Synchronous neural activity in scale-free network models versus random network models

Analytic relationship of relative synchronizability to network structure and motifs

Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time autoregressive processes and continuous-time Ornstein-Uhlenbeck dynamics on networks, which can represent linearizations of nonlinear systems. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronization specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks.

Small-world connectivity dictates collective endothelial cell signaling

Every blood vessel is lined by a single layer of highly specialized, yet adaptable and multifunctional endothelial cells. These cells, the endothelium, control vascular contractility, hemostasis, and inflammation and regulate the exchange of oxygen, nutrients, and waste products between circulating blood and tissue. To control each function, the endothelium processes endlessly arriving requests from multiple sources using separate clusters of cells specialized to detect specific stimuli. A well-developed but poorly understood communication system operates between cells to integrate multiple lines of information and coordinate endothelial responses. Here, the nature of the communication network has been addressed using single-cell Ca2+ imaging across thousands of endothelial cells in intact blood vessels. Cell activities were cross-correlated and compared to a stochastic model to determine network connections. Highly correlated Ca2+ activities occurred in scattered cell clusters, and network communication links between them exhibited unexpectedly short path lengths. The number of connections between cells (degree distribution) followed a power-law relationship revealing a scale-free network topology. The path length and degree distribution revealed an endothelial network with a “small-world” configuration. The small-world configuration confers particularly dynamic endothelial properties including high signal-propagation speed, stability, and a high degree of synchronizability. Local activation of small clusters of cells revealed that the short path lengths and rapid signal transmission were achieved by shortcuts via connecting extensions to nonlocal cells. These findings reveal that the endothelial network design is effective for local and global efficiency in the interaction of the cells and rapid and robust communication between endothelial cells in order to efficiently control cardiovascular activity.

In-vitro engineered human cerebral tissues mimic pathological circuit disturbances in 3D

In-vitro modeling of brain network disorders such as epilepsy remains a major challenge. A critical step is to develop an experimental approach that enables recapitulation of in-vivo-like three-dimensional functional complexity while allowing local modulation of the neuronal networks. Here, by promoting matrix-supported active cell reaggregation, we engineered multiregional cerebral tissues with intact 3D neuronal networks and functional interconnectivity characteristic of brain networks. Furthermore, using a multi-chambered tissue-culture chip, we show that our separated but interconnected cerebral tissues can mimic neuropathological signatures such as the propagation of epileptiform discharges.

A method is developed to engineer cerebral tissues with intact 3D neuronal networks, mimicking neuropathological signatures such as the propagation of epileptiform discharges, using a multi-chambered tissue culture chip.

Network Model With Reduced Metabolic Rate Predicts Spatial Synchrony of Neuronal Activity

In a cerebral hypometabolic state, cortical neurons exhibit slow synchronous oscillatory activity with sparse firing. How such a synchronization spatially organizes as the cerebral metabolic rate decreases have not been systemically investigated. We developed a network model of leaky integrate-and-fire neurons with an additional dependency on ATP dynamics. Neurons were scattered in a 2D space, and their population activity patterns at varying ATP levels were simulated. The model predicted a decrease in firing activity as the ATP production rate was lowered. Under hypometabolic conditions, an oscillatory firing pattern, that is, an ON-OFF cycle arose through a failure of sustainable firing due to reduced excitatory positive feedback and rebound firing after the slow recovery of ATP concentration. The firing rate oscillation of distant neurons developed at first asynchronously that changed into burst suppression and global synchronization as ATP production further decreased. These changes resembled the experimental data obtained from anesthetized rats, as an example of a metabolically suppressed brain. Together, this study substantiates a novel biophysical mechanism of neuronal network synchronization under limited energy supply conditions.

