Global phase-locking in finite populations of phase-coupled oscillators

We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling.Comment: 31 pages; to appear in SIAM journal of dynamical systems (SIADS

Graph Theory and Networks in Biology

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape

On equivalence classes in iterative learning control

This paper advocates a new approach to study the relation between causal iterative learning control (ILC) and conventional feedback control. Central to this approach is the introduction of the set of admissible pairs (of operators) defined with respect to a family of iterations. Considered are two problem settings: standard ILC, which does not include a current cycle feedback (CCF) term and CCF-ILC, which does. By defining an equivalence relation on the set of admissible pairs, it is shown that in the standard ILC problem there exists a bijective map between the induced equivalence classes and the set of all stabilizing controllers. This yields the well-known Youla parameterization as a corollary. These results do not extend in full generality to the case of CCF-ILC; though gain every admissible pair defines a stabilizing equivalent controller, the converse is no longer true in general

Conditions for the existence of fixed points in a finite system of Kuramoto oscillators.

We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling

Observations on the Stability Properties of Cooperative Systems

We extend two fundamental properties of positive linear time-invariant (LTI) systems to homogeneous cooperative systems. Specifically, we demonstrate that such systems are D-stable, meaning that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneous cooperative system is globally asymptotically stable (GAS) for any non-negative delay if and only if the system is GAS for zero delay

A Slow Axon Antidromic Blockade Hypothesis for Tremor Reduction via Deep Brain Stimulation

Parkinsonian and essential tremor can often be effectively treated by deep brain stimulation. We propose a novel explanation for the mechanism by which this technique ameliorates tremor: a reduction of the delay in the relevant motor control loops via preferential antidromic blockade of slow axons. The antidromic blockade is preferential because the pulses more rapidly clear fast axons, and the distribution of axonal diameters, and therefore velocities, in the involved tracts, is sufficiently long-tailed to make this effect quite significant. The preferential blockade of slow axons, combined with gain adaptation, results in a reduction of the mean delay in the motor control loop, which serves to stabilize the feedback system, thus ameliorating tremor. This theory, without any tuning, accounts for several previously perplexing phenomena, and makes a variety of novel predictions

Post break-up tectonic inversion across the southwestern cape of South Africa: new insights from apatite and zircon fission track thermochronometry

The south-west African margin is regarded as an example of a passive continental margin formed by continental rifting following a phase of lithospheric extension and thinning. Recent attention focused on this margin has included theoretical modelling studies of rift processes, plate kinematic studies of the opening geometry and timing, and empirical studies focused on documenting the crustal structure and offshore sedimentary record. Here, we examine the onshore geomorphic and tectonic response to rifting and breakup, with a specific focus on the SW Cape of South Africa. We present 75 new apatite and 8 new zircon fission track analyses from outcrop samples and onshore borehole profiles along the western margin of South Africa. The data are used to derive robust thermal histories that record two discrete phases of accelerated erosional cooling during the Early Cretaceous (150-130 Ma) and Late Cretaceous (100-80 Ma), respectively. Both periods of enhanced erosion are regional in extent, involved km-scale erosion, and extend well inland of the current escarpment zone, albeit with spatially variable intensity and style. The Late Cretaceous episode is also expressed more locally by tectonic reactivation and inversion of major faults causing km-scale differential displacement and erosion. The new AFT data do not exclude the possibility of modest surface uplift occurring during the Cenozoic, but they restrict the depth of regional Cenozoic erosion on the western margin to less than c. 1 km. The inferred pattern and chronology of erosion onshore is consistent with the key features and sediment accumulation patterns within the offshore Orange and Bredasdorp basins. It is suggested that the Late Cretaceous event was triggered by a combination of regional dynamic uplift augmented along the western margin and in the SW Cape by local tectonic forces arising from dextral displacement of the Falkland Plateau along the Falkland-Agulhas Fracture Zone

Global phase-locking in finite populations of phase-coupled oscillators

We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling