Impact of Adaptation Currents on Synchronization of Coupled Exponential Integrate-and-Fire Neurons

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies

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

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.Les populations d’oscillateurs impulsivement couplés constituent un paradigme simple et général pour étudier une multitude de phénomènes naturels, tels que les comportements collectifs des neurones, la synchronisation des cellules pacemaker du coeur, ou encore la dynamique des tremblements de terre. Dans ce contexte, les oscillateurs interagissent au sein du réseau par le biais d’un couplage instantané: quand un oscillateur décharge, il envoie vers les autres oscillateurs une impulsion qui incrémente instantanément leur état par une valeur constante. Diverses extensions du modèle ont été intensément étudiées dans la littérature, mais seulement dans le cas d’oscillateurs leaky integrate-and-fire. Afin de pallier cette restriction, le présent manuscrit traite de l’étude d’autres dynamiques integrate-and-fire: les dynamiques générales integrate-and-fire monotones et les dynamiques integrate-and-fire quadratiques. La contribution principale de la thèse met en évidence le comportement d’ensemble dichotomique selon lequel s’organisent les populations d’oscillateurs: soit les oscillateurs atteignent un état de synchronisation parfaite (taux de décharge lent), soit ils convergent vers une configuration de clustering en blocage de phase (taux de décharge rapide). Ce comportement dichotomique est établi aussi bien pour des populations finies que pour des populations infinies, ce qui démontre un parallèle élégant entre des systèmes en temps-discret dans des espaces de dimension finie et des systèmes en temps-continu dans des espaces de dimension infinie. La première partie du manuscrit se concentre sur l’étude des dynamiques integrate-and-fire monotones. Dans ce cadre, nous montrons que le comportement dichotomique résulte de la propriété de monotonicité des oscillateurs. Cette dernière induit une propriété de contraction globale, elle-même engendrant le comportement dichotomique. En outre, l’analyse révèle que la propriété de contraction est capturée par une norme 1, au lieu d’une norme quadratique (plus usuelle). Dans la seconde partie de la thèse, nous étudions le comportement d’ensemble d’oscillateurs integrate-and-fire quadratiques. Bien que la dynamique ne soit plus monotone, une propriété de monotonicité “en moyenne” implique que le comportement collectif est encore dichotomique. Alors qu’une analyse de stabilité globale s’avère être difficile et conduit à plusieurs conjectures, une analyse locale permet de prouver le comportement dichomique dans certaines situations (réseaux de petite taille, couplage faible, etc.). De plus, l’analyse locale prouve que des oscillateurs integrate-and-fire particuliers ne s’organisent pas suivant un comportement dichotomique, ce qui suggère que ce dernier n’est pas une caractéristique générale de tous les réseaux d’oscillateurs impulsivement couplés. En résumé, la thèse étudie le remarquable comportement dichotomique qui émerge des réseaux d’oscillateurs integrate-and-fire impulsivement couplés, mettant ainsi l’emphase sur les propriétés de stabilité desdits réseaux et développant les résultats théoriques nécessaires à l’étude mathématique des systèmes dynamiques correspondants

Coordination of multi-agent systems: stability via nonlinear Perron-Frobenius theory and consensus for desynchronization and dynamic estimation.

This thesis addresses a variety of problems that arise in the study of complex networks composed by multiple interacting agents, usually called multi-agent systems (MASs). Each agent is modeled as a dynamical system whose dynamics is fully described by a state-space representation. In the first part the focus is on the application to MASs of recent results that deal with the extensions of Perron-Frobenius theory to nonlinear maps. In the shift from the linear to the nonlinear framework, Perron-Frobenius theory considers maps being order-preserving instead of matrices being nonnegative. The main contribution is threefold. First of all, a convergence analysis of the iterative behavior of two novel classes of order-preserving nonlinear maps is carried out, thus establishing sufficient conditions which guarantee convergence toward a fixed point of the map: nonnegative row-stochastic matrices turns out to be a special case. Secondly, these results are applied to MASs, both in discrete and continuous-time: local properties of the agents' dynamics have been identified so that the global interconnected system falls into one of the above mentioned classes, thus guaranteeing its global stability. Lastly, a sufficient condition on the connectivity of the communication network is provided to restrict the set of equilibrium points of the system to the consensus points, thus ensuring the agents to achieve consensus. These results do not rely on standard tools (e.g., Lyapunov theory) and thus they constitute a novel approach to the analysis and control of multi-agent dynamical systems. In the second part the focus is on the design of dynamic estimation algorithms in large networks which enable to solve specific problems. The first problem consists in breaking synchronization in networks of diffusively coupled harmonic oscillators. The design of a local state feedback that achieves desynchronization in connected networks with arbitrary undirected interactions is provided. The proposed control law is obtained via a novel protocol for the distributed estimation of the Fiedler vector of the Laplacian matrix. The second problem consists in the estimation of the number of active agents in networks wherein agents are allowed to join or leave. The adopted strategy consists in the distributed and dynamic estimation of the maximum among numbers locally generated by the active agents and the subsequent inference of the number of the agents that took part in the experiment. Two protocols are proposed and characterized to solve the consensus problem on the time-varying max value. The third problem consists in the average state estimation of a large network of agents where only a few agents' states are accessible to a centralized observer. The proposed strategy projects the dynamics of the original system into a lower dimensional state space, which is useful when dealing with large-scale systems. Necessary and sufficient conditions for the existence of a linear and a sliding mode observers are derived, along with a characterization of their design and convergence properties