Formulation of Pruning Maps with Rhythmic Neural Firing

Rhythmic neural firing is thought to underlie the operation of neural function. This triggers the construction of dynamical network models to investigate how the rhythms interact with each other. Recently, an approach concerning neural path pruning has been proposed in a dynamical network system, in which critical neuronal connections are identified and adjusted according to the pruning maps, enabling neurons to produce rhythmic, oscillatory activity in simulation. Here, we construct a sort of homomorphic functions based on different rhythms of neural firing in network dynamics. Armed with the homomorphic functions, the pruning maps can be simply expressed in terms of interactive rhythms of neural firing and allow a concrete analysis of coupling operators to control network dynamics. Such formulation of pruning maps is applied to probe the consolidation of rhythmic patterns between layers of neurons in feedforward neural networks.

Recurrently connected and localized neuronal communities initiate coordinated spontaneous activity in neuronal networks

Developing neuronal systems intrinsically generate coordinated spontaneous activity that propagates by involving a large number of synchronously firing neurons. In vivo, waves of spikes transiently characterize the activity of developing brain circuits and are fundamental for activity-dependent circuit formation. In vitro, coordinated spontaneous spiking activity, or network bursts (NBs), interleaved within periods of asynchronous spikes emerge during the development of 2D and 3D neuronal cultures. Several studies have investigated this type of activity and its dynamics, but how a neuronal system generates these coordinated events remains unclear. Here, we investigate at a cellular level the generation of network bursts in spontaneously active neuronal cultures by exploiting high-resolution multielectrode array recordings and computational network modelling. Our analysis reveals that NBs are generated in specialized regions of the network (functional neuronal communities) that feature neuronal links with high cross-correlation peak values, sub-millisecond lags and that share very similar structural connectivity motifs providing recurrent interactions. We show that the particular properties of these local structures enable locally amplifying spontaneous asynchronous spikes and that this mechanism can lead to the initiation of NBs. Through the analysis of simulated and experimental data, we also show that AMPA currents drive the coordinated activity, while NMDA and GABA currents are only involved in shaping the dynamics of NBs. Overall, our results suggest that the presence of functional neuronal communities with recurrent local connections allows a neuronal system to generate spontaneous coordinated spiking activity events. As suggested by the rules used for implementing our computational model, such functional communities might naturally emerge during network development by following simple constraints on distance-based connectivity.

Coordinated spontaneous spiking activity is fundamental for the normal formation of brain circuits during development. However, how ensembles of neurons generate these events remains unclear. To address this question, in the present study, we investigated the network properties that might be required to a neuronal system for the generation of these spontaneous waves of spikes. We performed our study on spontaneously active neuronal cell cultures using high-resolution electrical recordings and a computational network model developed to reproduce our experimental data both quantitatively and qualitatively. Through the analysis of both experimental and simulated data, we found that network bursts are initiated in regions of the network, or “functional communities”, characterized by particular local connectivity properties. We also found that these regions can amplify the background asynchronous spiking activity preceding a network burst and, in this way, can give rise to coordinated spiking events.

As a whole, our results suggest the presence of functional communities of neurons in a developing neuronal system that might naturally emerge by following simple constraints on distance-based connectivity. These regions are most likely required for the generation of the spontaneous coordinated activity that can drive activity-dependent circuit formation.

Organization and control of epileptic circuits in temporal lobe epilepsy

When studying the pathological mechanisms of epilepsy, there are a seemingly endless number of approaches from the ultrastructural level-receptor expression by EM-to the behavioral level-comorbid depression in behaving animals. Epilepsy is characterized as a disorder of recurrent seizures, which are defined as "a transient occurrence of signs and/or symptoms due to abnormal excessive or synchronous neuronal activity in the brain" (Fisher et al., 2005). Such abnormal activity typically does not occur in a single isolated neuron; rather, it results from pathological activity in large groups-or circuits-of neurons. Here we choose to focus on two aspects of aberrant circuits in temporal lobe epilepsy: their organization and potential mechanisms to control these pathological circuits. We also look at two scales: microcircuits, ie, the relationship between individual neurons or small groups of similar neurons, and macrocircuits, ie, the organization of large-scale brain regions. We begin by summarizing the large body of literature that describes the stereotypical anatomical changes in the temporal lobe-ie, the anatomical basis of alterations in microcircuitry. We then offer a brief introduction to graph theory and describe how this type of mathematical analysis, in combination with computational neuroscience techniques and using parameters obtained from experimental data, can be used to postulate how microcircuit alterations may lead to seizures. We then zoom out and look at the changes which are seen over large whole-brain networks in patients and animal models, and finally we look to the future.

