BackgroundMany technological, biological, social, and information networks fall into the broad class of ‘small-world’ networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction (‘small/not-small’) rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model – the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified.Methodology/Principal FindingsWe defined a precise measure of ‘small-world-ness’ S based on the trade off between high local clustering and short path length. A network is now deemed a ‘small-world’ if S>1 - an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process.Conclusions/SignificanceWe have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing.

AbstractÑA selection problem arises whenever two or more competing systems seek simultaneous access to a restricted resource. Consideration of several selection architectures suggests there are significant advantages for systems which incorporate a central switching mechanism. We propose that the vertebrate basal ganglia have evolved as a centralised selection device, specialised to resolve conflicts over access to limited motor and cognitive resources. Analysis of basal ganglia functional architecture and its position within a wider anatomical framework suggests it can satisfy many of the requirements expected of an efficient selection mechanism.

We present a biologically plausible model of processing intrinsic to the basal ganglia based on the computational premise that action selection is a primary role of these central brain structures. By encoding the propensity for selecting a given action in a scalar value (the salience), it is shown that action selection may be recast in terms of signal selection. The generic properties of signal selection are defined and neural networks for this type of computation examined. A comparison between these networks and basal ganglia anatomy leads to a novel functional decomposition of the basal ganglia architecture into 'selection' and 'control' pathways. The former pathway performs the selection per se via a feedforward off-centre on-surround network. The control pathway regulates the action of the selection pathway to ensure its effective operation, and synergistically complements its dopaminergic modulation. The model contrasts with the prevailing functional segregation of basal ganglia into 'direct' and 'indirect' pathways.

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.

Neurophysiological studies have identified a number of brain regions critically involved in solving the problem of action selection or decision making. In the case of highly practiced tasks, these regions include cortical areas hypothesized to integrate evidence supporting alternative actions and the basal ganglia, hypothesized to act as a central switch in gating behavioral requests. However, despite our relatively detailed knowledge of basal ganglia biology and its connectivity with the cortex and numerical simulation studies demonstrating selective function, no formal theoretical framework exists that supplies an algorithmic description of these circuits. This article shows how many aspects of the anatomy and physiology of the circuit involving the cortex and basal ganglia are exactly those required to implement the computation defined by an asymptotically optimal statistical test for decision making: the multihypothesis sequential probability ratio test (MSPRT). The resulting model of basal ganglia provides a new framework for understanding the computation in the basal ganglia during decision making in highly practiced tasks. The predictions of the theory concerning the properties of particular neuronal populations are validated in existing experimental data. Further, we show that this neurobiologically grounded implementation of MSPRT outperforms other candidates for neural decision making, that it is structurally and parametrically robust, and that it can accommodate cortical mechanisms for decision making in a way that complements those in basal ganglia.