Models of cortical networks with long-range patchy projections

The cortex exhibits an intricate vertical and horizontal architecture, the latter often featuring spatially clustered projection patterns, so-called patches. Many network studies of cortical dynamics ignore such spatial structures and assume purely random wiring. Here, we focus on non-random network structures provided by long-range horizontal (patchy) connections that remain inside the gray matter. We investigate how the spatial arrangement of patchy projections influences global network topology and predict its impact on the activity dynamics of the network. Since neuroanatomical data on horizontal projections is rather sparse, we suggest and compare four candidate scenarios of how patchy connections may be established. To identify a set of characteristic network properties that enables us to pin down the differences between the resulting network models, we employ the framework of stochastic graph theory. We find that patchy projections provide an exceptionally efficient way of wiring, as the resulting networks tend to exhibit small-world properties with significantly reduced wiring costs. Furthermore, the eigenvalue spectra, as well as the structure of common in- and output of the networks suggest that different spatial connectivity patterns support distinct types of activity propagation.     

Towards a virtual C. elegans: a framework for simulation and visualization of the neuromuscular system in a 3D physical environment

The nematode C. elegans is the only animal with a known neuronal wiring diagram, or "connectome". During the last three decades, extensive studies of the C. elegans have provided wide-ranging data about it, but few systematic ways of integrating these data into a dynamic model have been put forward. Here we present a detailed demonstration of a virtual C. elegans aimed at integrating these data in the form of a 3D dynamic model operating in a simulated physical environment. Our current demonstration includes a realistic flexible worm body model, muscular system and a partially implemented ventral neural cord. Our virtual C. elegans demonstrates successful forward and backward locomotion when sending sinusoidal patterns of neuronal activity to groups of motor neurons. To account for the relatively slow propagation velocity and the attenuation of neuronal signals, we introduced "pseudo neurons" into our model to simulate simplified neuronal dynamics. The pseudo neurons also provide a good way of visualizing the nervous system's structure and activity dynamics.     

SynCAM 1 adhesion dynamically regulates synapse number and impacts plasticity and learning

Synaptogenesis is required for wiring neuronal circuits in the developing brain and continues to remodel adult networks. However, the molecules organizing synapse development and maintenance in?vivo remain incompletely understood. We now demonstrate that the immunoglobulin adhesion molecule SynCAM 1 dynamically alters synapse number and plasticity. Overexpression of SynCAM 1 in transgenic mice promotes excitatory synapse number, while loss of SynCAM 1 results in fewer excitatory synapses. By turning off SynCAM 1 overexpression in transgenic brains, we show that it maintains the newly induced synapses. SynCAM 1 also functions at mature synapses to alter their plasticity by regulating long-term depression. Consistent with these effects on neuronal connectivity, SynCAM 1 expression affects spatial learning, with knock-out mice learning better. The reciprocal effects of increased SynCAM 1 expression and loss reveal that this adhesion molecule contributes to the regulation of synapse number and plasticity, and impacts how neuronal networks undergo activity-dependent changes.     

Topological cluster analysis reveals the systemic organization of the Caenorhabditis elegans connectome

The modular organization of networks of individual neurons interwoven through synapses has not been fully explored due to the incredible complexity of the connectivity architecture. Here we use the modularity-based community detection method for directed, weighted networks to examine hierarchically organized modules in the complete wiring diagram (connectome) of Caenorhabditis elegans (C. elegans) and to investigate their topological properties. Incorporating bilateral symmetry of the network as an important cue for proper cluster assignment, we identified anatomical clusters in the C. elegans connectome, including a body-spanning cluster, which correspond to experimentally identified functional circuits. Moreover, the hierarchical organization of the five clusters explains the systemic cooperation (e.g., mechanosensation, chemosensation, and navigation) that occurs among the structurally segregated biological circuits to produce higher-order complex behaviors.     