Development and function of human cerebral cortex neural networks from pluripotent stem cells in vitro

A key aspect of nervous system development, including that of the cerebral cortex, is the formation of higher-order neural networks. Developing neural networks undergo several phases with distinct activity patterns in vivo, which are thought to prune and fine-tune network connectivity. We report here that human pluripotent stem cell (hPSC)-derived cerebral cortex neurons form large-scale networks that reflect those found in the developing cerebral cortex in vivo. Synchronised oscillatory networks develop in a highly stereotyped pattern over several weeks in culture. An initial phase of increasing frequency of oscillations is followed by a phase of decreasing frequency, before giving rise to non-synchronous, ordered activity patterns. hPSC-derived cortical neural networks are excitatory, driven by activation of AMPA- and NMDA-type glutamate receptors, and can undergo NMDA-receptor-mediated plasticity. Investigating single neuron connectivity within PSC-derived cultures, using rabies-based trans-synaptic tracing, we found two broad classes of neuronal connectivity: most neurons have small numbers (<10) of presynaptic inputs, whereas a small set of hub-like neurons have large numbers of synaptic connections (>40). These data demonstrate that the formation of hPSC-derived cortical networks mimics in vivo cortical network development and function, demonstrating the utility of in vitro systems for mechanistic studies of human forebrain neural network biology.

Summary: Human PSC-derived cerebral cortex neurons form large-scale functional networks that change over time and mimic those found in the developing cerebral cortex in vivo.

Extending the two faces of subjective randomness: From the gambler's and hot-hand fallacies toward a hierarchy of binary sequence perception

In this study, we examined perceptions of binary sequences under uncertainty in an attempt to depict a holistic and unifying framework. The first experiment applied a projection method that motivated participants to observe binary series and provide descriptions of their possible underlying mechanisms or processes. This procedure revealed four distinct perceptual categories: two previously studied categories of chance mechanisms and human performance, associated with the gambler's and hot-hand fallacies, and two newly identified categories-periods and processes and traits and preferences. The next three experiments tested the associations between the four categories and the alternation rates of the observed sequences under three categorical decisions structures: screening, discrimination, and classification. The results reveal the relativity of binary sequence perception. They show that the categories of chance mechanisms and periods and processes reflected rather stable perception across all tested conditions, whereas the other two categories were more susceptible to the context in which they were embedded. The findings support previous research on the gambler's fallacy and show that the hot-hand fallacy is confined to comparisons of human performance and chance mechanisms. A proposed developmental hierarchy suggests that all four categories embody basic cognitive structures that assist in detecting, decoding, and interpreting both inanimate and social aspects of the environment.

Multiplex networks of cortical and hippocampal neurons revealed at different timescales

Recent studies have emphasized the importance of multiplex networks--interdependent networks with shared nodes and different types of connections--in systems primarily outside of neuroscience. Though the multiplex properties of networks are frequently not considered, most networks are actually multiplex networks and the multiplex specific features of networks can greatly affect network behavior (e.g. fault tolerance). Thus, the study of networks of neurons could potentially be greatly enhanced using a multiplex perspective. Given the wide range of temporally dependent rhythms and phenomena present in neural systems, we chose to examine multiplex networks of individual neurons with time scale dependent connections. To study these networks, we used transfer entropy--an information theoretic quantity that can be used to measure linear and nonlinear interactions--to systematically measure the connectivity between individual neurons at different time scales in cortical and hippocampal slice cultures. We recorded the spiking activity of almost 12,000 neurons across 60 tissue samples using a 512-electrode array with 60 micrometer inter-electrode spacing and 50 microsecond temporal resolution. To the best of our knowledge, this preparation and recording method represents a superior combination of number of recorded neurons and temporal and spatial recording resolutions to any currently available in vivo system. We found that highly connected neurons ("hubs") were localized to certain time scales, which, we hypothesize, increases the fault tolerance of the network. Conversely, a large proportion of non-hub neurons were not localized to certain time scales. In addition, we found that long and short time scale connectivity was uncorrelated. Finally, we found that long time scale networks were significantly less modular and more disassortative than short time scale networks in both tissue types. As far as we are aware, this analysis represents the first systematic study of temporally dependent multiplex networks among individual neurons.

