The brainstem reticular formation is a small-world, not scale-free, network

Abstract

Recently, it has been demonstrated that several complex systems may have simple graph-theoretic characterizations as so-called 'small-world' and 'scale-free' networks. These networks have also been applied to the gross neural connectivity between primate cortical areas and the nervous system of Caenorhabditis elegans. Here, we extend this work to a specific neural circuit of the vertebrate brain-the medial reticular formation (RF) of the brainstem-and, in doing so, we have made three key contributions. First, this work constitutes the first model (and quantitative review) of this important brain structure for over three decades. Second, we have developed the first graph-theoretic analysis of vertebrate brain connectivity at the neural network level. Third, we propose simple metrics to quantitatively assess the extent to which the networks studied are small-world or scale-free. We conclude that the medial RF is configured to create small-world (implying coherent rapid-processing capabilities), but not scale-free, type networks under assumptions which are amenable to quantitative measurement.

The influence of ageing on complex brain networks: A graph theoretical analysis

“…The increase of the synchronization of theta band during the math task as shown by the main task effect is explained by the increase in WM function during math [Canolty et al, 2006;Humphries et al, 2006;Schack et al, 2002;Stam et al, 2002]. The more prominent enhancement of theta band synchronization during WM in adults could be explained as the result of the differences in the WM capacity.…”

“…The values of C and L were compared with the corresponding values of ensembles of twenty random graphs (C-s, L-s) using the ratios k 5 L/L-s and g 5 C/C-s. Subsequently, a summary measure of ''small-worldness'' can be defined by the ratio S 5 g/k, which will be greater than 1 for smallworld networks [Humphries et al, 2006;Stam and Reijneveld, 2007;. For statistics we used SPSS 14.0 for windows and we performed a repeated measures ANOVA for each frequency band and for the mean SL values averaged over all electrode pairs, C, L, C/C-s, L/L-s, and S taking the two situations (rest, math) as within subjects' factors for the two groups (children or students as between subject factors).…”

“…Recently, tools from graph theory have been used in several studies to investigate the general properties of the spatial structure of anatomical as well as functional networks as studied with EEG, MEG and fMRI [Achard et al, 2006;Basset and Bullmore, 2006;He et al, 2007;Humphries, 2006;Stam and Reijneveld, 2007;. The connectedness of the neuronal assemblies is usually assessed with two characteristic measures of the representations of the networks (graphs).…”

Application of directed transfer function and network formalism for the assessment of functional connectivity in working memory task

“…The small-world properties in the framework of unweighted networks are usually defined by means of cluster coefficient and path length, one of the assumptions being that there exists at least one path connecting any couple of nodes [1,4]. In specialist literature, several papers may be found where the small-world architecture was detected [47][48][49], as well as in other papers quoted in [5]. However, in some cases, small-world features were barely distinguishable from randomness [49,50].…”

Neuroticism and conscientiousness respectively positively and negatively correlated with the network characteristic path length in dorsal lateral prefrontal cortex: A resting‐state fNIRS study

“…The normalized characteristic path length is the ratio between the real and random characteristic path length: $\lambda =\frac{L_{p-\mathrm{normalreal}}}{L_{p-\mathrm{normalrand}}}.$ C p_ rand and L p_ rand denotes, respectively, the averaged clustering coefficient and characteristic path length of 100 matched random networks, which possess the same number of nodes, edges, and degree distribution with the real networks (Maslov & Sneppen, 2002; Sporns & Zwi, 2004). Typically, a small‐word network meets the conditions of $\gamma >1$ and λ  ≈ 1 (Watts & Strogatz, 1998), and therefore, the small‐world scalar σ  =  λ / γ is larger than 1 (Humphries, Gurney, & Prescott, 2006). …”