Self-avoiding walks and connective constants in small-world networks

Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's $u_n$ was obtained from numerical simulations as a function of the number of steps $n$ on the considered networks. The so-called connective constant, $\mu = \lim_{n \to \infty} u_n/u_{n-1}$, which characterizes the long-distance behavior of the walks, increases continuously with disorder strength (or rewiring probability, $p$). For small $p$, one has a linear relation $\mu = \mu_0 + a p$, $\mu_0$ and $a$ being constants dependent on the underlying lattice. Close to $p = 1$ one finds the behavior expected for random graphs. An analytical approach is given to account for the results derived from numerical simulations. Both methods yield results agreeing with each other for small $p$, and differ for $p$ close to 1, because of the different connectivity distributions resulting in both cases.Comment: 7 pages, 5 figure

Synaptic processes and immune-related pathways implicated in Tourette syndrome.

Tourette syndrome (TS) is a neuropsychiatric disorder of complex genetic architecture involving multiple interacting genes. Here, we sought to elucidate the pathways that underlie the neurobiology of the disorder through genome-wide analysis. We analyzed genome-wide genotypic data of 3581 individuals with TS and 7682 ancestry-matched controls and investigated associations of TS with sets of genes that are expressed in particular cell types and operate in specific neuronal and glial functions. We employed a self-contained, set-based association method (SBA) as well as a competitive gene set method (MAGMA) using individual-level genotype data to perform a comprehensive investigation of the biological background of TS. Our SBA analysis identified three significant gene sets after Bonferroni correction, implicating ligand-gated ion channel signaling, lymphocytic, and cell adhesion and transsynaptic signaling processes. MAGMA analysis further supported the involvement of the cell adhesion and trans-synaptic signaling gene set. The lymphocytic gene set was driven by variants in FLT3, raising an intriguing hypothesis for the involvement of a neuroinflammatory element in TS pathogenesis. The indications of involvement of ligand-gated ion channel signaling reinforce the role of GABA in TS, while the association of cell adhesion and trans-synaptic signaling gene set provides additional support for the role of adhesion molecules in neuropsychiatric disorders. This study reinforces previous findings but also provides new insights into the neurobiology of TS

Stochastic resonance in a locally excited system of bistable oscillators

Stochastic resonance is studied in a one-dimensional array of overdamped bistable oscillators in the presence of a local subthreshold periodic perturbation. The system can be treated as an ensemble of pseudospins tending to align parallel which are driven dynamically by an external periodic magnetic field. The oscillators are subjected to a dynamic white noise as well as to a static topological disorder. The latter is quantified by the fraction of randomly added long-range connections among ensemble elements. In the low connectivity regime the system displays an optimal global stochastic resonance response if a small-world network is formed. In the mean-field regime we explain strong changes in the dynamic disorder strength provoking a maximal stochastic resonance response via the variation of fraction of long-range connections by taking into account the ferromagnetic-paramagnetic phase transition of the pseudospins. The system size analysis shows only quantitative power-law type changes on increasing number of pseudospins. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2011