TY - JOUR

T1 - Stable synchrony in globally coupled integrate-and-fire oscillators

AU - Change, Yu Chuan

AU - Jonq, Juang

PY - 2008/12/1

Y1 - 2008/12/1

N2 - A model of integrate-and-fire oscillators is studied. In the special case of identical oscillators, the model was first proposed and analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645-1662]. We assume, as in Mirollo and Strogatz's model, that each oscillator xi evolves according to a map fi. Our main results are to demonstrate that the concavity structure of fi plays an important role in determining whether Peskin's second conjecture holds true. Specifically, the following statements are proved. First, the system of convex oscillators (i.e., f i″ < 0 for all i), in general, synchronizes when the oscillators are not quite identical. Second, the system of a certain class of concave oscillators (i.e., fi″ > 0 for all i) will not achieve synchrony for initial conditions in a set of positive measure when the oscillators are nearly identical. Third, the system of concave oscillators may achieve synchrony under certain sufficient conditions, provided that the oscillators are not quite nonidentical and that its concavity is small.

AB - A model of integrate-and-fire oscillators is studied. In the special case of identical oscillators, the model was first proposed and analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645-1662]. We assume, as in Mirollo and Strogatz's model, that each oscillator xi evolves according to a map fi. Our main results are to demonstrate that the concavity structure of fi plays an important role in determining whether Peskin's second conjecture holds true. Specifically, the following statements are proved. First, the system of convex oscillators (i.e., f i″ < 0 for all i), in general, synchronizes when the oscillators are not quite identical. Second, the system of a certain class of concave oscillators (i.e., fi″ > 0 for all i) will not achieve synchrony for initial conditions in a set of positive measure when the oscillators are nearly identical. Third, the system of concave oscillators may achieve synchrony under certain sufficient conditions, provided that the oscillators are not quite nonidentical and that its concavity is small.

KW - Concavity

KW - Integrate-and-fire

KW - Nonidentical oscillators

KW - Stable synchrony

UR - http://www.scopus.com/inward/record.url?scp=70349643128&partnerID=8YFLogxK

U2 - 10.1137/070709220

DO - 10.1137/070709220

M3 - Article

AN - SCOPUS:70349643128

SN - 1536-0040

VL - 7

SP - 1445

EP - 1476

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 4

ER -