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.

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 basal ganglia (BG) have long been implicated in both motor function and dysfunction. It has been proposed that the BG form a centralized action selection circuit, resolving conflict between multiple neural systems competing for access to the final common motor pathway. We present a new spiking neuron model of the BG circuitry to test this proposal, incorporating all major features and many physiologically plausible details. We include the following: effects of dopamine in the subthalamic nucleus (STN) and globus pallidus (GP), transmission delays between neurons, and specific distributions of synaptic inputs over dendrites. All main parameters were derived from experimental studies. We find that the BG circuitry supports motor program selection and switching, which deteriorates under dopamine-depleted and dopamine-excessive conditions in a manner consistent with some pathologies associated with those dopamine states. We also validated the model against data describing oscillatory properties of BG. We find that the same model displayed detailed features of both ␥-band

The basal ganglia are often conceptualised as three parallel domains that include all the constituent nuclei. The 'ventral domain' appears to be critical for learning flexible behaviours for exploration and foraging, as it is the recipient of converging inputs from amygdala, hippocampal formation and prefrontal cortex, putatively centres for stimulus evaluation, spatial navigation, and planning/contingency, respectively. However, compared to work on the dorsal domains, the rich potential for quantitative theories and models of the ventral domain remains largely untapped, and the purpose of this review is to provide the stimulus for this work. We systematically review the ventral domain's structures and internal organisation, and propose a functional architecture as the basis for computational models. Using a full schematic of the structure of inputs to the ventral striatum (nucleus accumbens core and shell), we argue for the existence of many identifiable processing channels on the basis of unique combinations of afferent inputs. We then identify the potential information represented in these channels by reconciling a broad range of studies from the hippocampal, amygdala and prefrontal cortex literatures with known properties of the ventral striatum from lesion, pharmacological, and electrophysiological studies. Dopamine's key role in learning is reviewed within the three current major computational frameworks; we also show that the shell-based basal ganglia sub-circuits are well placed to generate the phasic burst and dip responses of dopaminergic neurons. We detail dopamine's modulation of ventral basal ganglia's inputs by its actions on pre-synaptic terminals and post-synaptic membranes in the striatum, arguing that the complexity of these effects hint at computational roles for dopamine beyond current ideas. The ventral basal ganglia are revealed as a constellation of multiple functional systems for the learning and selection of flexible behaviours and of behavioural strategies, sharing the common operations of selection-by-disinhibition and of dopaminergic modulation.

The existence of multiple parallel loops connecting sensorimotor systems to the basal ganglia has given rise to proposals that these nuclei serve as a selection mechanism resolving competitions between the alternative actions available in a given context. A strong test of this hypothesis is to require a computational model of the basal ganglia to generate integrated selection sequences in an autonomous agent, we therefore describe a robot architecture into which such a model is embedded, and require it to control action selection in a robotic task inspired by animal observations. Our results demonstrate effective action selection by the embedded model under a wide range of sensory and motivational conditions. When confronted with multiple, high salience alternatives, the robot also exhibits forms of behavioral disintegration that show similarities to animal behavior in conflict situations. The model is shown to cast light on recent neurobiological findings concerning behavioral switching and sequencing.