Variability v.s. synchronicity of neuronal activity in local cortical network models with different wiring topologies

Dynamical behavior of a biological neuronal network depends significantly on the spatial pattern of synaptic connections among neurons. While neuronal network dynamics has extensively been studied with simple wiring patterns, such as all-to-all or random synaptic connections, not much is known about the activity of networks with more complicated wiring topologies. Here, we examined how different wiring topologies may influence the response properties of neuronal networks, paying attention to irregular spike firing, which is known as a characteristic of in vivo cortical neurons, and spike synchronicity. We constructed a recurrent network model of realistic neurons and systematically rewired the recurrent synapses to change the network topology, from a localized regular and a “small-world” network topology to a distributed random network topology. Regular and small-world wiring patterns greatly increased the irregularity or the coefficient of variation (Cv) of output spike trains, whereas such an increase was small in random connectivity patterns. For given strength of recurrent synapses, the firing irregularity exhibited monotonous decreases from the regular to the random network topology. By contrast, the spike coherence between an arbitrary neuron pair exhibited a non-monotonous dependence on the topological wiring pattern. More precisely, the wiring pattern to maximize the spike coherence varied with the strength of recurrent synapses. In a certain range of the synaptic strength, the spike coherence was maximal in the small-world network topology, and the long-range connections introduced in this wiring changed the dependence of spike synchrony on the synaptic strength moderately. However, the effects of this network topology were not really special in other properties of network activity. Action Editor: Xiao-Jing Wang     

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.     

The brain's polymath: Emerging roles of microglia throughout brain development

Microglia, the resident brain immune cells, have garnered a reputation as major effectors of circuit wiring due to their ability to prune synapses. Other roles of microglia in regulating neuronal circuit development have so far received comparatively less attention. Here, we review the latest studies that have contributed to our increased understanding of how microglia regulate brain wiring beyond their role in synapse pruning. We summarize recent findings showing that microglia regulate neuronal numbers and influence neuronal connectivity through a bidirectional communication between microglia and neurons, processes regulated by neuronal activity and the remodeling of the extracellular matrix. Finally, we speculate on the potential contribution of microglia to the development of functional networks and propose an integrative view of microglia as active elements of neural circuits.     

Topology of gap junction networks in C. elegans

Gap junctions are prevalent in every nervous system, but their role in information processing remains largely unknown. In C. elegans, the role of gap junctional communication in touch sensitivity has been demonstrated. In this animal, the entire complement of gap junctions in the nervous system is documented, therefore providing a good model for the computational investigation of circuit functions of gap junctions.We explored several hypotheses about the role of gap junctions in the nervous system of C. elegans by systematically analysing an anatomical database with recursive algorithms. We find that gap junctions connect different sets of neurons from those connected by chemical synapses. In addition, when analysing the topology of the gap-junction networks, we find that, surprisingly, most (92%) neurons in the worm are linked in a single gap-junction network. The worm nervous system can only be divided into smaller networks by assuming that two or more gap junctions are necessary for functional coupling or that neural activity has limited propagation. However, these groups, and others identified using algorithms with subsets or combinations of restrictive criteria, do not correspond to any known circuits identified in genetic and behavioral studies. Finally, we notice that the function of some gap junctions appears linked to their precise location on the neuronal processes. We propose that the location of the gap junctions within the neuron determines their functional role.     

Local cortical circuit model inferred from power-law distributed neuronal avalanches

How cortical neurons process information crucially depends on how their local circuits are organized. Spontaneous synchronous neuronal activity propagating through neocortical slices displays highly diverse, yet repeatable, activity patterns called “neuronal avalanches”. They obey power-law distributions of the event sizes and lifetimes, presumably reflecting the structure of local circuits developed in slice cultures. However, the explicit network structure underlying the power-law statistics remains unclear. Here, we present a neuronal network model of pyramidal and inhibitory neurons that enables stable propagation of avalanche-like spiking activity. We demonstrate a neuronal wiring rule that governs the formation of mutually overlapping cell assemblies during the development of this network. The resultant network comprises a mixture of feedforward chains and recurrent circuits, in which neuronal avalanches are stable if the former structure is predominant. Interestingly, the recurrent synaptic connections formed by this wiring rule limit the number of cell assemblies embeddable in a neuron pool of given size. We investigate how the resultant power laws depend on the details of the cell-assembly formation as well as on the inhibitory feedback. Our model suggests that local cortical circuits may have a more complex topological design than has previously been thought. Competing financial interests: The authors declare that they have no competing financial interests. Action Editor: Peter Latham     

