Large assemblies of oscillator units can spontaneously evolve to a state of large scale organization. Synchronization is the best known phenomenon of this kind, where after some transient regime a coherent oscillatory activity of the set of os-cillators emerges. This interesting phenomenon is quite common in many differ-ent disciplines such as engineering [54], physics [13, 30] and [48], chemistry [31], as well as biology [53]. For example, southeastern fireflies, where thousands of individuals gathered on trees flash in unison. Other examples of biological oscil-lators are the rhythmic activity of cells of the heart pacemaker [25, 35, 39] and [51], of cells of pancreas [45] and [46], and of neural networks [11, 18, 39, 41]
and [47]. Synchronization of oscillators has been studied in both phase-coupled [3, 4, 5, 6, 14, 15, 16, 17, 26, 27, 29, 32, 33, 34, 37, 40, 49, 50, 52] and [56], where the interaction between the oscillators is smooth and continuous in time, and pulsed-coupled models [1, 7, 9, 10, 19, 20, 21, 23, 24, 28, 31, 36, 42, 43, 44] and [55], where the membrane voltage is discontinuously reset to a fixed value once it reaches a cer-tain threshold. It should be noted that pulse-coupled models are of greater relevance for neuroscience applications since synaptic coupling is often spike mediated.
The purpose of this thesis is to study synchronization in locally coupled and-fire oscillators. We begin with describing the Peskin’s model [39] of n integrate-and-fire oscillators. Let the state of the i-th oscillator be denoted by xi, where xi
are subject to the dynamics dxdti = −rixi + Ii, 0 ≤ xi ≤ 1, i = 1, 2, · · · , n with input Ii> 0, a normalized threshold 1 and leakiness ri≥ 0. When xj = 1, the jth oscillators fires and xj jumps back to zero. As a consequence of the firing of jth oscillator, the activation of any other oscillator i is incremented by the coupling ²ij. If ²ij 6= 0 for all i 6= j, then the system of n such oscillators is said to be globally coupled. Otherwise, it is said to be locally coupled. This model was later generalized by Mirollo and Strogatz [36]. Specifically, they assumed that the state variable xi
evolve according to a map fi. When xi reaches the threshold, the oscillator fires and xi jumps back instantly to zero, and the activation of any other oscillator j is incremented by the positive coupling ²ji. Specifically, xi evolve according to xi = fi(φi), where fi : [0, 1] → [0, 1] is smooth, and strictly increasing, i.e., fi0 > 0 on (0, 1). Here φi is a phase variable so that (i) dφdti = T1
i, where Ti is the cycle period for oscillator xi when evolving freely, (ii) φi= 0 when the oscillator is at its lowest state xi= 0, and (iii) φi≡ 1 at the end of cycle when the oscillator reaches the threshold xi= 1. Therefore, fi satisfy fi(0) = 0, fi(1) = 1. These maps fi are to be called evolution maps. The inverses of fi are to be denoted by gi. If fi≡ f ,
1
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conjectured that, first, for identical oscillators that are globally coupled, the system approaches a state in which all oscillators are firing synchronously for almost all initial conditions and that, second, this remains true even when the oscillators are not quite identical. The first part of the conjecture was essentially proved by Mirollo and Strogatz [36] with convex oscillators (i.e., fi00 < 0). The second part of Peskin’s conjecture was verified by Senn and Urbanczik [44] with flat oscillators (i.e., fi00 ≡ 0). The key feature in those proofs rely on the non-concavity of the evolution functions fi. However, Bottani [8] numerical showed that even concave oscillators (i.e., fi00 > 0) can synchronize provided that the concavity is not too large. Recently Chang and Juang [12] proves the second part of Peskin’s conjecture for the system of convex oscillators. Moreover, they also reconfirm Mirollo and Strogatz’s observation that convexity of the oscillators indeed plays an important role in achieving synchrony. Specifically, they show that for concave oscillators if they are ”nearly” identical, then no synchronization is to occur for initial conditions in a set of positive measure. That is to say, in general, concave oscillators may synchronize for almost all initial conditions only if they are not that identical.