Parallel discrete-event simulation schemes with heterogeneous processing elements

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind.

GABA networks destabilize genetic oscillations in the circadian pacemaker

Systems of coupled oscillators abound in nature. How they establish stable phase relationships under diverse conditions is fundamentally important. The mammalian suprachiasmatic nucleus (SCN) is a self-sustained, synchronized network of circadian oscillators that coordinates daily rhythms in physiology and behavior. To elucidate the underlying topology and signaling mechanisms that modulate circadian synchrony, we discriminated the firing of hundreds of SCN neurons continuously over days. Using an analysis method to identify functional interactions between neurons based on changes in their firing, we characterized a GABAergic network comprised of fast, excitatory, and inhibitory connections that is both stable over days and changes in strength with time of day. By monitoring PERIOD2 protein expression, we provide the first evidence that these millisecond-level interactions actively oppose circadian synchrony and inject jitter into daily rhythms. These results provide a mechanism by which circadian oscillators can tune their phase relationships under different environmental conditions.

From neural plate to cortical arousal-a neuronal network theory of sleep derived from in vitro "model" systems for primordial patterns of spontaneous bioelectric activity in the vertebrate central nervous system

In the early 1960s intrinsically generated widespread neuronal discharges were discovered to be the basis for the earliest motor behavior throughout the animal kingdom. The pattern generating system is in fact programmed into the developing nervous system, in a regionally specific manner, already at the early neural plate stage. Such rhythmically modulated phasic bursts were next discovered to be a general feature of developing neural networks and, largely on the basis of experimental interventions in cultured neural tissues, to contribute significantly to their morpho-physiological maturation. In particular, the level of spontaneous synchronized bursting is homeostatically regulated, and has the effect of constraining the development of excessive network excitability. After birth or hatching, this "slow-wave" activity pattern becomes sporadically suppressed in favor of sensory oriented "waking" behaviors better adapted to dealing with environmental contingencies. It nevertheless reappears periodically as "sleep" at several species-specific points in the diurnal/nocturnal cycle. Although this "default" behavior pattern evolves with development, its essential features are preserved throughout the life cycle, and are based upon a few simple mechanisms which can be both experimentally demonstrated and simulated by computer modeling. In contrast, a late onto- and phylogenetic aspect of sleep, viz., the intermittent "paradoxical" activation of the forebrain so as to mimic waking activity, is much less well understood as regards its contribution to brain development. Some recent findings dealing with this question by means of cholinergically induced "aroused" firing patterns in developing neocortical cell cultures, followed by quantitative electrophysiological assays of immediate and longterm sequelae, will be discussed in connection with their putative implications for sleep ontogeny.

Collective dynamics in sparse networks

The microscopic and macroscopic dynamics of random networks is investigated in the strong-dilution limit (i.e., for sparse networks). By simulating chaotic maps, Stuart-Landau oscillators, and leaky integrate-and-fire neurons, we show that a finite connectivity (of the order of a few tens) is able to sustain a nontrivial collective dynamics even in the thermodynamic limit. Although the network structure implies a nonadditive dynamics, the microscopic evolution is extensive (i.e., the number of active degrees of freedom is proportional to the number of network elements).