Wave speed in excitable random networks with spatially constrained connections

Very fast oscillations (VFO) in neocortex are widely observed before epileptic seizures, and there is growing evidence that they are caused by networks of pyramidal neurons connected by gap junctions between their axons. We are motivated by the spatio-temporal waves of activity recorded using electrocorticography (ECoG), and study the speed of activity propagation through a network of neurons axonally coupled by gap junctions. We simulate wave propagation by excitable cellular automata (CA) on random (Erdös-Rényi) networks of special type, with spatially constrained connections. From the cellular automaton model, we derive a mean field theory to predict wave propagation. The governing equation resolved by the Fisher-Kolmogorov PDE fails to describe wave speed. A new (hyperbolic) PDE is suggested, which provides adequate wave speed that saturates with network degree , in agreement with intuitive expectations and CA simulations. We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length. When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments rather than on mean degree , which is explained by general network theory. The wave speeds are strikingly similar in a diverse set of networks, including regular, Poisson, exponential and power law distributions, supporting our theory for various network topologies. Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.     

Reconstructing the three-dimensional GABAergic microcircuit of the striatum

A system's wiring constrains its dynamics, yet modelling of neural structures often overlooks the specific networks formed by their neurons. We developed an approach for constructing anatomically realistic networks and reconstructed the GABAergic microcircuit formed by the medium spiny neurons (MSNs) and fast-spiking interneurons (FSIs) of the adult rat striatum. We grew dendrite and axon models for these neurons and extracted probabilities for the presence of these neurites as a function of distance from the soma. From these, we found the probabilities of intersection between the neurites of two neurons given their inter-somatic distance, and used these to construct three-dimensional striatal networks. The MSN dendrite models predicted that half of all dendritic spines are within 100μm of the soma. The constructed networks predict distributions of gap junctions between FSI dendrites, synaptic contacts between MSNs, and synaptic inputs from FSIs to MSNs that are consistent with current estimates. The models predict that to achieve this, FSIs should be at most 1% of the striatal population. They also show that the striatum is sparsely connected: FSI-MSN and MSN-MSN contacts respectively form 7% and 1.7% of all possible connections. The models predict two striking network properties: the dominant GABAergic input to a MSN arises from neurons with somas at the edge of its dendritic field; and FSIs are inter-connected on two different spatial scales: locally by gap junctions and distally by synapses. We show that both properties influence striatal dynamics: the most potent inhibition of a MSN arises from a region of striatum at the edge of its dendritic field; and the combination of local gap junction and distal synaptic networks between FSIs sets a robust input-output regime for the MSN population. Our models thus intimately link striatal micro-anatomy to its dynamics, providing a biologically grounded platform for further study.     

Structure-Dynamics Relationships in Bursting Neuronal Networks Revealed Using a Prediction Framework

The question of how the structure of a neuronal network affects its functionality has gained a lot of attention in neuroscience. However, the vast majority of the studies on structure-dynamics relationships consider few types of network structures and assess limited numbers of structural measures. In this in silico study, we employ a wide diversity of network topologies and search among many possibilities the aspects of structure that have the greatest effect on the network excitability. The network activity is simulated using two point-neuron models, where the neurons are activated by noisy fluctuation of the membrane potential and their connections are described by chemical synapse models, and statistics on the number and quality of the emergent network bursts are collected for each network type. We apply a prediction framework to the obtained data in order to find out the most relevant aspects of network structure. In this framework, predictors that use different sets of graph-theoretic measures are trained to estimate the activity properties, such as burst count or burst length, of the networks. The performances of these predictors are compared with each other. We show that the best performance in prediction of activity properties for networks with sharp in-degree distribution is obtained when the prediction is based on clustering coefficient. By contrast, for networks with broad in-degree distribution, the maximum eigenvalue of the connectivity graph gives the most accurate prediction. The results shown for small () networks hold with few exceptions when different neuron models, different choices of neuron population and different average degrees are applied. We confirm our conclusions using larger () networks as well. Our findings reveal the relevance of different aspects of network structure from the viewpoint of network excitability, and our integrative method could serve as a general framework for structure-dynamics studies in biosciences.     

