Mark D. Humphries

Network 'small-world-ness': A quantitative method for determining canonical network equivalence

Background Many 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 networks 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 Findings We 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/Significance We 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.

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

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.

A physiologically plausible model of action selection and oscillatory activity in the 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 γ-band (30–80 Hz) and slow (∼1 Hz) oscillatory phenomena reported by Brown et al. (2002) and Magill et al. (2001), respectively. Only the parameters required to mimic experimental conditions (e.g., anesthetic) or manipulations (e.g., lesions) were changed. From the results, we derive the following novel predictions about the STN–GP feedback loop: (1) the loop is functionally decoupled by tonic dopamine under normal conditions and recoupled by dopamine depletion; (2) the loop does not show pacemaking activity under normal conditions in vivo (but does after combined dopamine depletion and cortical lesion); (3) the loop has a resonant frequency in the γ-band.

2009 Special Issue: Dopamine-modulated dynamic cell assemblies generated by the GABAergic striatal microcircuit

The striatum, the principal input structure of the basal ganglia, is crucial to both motor control and learning. It receives convergent input from all over the neocortex, hippocampal formation, amygdala and thalamus, and is the primary recipient of dopamine in the brain. Within the striatum is a GABAergic microcircuit that acts upon these inputs, formed by the dominant medium-spiny projection neurons (MSNs) and fast-spiking interneurons (FSIs). There has been little progress in understanding the computations it performs, hampered by the non-laminar structure that prevents identification of a repeating canonical microcircuit. We here begin the identification of potential dynamically-defined computational elements within the striatum. We construct a new three-dimensional model of the striatal microcircuits connectivity, and instantiate this with our dopamine-modulated neuron models of the MSNs and FSIs. A new model of gap junctions between the FSIs is introduced and tuned to experimental data. We introduce a novel multiple spike-train analysis method, and apply this to the outputs of the model to find groups of synchronised neurons at multiple time-scales. We find that, with realistic in vivo background input, small assemblies of synchronised MSNs spontaneously appear, consistent with experimental observations, and that the number of assemblies and the time-scale of synchronisation is strongly dependent on the simulated concentration of dopamine. We also show that feed-forward inhibition from the FSIs counter-intuitively increases the firing rate of the MSNs. Such small cell assemblies forming spontaneously only in the absence of dopamine may contribute to motor control problems seen in humans and animals following a loss of dopamine cells.

The role of intra-thalamic and thalamocortical circuits in action selection

We previously proposed that the basal ganglia (BG) play a crucial role in action selection. Quantitative analysis and simulation of a computational model of the intrinsic BG demonstrated that its output was consistent with this proposition. Here we build on that model by embedding it into a wider circuit containing the motor thalamocortical loop and thalamic reticular nucleus (TRN). Simulation of this extended model showed that the additions gave five main results which are desirable in a selection/switching mechanism. First, low salience actions (i.e. those with low urgency) could be selected. Second, the range of salience values over which actions could be switched between was increased. Third, the contrast between the selected and non-selected actions was enhanced via improved differentiation of outputs from the BG. Fourth, transient increases in the salience of a non-selected action were prevented from interrupting the ongoing action, unless the transient was of sufficient magnitude. Finally, the selection of the ongoing action persisted when a new closely matched salience action became active. The first result was facilitated by the thalamocortical loop; the rest were dependent on the presence of the TRN. Thus, we conclude that the results are consistent with these structures having clearly defined functions in action selection.

Dopaminergic control of the exploration-exploitation trade-off via the basal ganglia

We continuously face the dilemma of choosing between actions that gather new information or actions that exploit existing knowledge. This “exploration-exploitation” trade-off depends on the environment: stability favors exploiting knowledge to maximize gains; volatility favors exploring new options and discovering new outcomes. Here we set out to reconcile recent evidence for dopamine’s involvement in the exploration-exploitation trade-off with the existing evidence for basal ganglia control of action selection, by testing the hypothesis that tonic dopamine in the striatum, the basal ganglia’s input nucleus, sets the current exploration-exploitation trade-off. We first advance the idea of interpreting the basal ganglia output as a probability distribution function for action selection. Using computational models of the full basal ganglia circuit, we showed that, under this interpretation, the actions of dopamine within the striatum change the basal ganglia’s output to favor the level of exploration or exploitation encoded in the probability distribution. We also found that our models predict striatal dopamine controls the exploration-exploitation trade-off if we instead read-out the probability distribution from the target nuclei of the basal ganglia, where their inhibitory input shapes the cortical input to these nuclei. Finally, by integrating the basal ganglia within a reinforcement learning model, we showed how dopamine’s effect on the exploration-exploitation trade-off could be measurable in a forced two-choice task. These simulations also showed how tonic dopamine can appear to affect learning while only directly altering the trade-off. Thus, our models support the hypothesis that changes in tonic dopamine within the striatum can alter the exploration-exploitation trade-off by modulating the output of the basal ganglia.