The remarkable, yet not extraordinary, human brain as a scaled-up primate brain and its associated cost

Neuroscientists have become used to a number of "facts" about the human brain: It has 100 billion neurons and 10- to 50-fold more glial cells; it is the largest-than-expected for its body among primates and mammals in general, and therefore the most cognitively able; it consumes an outstanding 20% of the total body energy budget despite representing only 2% of body mass because of an increased metabolic need of its neurons; and it is endowed with an overdeveloped cerebral cortex, the largest compared with brain size. These facts led to the widespread notion that the human brain is literally extraordinary: an outlier among mammalian brains, defying evolutionary rules that apply to other species, with a uniqueness seemingly necessary to justify the superior cognitive abilities of humans over mammals with even larger brains. These facts, with deep implications for neurophysiology and evolutionary biology, are not grounded on solid evidence or sound assumptions, however. Our recent development of a method that allows rapid and reliable quantification of the numbers of cells that compose the whole brain has provided a means to verify these facts. Here, I review this recent evidence and argue that, with 86 billion neurons and just as many nonneuronal cells, the human brain is a scaled-up primate brain in its cellular composition and metabolic cost, with a relatively enlarged cerebral cortex that does not have a relatively larger number of brain neurons yet is remarkable in its cognitive abilities and metabolism simply because of its extremely large number of neurons.

Complexity in a brain-inspired agent-based model

An agent-based model consists of a set of agents representing the components of a system. These agents interact with each other according to rules designed with knowledge of the system in mind. Although rules control the low-level interactions of agents, these models often exhibit emergent behavior at the system level. We apply the agent-based modeling framework to functional brain imaging data. In this model, agents are defined by network nodes and represent brain regions, and links representing functional connectivity between nodes dictate which agents interact. A link between two regions may be positive or negative, depending on the correlation in functional activity between the two regions. Agents are either active or inactive, and systematically update based on the activity of their immediate neighbors. Their dynamics are observed over a certain time period starting from predetermined initial configurations. While the information received by each node is limited by the number of other nodes connected to it, we have shown that this model is capable of producing emergent behavior dependent on global information transfer. Specifically, the system is capable of solving well-described test problems, such as the density classification and synchronization problems. The model is capable of producing a wide range of behaviors varying greatly in complexity, including oscillations with cycles ranging from a few steps to hundreds, and non-repeating patterns over hundreds of thousands of time steps. We believe this wide dynamic range may impart the potential for this system to produce a myriad of brain-like functional states.

Coherent periodic activity in excitatory Erdös-Renyi neural networks: the role of network connectivity

In this article, we investigate the role of connectivity in promoting coherent activity in excitatory neural networks. In particular, we would like to understand if the onset of collective oscillations can be related to a minimal average connectivity and how this critical connectivity depends on the number of neurons in the networks. For these purposes, we consider an excitatory random network of leaky integrate-and-fire pulse coupled neurons. The neurons are connected as in a directed Erdös-Renyi graph with average connectivity <k> scaling as a power law with the number of neurons in the network. The scaling is controlled by a parameter γ, which allows to pass from massively connected to sparse networks and therefore to modify the topology of the system. At a macroscopic level, we observe two distinct dynamical phases: an asynchronous state corresponding to a desynchronized dynamics of the neurons and a regime of partial synchronization (PS) associated with a coherent periodic activity of the network. At low connectivity, the system is in an asynchronous state, while PS emerges above a certain critical average connectivity <k>(c). For sufficiently large networks, <k>(c) saturates to a constant value suggesting that a minimal average connectivity is sufficient to observe coherent activity in systems of any size irrespectively of the kind of considered network: sparse or massively connected. However, this value depends on the nature of the synapses: reliable or unreliable. For unreliable synapses, the critical value required to observe the onset of macroscopic behaviors is noticeably smaller than for reliable synaptic transmission. Due to the disorder present in the system, for finite number of neurons we have inhomogeneities in the neuronal behaviors, inducing a weak form of chaos, which vanishes in the thermodynamic limit. In such a limit, the disordered systems exhibit regular (non chaotic) dynamics and their properties correspond to that of a homogeneous fully connected network for any γ-value. Apart for the peculiar exception of sparse networks, which remain intrinsically inhomogeneous at any system size.

