Synchronization Properties of Network Motifs

We address the problem of understanding the variable abundance of 3-node and 4-node subgraphs (motifs) in complex networks from a dynamical point of view. As a criterion in the determination of the functional significance of a n-node subgraph, we propose an analytic method to measure the stability of the synchronous state (SSS) the subgraph displays. We show that, for undirected graphs, the SSS is correlated with the relative abundance, while in directed graphs the correlation exists only for some specific motifs.Comment: 7 pages, 3 figure

Gershgorin disks for multiple eigenvalues of non-negative matrices

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper

Synchronisation in networks of delay-coupled type-I excitable systems

We use a generic model for type-I excitability (known as the SNIPER or SNIC model) to describe the local dynamics of nodes within a network in the presence of non-zero coupling delays. Utilising the method of the Master Stability Function, we investigate the stability of the zero-lag synchronised dynamics of the network nodes and its dependence on the two coupling parameters, namely the coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model for type-II excitability), there are parameter ranges where the stability of synchronisation depends on the coupling strength and delay time. One important implication of these results is that there exist complex networks for which the adding of inhibitory links in a small-world fashion may not only lead to a loss of stable synchronisation, but may also restabilise synchronisation or introduce multiple transitions between synchronisation and desynchronisation. To underline the scope of our results, we show using the Stuart-Landau model that such multiple transitions do not only occur in excitable systems, but also in oscillatory ones.Comment: 10 pages, 9 figure

Developmental dysplasia of the hip in cerebral palsy–surgical treatment

Developmental dysplasia of the hip (DDH) in cerebral palsy (CP) is very rare, and very little clinical data is available. We have analysed the results of open reduction of the hip (Howorth), acetabuloplasty (Salter, Pemberton), pelvic osteotomy (Chiari) and femoral osteotomy in the treatment of the DDH in CP patients. Radiographic assessment was based on the Severin grading classification system and measurements of migration percentage (MP) before and after surgery. Clinical results were analysed using the Ponsetti classification system in ambulatory patients. A total of 45 hips in 31 patients were treated surgically. The average age of the patients was 5.2 years (range: 2–16 years). The average follow-up was 9.6 years (range: 3–28 years). Analysis of the radiographic results according to Severin placed 35 hips (77.8%) in groups 1 and 2 and ten hips (22.2%) in groups 3 and higher. The average preoperative MP value was 78.7% and the average postoperative MP was 15.2%. Redislocation occurred in three hips. Clinical results were disappointing: based on Ponsetti assessment 14 hips (36.8%) were classified in the first three groups and 24 hips (63.2%) in the last three groups. This analysis suggests that these surgical procedures could be applied in treating DDH in CP