Indeed, they further prove that the imbalance between the speeds and/or coupling strengthens of the oscillators induces the synchronization of the system provided that the concavity of the evolution maps is sufficiently small. The last part of their results verifies the numerical observation of Bottani [8].
In the celebrated paper of Mirollo and Strogatz, they also raise an open question.
How would the dynamics be affected if one replaces the all-to-all coupling with more local interactions, e.g., between the nearest neighbors or on a ring of d-dimensional chain, or more general graph [2, 22, 38]? Would the system still always end up firing in unison, or would more complex modes of organization become possible? In this thesis, we prove that for the identical system of four convex oscillators being the nearest neighbor coupling with periodic boundary conditions, the system always
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ends up firing in unison. See Fig1.1 for more explanation of such coupling. The locally coupling rules mean that, for instance, if oscillator 2 reaches the threshold, then its nearest neighbors, oscillator 3 and 1 receive the coupling strength ².
We next describe the dynamics of such system. Without loss of generality, we let the speed 1
T of each oscillator be one. Let Φ0 = (φ01, φ02, φ03, φ04) be the initial condition, which denote the phase position of four oscillators labeled 1,2,3,4, re-spectively. Suppose φ04 is the first one reaching the threshold. Then the resulting phase position of the first three oscillators are, respectively,
φ1i = g(f (1 − φ04+ φ0i) + ²), i = 1, 3 φ12= 1 − φ04+ φ02.
provided that f (1 − φ04+ φ0i) + ² < 1, i = 1, 3 and 1 − φ04+ φ02 < 1. Suppose, in addition, that f (1 − φ04+ φ01) + ² ≥ 1. That is to say that the first oscillator also reach the threshold after receiving the coupling strength ² from the fourth oscillator.
Then φ13= g(f (1 − φ04+ φ03) + ²) and φ12= g(f (1 − φ04+ φ02) + ²). Such chain reaction might continue if the second and/or the third oscillator also reach the threshold due to the earlier chain reaction.
Now we define the firing map that describes the changing of the phase of oscil-lators after one firing (i.e., some osciloscil-lators reaching the threshold).
Definition 1.1. Let S = {φ = (φ1, φ2, φ3, φ4) : 0 = min
1≤i≤4{φi} ≤ max
1≤i≤4{φi} < 1}.
Then the firing map h is defined as the mapping from S to S satisfying h(φ) = φnew,
where φnew is the new phase of oscillators from the original phase φ after another shortest time that causes some oscillator to reach the threshold and right after the time that the spike from the fired oscillator is achieved to the corresponding oscilla-tors.
We remark that from above definition, h(S) ⊆ S and thus iterations under h is well-defined. That is, hi(φi−11 , φi−12 , φi−13 , φi−14 ) =: (φi1, φi2, φi3, φi4) is well-defined for all i > 0. In the following, for the reason of clarity, we denote h to be hi if firing map h is acted on the phase (φi−11 , φi−12 , φi−13 , φi−14 ).
Definition 1.2. Define function Fm(φ) = g(min[f (φ)+m², 1]) in the interval [0, 1], where g is the inverse function of f .
Lemma 1.1. Let f00< 0.
(1) If φ1< φ2, then Fm(φ1) ≤ Fm(φ2).
If φ1< φ2 and Fm(φ1) < 1, then Fm(φ1) < Fm(φ2).
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(2) If 0 < Fm(φ) < 1, then dFm(φ)
where I > r > 0. The evolution map f and its inverse function g are given in the following, respectively
Definition 1.4.
N1= {φ ∈ S1: after iterations, the states never reach synchronization}.
N2= {φ ∈ S2: after iterations, the states never reach synchronization}.
N3= {φ ∈ S3: after iterations, the states never reach synchronization}.
2. Three Neurons Having Initial State Zero