Emergent functional properties of neuronal networks with controlled topology

The interplay between anatomical connectivity and dynamics in neural networks plays a key role in the functional properties of the brain and in the associated connectivity changes induced by neural diseases. However, a detailed experimental investigation of this interplay at both cellular and population scales in the living brain is limited by accessibility. Alternatively, to investigate the basic operational principles with morphological, electrophysiological and computational methods, the activity emerging from large in vitro networks of primary neurons organized with imposed topologies can be studied. Here, we validated the use of a new bio-printing approach, which effectively maintains the topology of hippocampal cultures in vitro and investigated, by patch-clamp and MEA electrophysiology, the emerging functional properties of these grid-confined networks. In spite of differences in the organization of physical connectivity, our bio-patterned grid networks retained the key properties of synaptic transmission, short-term plasticity and overall network activity with respect to random networks. Interestingly, the imposed grid topology resulted in a reinforcement of functional connections along orthogonal directions, shorter connectivity links and a greatly increased spiking probability in response to focal stimulation. These results clearly demonstrate that reliable functional studies can nowadays be performed on large neuronal networks in the presence of sustained changes in the physical network connectivity.

Maturation of a recurrent excitatory neocortical circuit by experience-dependent unsilencing of newly formed dendritic spines

Local recurrent excitatory circuits are ubiquitous in neocortex, yet little is known about their development or architecture. Here we introduce a quantitative technique for efficient single-cell resolution circuit mapping using 2-photon (2P) glutamate uncaging and analyze experience-dependent neonatal development of the layer 4 barrel cortex local excitatory circuit. We show that sensory experience specifically drives a 3-fold increase in connectivity at postnatal day (P) 9, producing a highly recurrent network. A profound dendritic spinogenesis occurs concurrent with the connectivity increase, but this is not experience dependent. However, in experience-deprived cortex, a much greater proportion of spines lack postsynaptic AMPA receptors (AMPARs) and synaptic connectivity via NMDA receptors (NMDARs) is the same as in normally developing cortex. Thus we describe a approach for quantitative circuit mapping and show that sensory experience sculpts an intrinsically developing template network, which is based on NMDAR-only synapses, by driving AMPARs into newly formed silent spines.

Sustained activity in hierarchical modular neural networks: self-organized criticality and oscillations

Cerebral cortical brain networks possess a number of conspicuous features of structure and dynamics. First, these networks have an intricate, non-random organization. In particular, they are structured in a hierarchical modular fashion, from large-scale regions of the whole brain, via cortical areas and area subcompartments organized as structural and functional maps to cortical columns, and finally circuits made up of individual neurons. Second, the networks display self-organized sustained activity, which is persistent in the absence of external stimuli. At the systems level, such activity is characterized by complex rhythmical oscillations over a broadband background, while at the cellular level, neuronal discharges have been observed to display avalanches, indicating that cortical networks are at the state of self-organized criticality (SOC). We explored the relationship between hierarchical neural network organization and sustained dynamics using large-scale network modeling. Previously, it was shown that sparse random networks with balanced excitation and inhibition can sustain neural activity without external stimulation. We found that a hierarchical modular architecture can generate sustained activity better than random networks. Moreover, the system can simultaneously support rhythmical oscillations and SOC, which are not present in the respective random networks. The mechanism underlying the sustained activity is that each dense module cannot sustain activity on its own, but displays SOC in the presence of weak perturbations. Therefore, the hierarchical modular networks provide the coupling among subsystems with SOC. These results imply that the hierarchical modular architecture of cortical networks plays an important role in shaping the ongoing spontaneous activity of the brain, potentially allowing the system to take advantage of both the sensitivity of critical states and the predictability and timing of oscillations for efficient information processing.

