On the dichotomic collective behaviors of large populations of pulse-coupled firing oscillators

Doctoral thesis (2011)

The study of populations of pulse-coupled firing oscillators is a general and simple paradigm to investigate a wealth of natural phenomena, including the collective behaviors of neurons, the ... [more ▼]

The study of populations of pulse-coupled firing oscillators is a general and simple paradigm to investigate a wealth of natural phenomena, including the collective behaviors of neurons, the synchronization of cardiac pacemaker cells, or the dynamics of earthquakes. In this framework, the oscillators of the network interact through an instantaneous impulsive coupling: whenever an oscillator fires, it sends out a pulse which instantaneously increments the state of the other oscillators by a constant value. There is an extensive literature on the subject, which investigates various model extensions, but only in the case of leaky integrate-and-fire oscillators. In contrast, the present dissertation addresses the study of other integrate-and-fire dynamics: general monotone integrate-and-fire dynamics and quadratic integrate-and-fire dynamics. The main contribution of the thesis highlights that the populations of oscillators exhibit a dichotomic collective behavior: either the oscillators achieve perfect synchrony (slow firing frequency) or the oscillators converge toward a phase-locked clustering configuration (fast firing frequency). The dichotomic behavior is established both for finite and infinite populations of oscillators, drawing a strong parallel between discrete-time systems in finite-dimensional spaces and continuous-time systems in infinite-dimensional spaces. The first part of the dissertation is dedicated to the study of monotone integrate-and-fire dynamics. We show that the dichotomic behavior of the oscillators results from the monotonicity property of the dynamics: the monotonicity property induces a global contraction property of the network, that forces the dichotomic behavior. Interestingly, the analysis emphasizes that the contraction property is captured through a 1-norm, instead of a (more common) quadratic norm. In the second part of the dissertation, we investigate the collective behavior of quadratic integrate-and-fire oscillators. Although the dynamics is not monotone, an “average” monotonicity property ensures that the collective behavior is still dichotomic. However, a global analysis of the dichotomic behavior is elusive and leads to a standing conjecture. A local stability analysis circumvents this issue and proves the dichotomic behavior in particular situations (small networks, weak coupling, etc.). Surprisingly, the local stability analysis shows that specific integrate-and-fire oscillators exhibit a non-dichotomic behavior, thereby suggesting that the dichotomic behavior is not a general feature of every network of pulse-coupled oscillators. The present thesis investigates the remarkable dichotomic behavior that emerges from networks of pulse-coupled integrate-and-fire oscillators, putting emphasis on the stability properties of these particular networks and developing theoretical results for the analysis of the corresponding dynamical systems. [less ▲]

Erratum: “Clustering behaviors in networks of integrate-and-fire oscillators” [Chaos 18, 037122 (2008)]

in Chaos (2009), 19

Phénomènes dynamiques de synchronisation par couplage impulsif

Master's dissertation (2007)

Synchronization phenomena are extremely important and omnipresent in nature. So they are part of the most observed biological behaviours. For instance, they point up the way of working of some brain ... [more ▼]

Synchronization phenomena are extremely important and omnipresent in nature. So they are part of the most observed biological behaviours. For instance, they point up the way of working of some brain mechanisms but also the existence of disorders in neuroscience. The main line of the study consists in analysing the different behaviours of a neuronal network in order to describe the conditions in which synchronization appears. Each neuron is simulated by the quadratic integrate-and-fire model (QIF), i.e. by an hybrid integrator. The state variable -the membrane potential- moves between two threshold values. When the high threshold level is reached, the variable is reset to the low one. The coupling between the neurons is impulsive: when a reset occurs, the membrane potential of every connected neurons is increased by a discrete value. This kind of coupling applied to the QIF model is an original feature of the work as litterature is poor on the subject. Moreover, it allows to observe the behaviours of populations including both oscillatory neurons and excitable ones. After a short introduction about the mathematical models of neurons (chapter 2), some analytical tools are first developped in basic cases with two neurons interconnected (chapter 3). The model is then applied in order to simulate the behaviour of more complicated situations (chapter 4 and 5). Finally, heterogeneous neuronal populations are analysed (chapter 6). Throughout the study, the model is compared with the Kuramoto one and their similarities are pointed out. [less ▲]