Beyond the connectome: how neuromodulators shape neural circuits

Powerful ultrastructural tools are providing new insights into neuronal circuits, revealing a wealth of anatomically-defined synaptic connections. These wiring diagrams are incomplete, however, because functional connectivity is actively shaped by neuromodulators that modify neuronal dynamics, excitability, and synaptic function. Studies of defined neural circuits in crustaceans, C. elegans, Drosophila, and the vertebrate retina have revealed the ability of modulators and sensory context to reconfigure information processing by changing the composition and activity of functional circuits. Each ultrastructural connectivity map encodes multiple circuits, some of which are active and some of which are latent at any given time.     

Beyond the wiring diagram: signalling through complex neuromodulator networks

During the computations performed by the nervous system, its ‘wiring diagram’—the map of its neurons and synaptic connections—is dynamically modified and supplemented by multiple actions of neuromodulators that can be so complex that they can be thought of as constituting a biochemical network that combines with the neuronal network to perform the computation. Thus, the neuronal wiring diagram alone is not sufficient to specify, and permit us to understand, the computation that underlies behaviour. Here I review how such modulatory networks operate, the problems that their existence poses for the experimental study and conceptual understanding of the computations performed by the nervous system, and how these problems may perhaps be solved and the computations understood by considering the structural and functional ‘logic’ of the modulatory networks.     

Shapes of astrocyte networks in the juvenile brain

The high level of intercellular communication mediated by gap junctions between astrocytes indicates that, besides individual astrocytic domains, a second level of organization might exist for these glial cells as they form communicating networks. Therefore,the contribution of astrocytes to brain function should also be considered to result from coordinated groups of cells. To evaluate the shape and extent of these networks we have studied the expression of connexin 43, a major gap junction protein in astrocytes, and the intercellular diffusion of gap junction tracers in two structures of the developing brain, the hippocampus and the cerebral cortex. We report that the shape of astrocytic networks depends on their location within neuronal compartments ina defined brain structure. Interestingly, not all astrocytes are coupled, which indicates that connections within these networks are restricted. As gap junctional communication in astrocytes is reported to contribute to several glial functions, differences in the shape of astrocytic networks might have consequences on neuronal activity and survival.     

A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

Coemergence of regularity and complexity during neural network development

With the growing recognition that rhythmic and oscillatory patterns are widespread in the brain and play important roles in all aspects of the function of our nervous system, there has been a resurgence of interest in neuronal synchronized bursting activity. Here, we were interested in understanding the development of synchronized bursts as information-bearing neuronal activity patterns. For that, we have monitored the morphological organization and spontaneous activity of neuronal networks cultured on multielectrode-arrays during their self-executed evolvement from a mixture of dissociated cells into an active network. Complex collective network electrical activity evolved from sporadic firing patterns of the single neurons. On the system (network) level, the activity was marked by bursting events with interneuronal synchronization and nonarbitrary temporal ordering. We quantified these individual-to-collective activity transitions using newly-developed system level quantitative measures of time series regularity and complexity. We found that individual neuronal activity before synchronization was characterized by high regularity and low complexity. During neuronal wiring, there was a transient period of reorganization marked by low regularity, which then leads to coemergence of elevated regularity and functional (nonstochastic) complexity. We further investigated the morphology-activity interplay by modeling artificial neuronal networks with different topological organizations and connectivity schemes. The simulations support our experimental results by showing increased levels of complexity of neuronal activity patterns when neurons are wired up and organized in clusters (similar to mature real networks), as well as network-level activity regulation once collective activity forms.     

