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By E. Krause

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2 Â1 / ti / . Â2 Â1 / C . 1 e . 1 e . tiC1 Âk / . 1 . 1 e . Â2 Â1 / . 1 e . Âj Âj 1 C / C C l /e 1/ . 1 e . tiC1 Âk / . tiC1 Âk / C /: The last two formulas describe the dependence of uiC1 on ui : One can easily find a similar relation for the case when i is odd. 1 The Solution of the Second Peskin Conjecture and Developments 39 Set ıi . 1 . tiC1 ti / e . C r/ k X . 1 e . 1 . C l/ k X e e . tiC1 ti / . 77) jD1 Now, recall the map LD defined in the last section. NtiC1 / D ˚1 . 81) where ˚1 . NtiC1 .

T; 0; v / of u; to achieve the threshold. Since all oscillators fire within the equation u0 D S an interval of length T and the distance between two firing moments of an oscillator Q we can conclude the validity of the following theorem. is not less than T; Theorem 7. 67) are valid. U. Akhmet 1 ω a 0 a2 a4 a5 a3 a1 1 Fig. 2 Nonidentical Oscillators: The General Case To make our investigation closer to real-world problems, one has to consider an ensemble of nonidentical oscillators. S C i/ . 70) In what follows, we call the real numbers ; i ; i ; i ; i ; parameters, assuming the first one is positive.

Peskin’s [50] two famous conjectures were developed for further applications. One important additional question is whether continuous or piecewise continuous couplings synchronize the model. This chapter contains sufficient conditions to answer that question in the affirmative. The investigation is based on a specially constructed map. One can remark that the systems investigated in this chapter are, in fact, cooperative discontinuous systems [26–31] with monotone dynamics [54]. Consequently, by applying the methods of dynamical systems with discontinuities at variable moments [2], one can obtain more results concerning biological processes in the future.

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