f ( A ) sin θ . (2)Four representative pairs of functions f , ( A ) were studied indetail in and the corresponding coupling functions Q ( θ , A ) were depicted in Fig. 2 of Ref. . With the aim of comparingthem, we obtain the PRC for each of the four coupling func-tions Q considered in .The PRC value depends on the timing as well as on thespecific shape of the stimulus, which is not necessarily weakor brief . Numerically, we obtain the PRC measuring theeffect on one oscillator’s phase of a pulse generated by an-other oscillator. This means that the two oscillators are uni-directionally coupled (i.e., a master-slave configuration). We adopt ω = . Moreover, wefollow and use the same 2 π -periodic symmetric unimodalpulse function P ( θ ) . It vanishes at θ = ± π , and a free pa-rameter r < P : The height ofthe pulse is P ( ) = / ( − r ) , and lim r → P ( θ ) = πδ ( θ ) . Inthis comment we consider two different pulse widths: r = . ), and r = .
99 corresponding to an ex-tremely narrow pulse.The simulation starts at time t = θ in . Then, we let it to evolve un-der the influence of the forcing oscillator. The phase of thisone grows linearly, such that the input felt by the first oscilla-tor is A ( t ) = ε P ( t − π ) . Parameter ε determines the strengthof the stimulus. The simulation runs from t = t = π ,since A exactly vanishes at these times. Note that we donot need to run the simulation further since phase oscilla-tors are governed by first-order differential equations. For agiven ε value, we measure the phase shift at t = π such thatPRC ( θ , ε ) = θ ( t = π ) − ( θ in + π ) . The phase θ in the ar-gument of the PRC is the phase value when A attains its max-imum, assuming no input exists: θ = θ in + π . The resultsare shown in Figs. 1 and 2 for a set of ε values; in each panelfor one particular coupling function Q ( θ , A ) already adoptedin . In all panels, the corresponding iPRC is shown as a refer-ence. Note that the normalization of the y -axis in Figs. 1 and2 includes a 2 π factor —in addition to ε — because this is theintegral of the pulse over an interval of length 2 π . Figures 1and 2 are quite similar, though for r = . ε value.The comparison of the PRCs in this comment with thecorresponding Q functions in Fig. 2 of Ref. evidences that Q ( θ , A ) is not simply the PRC. Indeed, Q in (2) has only thefirst harmonic in θ , whereas the non-infinitesimal PRCs inthe figures display additional Fourier components. In spiteof these dissimilarities, simple visual inspection indicates thatthe PRC strongly resembles the coupling function Q in allfour cases. For example, we observe the same loss of non-negativeness of the (type-I) iPRC as A increases in panel (a), a r X i v : . [ n li n . AO ] J a n -2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (a) iPRC ε =0.003 ε =0.01 ε =0.03 ε =0.1 ε =0.3 -2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (b)-2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (c) -2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (d) FIG. 1. PRCs for cases (a-d) in Ref. .The set of five ε values used in indicatedin panel (a). The coupling function (2) ineach panel is: (a) f = A / ( + A ) = f / A ;(b) f = A / ( + A ) = A f ; (c) f = f = A ( − A ) / ( + A ) ; (d) f = f = A ( A − ) / ( + A ) . The pulse acting on theoscillator is P ( t − π ) = ( − r )[ + cos ( t − π )] − r cos ( t − π )+ r ,with r = . -2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (a) iPRC ε =0.003 ε =0.01 ε =0.03 ε =0.1 ε =0.3 -2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (b)-2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (c) -2-1 0 1 2 0 π π P RC ( θ , ε ) / ( π ε ) θ (d) FIG. 2. The same as Fig. 1 with r = . or the transition from a synchronizing iPRC to a desynchro-nizing PRC for large enough A in panel (c). Summarizing,our simulations confirm that the main attributes of the cou-pling function Q are shared by the non-infinitesimal PRC. ACKNOWLEDGMENTS
We acknowledge support by the Agencia Estatal de Inves-tigación and Fondo Europeo de Desarrollo Regional underProject No. FIS2016-74957-P (AEI/FEDER, EU).
REFERENCES D. Pazó and R. Gallego, “The Winfree model with non-infinitesimal phase-response curve: Ott-Antonsen theory,” Chaos , 073139 (2020). A. T. Winfree, “Biological rhythms and the behavior of populations of cou-pled oscillators.” J. Theor. Biol. , 15–42 (1967). A. T. Winfree,
The Geometry of Biological Time (Springer, New York, 1980). M. Rosenblum and A. Pikovsky, “Numerical phase reduction beyond thefirst order approximation,” Chaos , 011105 (2019). I. León and D. Pazó, “Phase reduction beyond the first order: The case of themean-field complex Ginzburg-Landau equation,” Phys. Rev. E , 012211(2019). E. M. Izhikevich,