Innate synchronous oscillations in freely-organized small neuronal circuits

Background

Information processing in neuronal networks relies on the network''s ability to generate temporal patterns of action potentials. Although the nature of neuronal network activity has been intensively investigated in the past several decades at the individual neuron level, the underlying principles of the collective network activity, such as the synchronization and coordination between neurons, are largely unknown. Here we focus on isolated neuronal clusters in culture and address the following simple, yet fundamental questions: What is the minimal number of cells needed to exhibit collective dynamics? What are the internal temporal characteristics of such dynamics and how do the temporal features of network activity alternate upon crossover from minimal networks to large networks?

Methodology/Principal Findings

We used network engineering techniques to induce self-organization of cultured networks into neuronal clusters of different sizes. We found that small clusters made of as few as 40 cells already exhibit spontaneous collective events characterized by innate synchronous network oscillations in the range of 25 to 100 Hz. The oscillation frequency of each network appeared to be independent of cluster size. The duration and rate of the network events scale with cluster size but converge to that of large uniform networks. Finally, the investigation of two coupled clusters revealed clear activity propagation with master/slave asymmetry.

Conclusions/Significance

The nature of the activity patterns observed in small networks, namely the consistent emergence of similar activity across networks of different size and morphology, suggests that neuronal clusters self-regulate their activity to sustain network bursts with internal oscillatory features. We therefore suggest that clusters of as few as tens of cells can serve as a minimal but sufficient functional network, capable of sustaining oscillatory activity. Interestingly, the frequencies of these oscillations are similar those observed in vivo.     

Signal propagation in feedforward neuronal networks with unreliable synapses

In this paper, we systematically investigate both the synfire propagation and firing rate propagation in feedforward neuronal network coupled in an all-to-all fashion. In contrast to most earlier work, where only reliable synaptic connections are considered, we mainly examine the effects of unreliable synapses on both types of neural activity propagation in this work. We first study networks composed of purely excitatory neurons. Our results show that both the successful transmission probability and excitatory synaptic strength largely influence the propagation of these two types of neural activities, and better tuning of these synaptic parameters makes the considered network support stable signal propagation. It is also found that noise has significant but different impacts on these two types of propagation. The additive Gaussian white noise has the tendency to reduce the precision of the synfire activity, whereas noise with appropriate intensity can enhance the performance of firing rate propagation. Further simulations indicate that the propagation dynamics of the considered neuronal network is not simply determined by the average amount of received neurotransmitter for each neuron in a time instant, but also largely influenced by the stochastic effect of neurotransmitter release. Second, we compare our results with those obtained in corresponding feedforward neuronal networks connected with reliable synapses but in a random coupling fashion. We confirm that some differences can be observed in these two different feedforward neuronal network models. Finally, we study the signal propagation in feedforward neuronal networks consisting of both excitatory and inhibitory neurons, and demonstrate that inhibition also plays an important role in signal propagation in the considered networks.     

Electrical Transmission between Mammalian Neurons is Supported by a Small Fraction of Gap Junction Channels

Electrical synapses formed by gap junctions between neurons create networks of electrically coupled neurons in the mammalian brain, where these networks have been found to play important functional roles. In most cases, interneuronal gap junctions occur at remote dendro-dendritic contacts, making difficult accurate characterization of their physiological properties and correlation of these properties with their anatomical and morphological features of the gap junctions. In the mesencephalic trigeminal (MesV) nucleus where neurons are readily accessible for paired electrophysiological recordings in brain stem slices, our recent data indicate that electrical transmission between MesV neurons is mediated by connexin36 (Cx36)-containing gap junctions located at somato-somatic contacts. We here review evidence indicating that electrical transmission between these neurons is supported by a very small fraction of the gap junction channels present at cell-cell contacts. Acquisition of this evidence was enabled by the unprecedented experimental access of electrical synapses between MesV neurons, which allowed estimation of the average number of open channels mediating electrical coupling in relation to the average number of gap junction channels present at these contacts. Our results indicate that only a small proportion of channels (~0.1?%) appear to be conductive. On the basis of similarities with other preparations, we postulate that this phenomenon might constitute a general property of vertebrate electrical synapses, reflecting essential aspects of gap junction function and maintenance.