Is there a brainstem substrate for action selection

The search for the neural substrate of vertebrate action selection has focused on structures in the forebrain and midbrain, and particularly on the group of sub-cortical nuclei known as the basal ganglia. Yet, the behavioural repertoire of decerebrate and neonatal animals suggests the existence of a relatively self-contained neural substrate for action selection in the brainstem. We propose that the medial reticular formation (mRF) is the substrates main component and review evidence showing that the mRFs inputs, outputs and intrinsic organization are consistent with the requirements of an action-selection system. The internal architecture of the mRF is composed of interconnected neuron clusters. We present an anatomical model which suggests that the mRFs intrinsic circuitry constitutes a small-world network and extend this result to show that it may have evolved to reduce axonal wiring. Potential configurations of action representation within the internal circuitry of the mRF are then assessed by computational modelling. We present new results demonstrating that each clusters output is most likely to represent activation of a component action; thus, coactivation of a set of these clusters would lead to the coordinated behavioural response observed in the animal. Finally, we consider the potential integration of the basal ganglia and mRF substrates for selection and suggest that they may collectively form a layered/hierarchical control system.

Spike-Train Communities: Finding Groups of Similar Spike Trains

Identifying similar spike-train patterns is a key element in understanding neural coding and computation. For single neurons, similar spike patterns evoked by stimuli are evidence of common coding. Across multiple neurons, similar spike trains indicate potential cell assemblies. As recording technology advances, so does the urgent need for grouping methods to make sense of large-scale datasets of spike trains. Existing methods require specifying the number of groups in advance, limiting their use in exploratory analyses. I derive a new method from network theory that solves this key difficulty: it self-determines the maximum number of groups in any set of spike trains, and groups them to maximize intragroup similarity. This method brings us revealing new insights into the encoding of aversive stimuli by dopaminergic neurons, and the organization of spontaneous neural activity in cortex. I show that the characteristic pause response of a rats dopaminergic neuron depends on the state of the superior colliculus: when it is inactive, aversive stimuli invoke a single pattern of dopaminergic neuron spiking; when active, multiple patterns occur, yet the spike timing in each is reliable. In spontaneous multineuron activity from the cortex of anesthetized cat, I show the existence of neural ensembles that evolve in membership and characteristic timescale of organization during global slow oscillations. I validate these findings by showing that the method both is remarkably reliable at detecting known groups and can detect large-scale organization of dynamics in a model of the striatum.

A new framework for cortico-striatal plasticity: behavioural theory meets in vitro data at the reinforcement-action interface.

A computational model yields new insights into the bewildering complexity of cortico-striatal plasticity and its rationale for supporting operant learning.

Capturing dopaminergic modulation and bimodal membrane behaviour of striatal medium spiny neurons in accurate, reduced models

Loss of dopamine from the striatum can cause both profound motor deficits, as in Parkinsons disease, and disrupt learning. Yet the effect of dopamine on striatal neurons remains a complex and controversial topic, and is in need of a comprehensive framework. We extend a reduced model of the striatal medium spiny neuron (MSN) to account for dopaminergic modulation of its intrinsic ion channels and synaptic inputs. We tune our D1 and D2 receptor MSN models using data from a recent large-scale compartmental model. The new models capture the input–output relationships for both current injection and spiking input with remarkable accuracy, despite the order of magnitude decrease in system size. They also capture the paired pulse facilitation shown by MSNs. Our dopamine models predict that synaptic effects dominate intrinsic effects for all levels of D1 and D2 receptor activation. We analytically derive a full set of equilibrium points and their stability for the original and dopamine modulated forms of the MSN model. We find that the stability types are not changed by dopamine activation, and our models predict that the MSN is never bistable. Nonetheless, the MSN models can produce a spontaneously bimodal membrane potential similar to that recently observed in vitro following application of NMDA agonists. We demonstrate that this bimodality is created by modelling the agonist effects as slow, irregular and massive jumps in NMDA conductance and, rather than a form of bistability, is due to the voltage-dependent blockade of NMDA receptors. Our models also predict a more pronounced membrane potential bimodality following D1 receptor activation. This work thus establishes reduced yet accurate dopamine-modulated models of MSNs, suitable for use in large-scale models of the striatum. More importantly, these provide a tractable framework for further study of dopamines effects on computation by individual neurons.