Transience Bounds for Distributed Algorithms

International audienceA large variety of distributed systems, like some classical synchronizers, routers, or schedulers, have been shown to have a periodic behavior after an initial transient phase (Malka and Rajsbaum, WDAG 1991). In fact, each of these systems satisfies recurrence relations that turn out to be linear as soon as we consider max-plus or min-plus algebra. In this paper, we give a new proof that such systems are eventually periodic and a new upper bound on the length of the initial transient phase. Interestingly, this is the first asymptotically tight bound that is linear in the system size for various classes of systems. Another significant benefit of our approach lies in the straightforwardness of arguments: The proof is based on an easy convolution lemma borrowed from Nachtigall (Math. Method. Oper. Res. 46) instead of purely graph-theoretic arguments and involved path reductions found in all previous proofs

Functional connectivity and neuronal dynamics: insights from computational methods

International audienceBrain functions rely on flexible communication between microcircuits in distinct cortical regions. The mechanisms underlying flexible information routing are still, however, largely unknown. Here, we hypothesize that the emergence of a multiplicity of possible information routing patterns is due to the richness of the complex dynamics that can be supported by an underlying structural network. Analyses of circuit computational models of interacting brain areas suggest that different dynamical states associated with a given connectome mechanistically implement different information routing patterns between system's components. As a result, a fast, network-wide and self-organized reconfiguration of information routing patterns-and Functional Connectivity networks, seen as their proxy-can be achieved by inducing a transition between the available intrinsic dynamical states. We present here a survey of theoretical and modelling results, as well as of sophisticated metrics of Functional Connectivity which are compliant with the daunting task of characterizing dynamic routing from neural data. Theory: Function follows dynamics, rather than structure Neuronal activity conveys information, but which target should this information be-pushed‖ to, or which source should new information be-pulled‖ from? The problem of dynamic information routing is ubiquitous in a distributed information processing system as the brain. Brain functions in general require the control of distributed networks of interregional communication on fast timescales compliant with behavior, but incompatible with plastic modifications of connectivity tracts (Bressler & Kelso, 2001; Varela et al., 2001). This argument led to notions of connectivity based on information exchange-or more generically, an-interaction‖-between brain regions or neuronal populations, rather than based on the underlying STRUCTURAL CONNECTIVITY (SC, i.e. anatomic). An entire zoo of data-driven metrics has been introduced in the literature and this chapter will review some of them. Notwithstanding, they track simple correlation, or directed causal influence (Friston, 2011) or information transfer (Wibral et al., 2014) between time-series of activity. Thes

Floquet Chern Insulators of Light

Achieving topologically-protected robust transport in optical systems has recently been of great interest. Most topological photonic structures can be understood by solving the eigenvalue problem of Maxwell's equations for a static linear system. Here, we extend topological phases into dynamically driven nonlinear systems and achieve a Floquet Chern insulator of light in nonlinear photonic crystals (PhCs). Specifically, we start by presenting the Floquet eigenvalue problem in driven two-dimensional PhCs and show it is necessarily non-Hermitian. We then define topological invariants associated with Floquet bands using non-Hermitian topological band theory, and show that topological band gaps with non-zero Chern number can be opened by breaking time-reversal symmetry through the driving field. Furthermore, we show that topological phase transitions between Floquet Chern insulators and normal insulators occur at synthetic Weyl points in a three-dimensional parameter space consisting of two momenta and the driving frequency. Finally, we numerically demonstrate the existence of chiral edge states at the interfaces between a Floquet Chern insulator and normal insulators, where the transport is non-reciprocal and uni-directional. Our work paves the way to further exploring topological phases in driven nonlinear optical systems and their optoelectronic applications, and our method of inducing Floquet topological phases is also applicable to other wave systems, such as phonons, excitons, and polaritons