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.

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

No abstract

Unexpected stimuli which are behaviourally significant have the capacity to evoke a short latency, short duration burst of firing in mesencephalic dopamine neurones. An influential interpretation of the experimental data characterising this response proposes that dopamine neurones play a critical role in reinforcement learning by signalling errors in the prediction of future reward. In the present viewpoint we propose a different functional role for the short latency dopamine response in the mechanisms of associative learning.We suggest that the initial burst of dopaminergic firing may represent an essential component in the process of switching attentional and behavioural selections to unexpected, behaviourally important stimuli. This switching response could be a critical prerequisite for associative learning and may be part of a general short latency reaction, mediated by catecholamines, which prepares the organism to react appropriately to biologically significant events. Introduction:ÒAny act which in a given situation produces satisfaction becomes associated with that situation so that when the situation recurs the act is more likely than before to recur alsoÓ. Although the effects of positive and negative reinforcement on behaviour have been known for centuries, Thorndike 1 in this statement formalised the linking of action to situation on the basis of outcome. It also emphasises two of the principal functions of rewarding or appetitive stimuli: to produce satisfaction (hedonia) and to adjust the probabilities of selecting immediately preceding actions. A third, often recognised function of rewarding stimuli is to elicit approach and consummatory behaviour 2 . While the neural mechanisms mediating any of these processes have yet to be identified in detail, much evidence points to the vertebrate basal ganglia playing a central role 3 . Numerous investigations of this system using a wide range of experimental techniques suggest that ascending dopaminergic projections from the ventral midbrain (substantia nigra pars compacta (SNc) and the ventral tegmental area (VTA)) to the striatum (caudate, putamen and nucleus accumbens) provide essential signals for reinforcement learning 2, 4, 5 . Currently, a popular view is that dopaminergic input to the striatum provides the reinforcement signal required to adjust the probabilities of subsequent action selection [4][5][6][7] . A particularly important and influential part of the evidence supporting this view concerns the short latency, short duration response of dopamine cells observed after the unexpected presentation of a behaviourally significant stimulus 2,8 . This response has been widely interpreted as providing the system with a reinforcement prediction error signal 5, 9 . We will, however, argue that the short latency burst of dopamine activity could have a rather different functional role. Specifically, we suggest that the short latency response may represent an important component of the processes responsible for re-allocating attentional and behavioural re...