Dissecting functional connectivity of neuronal microcircuits: experimental and theoretical insights

Structure-function studies of neuronal networks have recently benefited from considerable progress in different areas of investigation. Advances in molecular genetics and imaging have allowed for the dissection of neuronal connectivity with unprecedented detail whereas in vivo recordings are providing much needed clues as to how sensory, motor and cognitive function is encoded in neuronal firing. However, bridging the gap between the cellular and behavioral levels will ultimately require an understanding of the functional organization of the underlying neuronal circuits. One way to unravel the complexity of neuronal networks is to understand how their connectivity emerges during brain maturation. In this review, we will describe how graph theory provides experimentalists with novel concepts that can be used to describe and interpret these developing connectivity schemes.

Dynamics of excitable neural networks with heterogeneous connectivity

A central issue of neuroscience is to understand how neural units integrates internal and external signals to create coherent states. Recently, it has been shown that the sensitivity and dynamic range of neural assemblies are optimal at a critical coupling among its elements. Complex architectures of connections seem to play a constructive role on the reliable coordination of neural units. Here we show that, the synchronizability and sensitivity of excitable neural networks can be tuned by diversity in the connections strengths. We illustrate our findings for weighted networks with regular, random and complex topologies. Additional comparisons of real brain networks support previous studies suggesting that heterogeneity in the connectivity may play a constructive role on information processing. These findings provide insights into the relationship between structure and function of neural circuits.

Connectivity-driven white matter scaling and folding in primate cerebral cortex

Larger brains have an increasingly folded cerebral cortex whose white matter scales up faster than the gray matter. Here we analyze the cellular composition of the subcortical white matter in 11 primate species, including humans, and one Scandentia, and show that the mass of the white matter scales linearly across species with its number of nonneuronal cells, which is expected to be proportional to the total length of myelinated axons in the white matter. This result implies that the average axonal cross-section area in the white matter, a, does not scale significantly with the number of neurons in the gray matter, N. The surface area of the white matter increases with N(0.87), not N(1.0). Because this surface can be defined as the product of N, a, and the fraction n of cortical neurons connected through the white matter, we deduce that connectivity decreases in larger cerebral cortices as a slowly diminishing fraction of neurons, which varies with N(-0.16), sends myelinated axons into the white matter. Decreased connectivity is compatible with previous suggestions that neurons in the cerebral cortex are connected as a small-world network and should slow down the increase in global conduction delay in cortices with larger numbers of neurons. Further, a simple model shows that connectivity and cortical folding are directly related across species. We offer a white matter-based mechanism to account for increased cortical folding across species, which we propose to be driven by connectivity-related tension in the white matter, pulling down on the gray matter.

Mapping from structure to dynamics: a unified view of dynamical processes on networks

Although it is unambiguously agreed that structure plays a fundamental role in shaping the collective dynamics of complex systems, how structure determines dynamics exactly still remains unclear. We investigate a general computational transformation by which we can map the network topology directly to the dynamical patterns emergent on it-independent of the nature of the dynamical processes. Remarkably, we find that many seemingly different dynamical processes on networks, such as coupled oscillators, ensemble neuron firing, epidemic spreading and diffusion can all be understood and unified through this same procedure. Utilizing the inherent multiscale nature of this structure-dynamics transformation, we further define a multiscale complexity measure, which can quantify the functional diversity a general network can support at different organization levels using only its structure. We find that a wide variety of topological features observed in real networks, such as modularity, hierarchy, degree heterogeneity and mixing all result in higher complexity. This result suggests that the demand for functional diversity is driving the structural evolution of physical networks.

Stochastic cellular automata model of neural networks

We propose a stochastic dynamical model of noisy neural networks with complex architectures and discuss activation of neural networks by a stimulus, pacemakers, and spontaneous activity. This model has a complex phase diagram with self-organized active neural states, hybrid phase transitions, and a rich array of behaviors. We show that if spontaneous activity (noise) reaches a threshold level then global neural oscillations emerge. Stochastic resonance is a precursor of this dynamical phase transition. These oscillations are an intrinsic property of even small groups of 50 neurons.

Neural synchrony in cortical networks: history, concept and current status

Following the discovery of context-dependent synchronization of oscillatory neuronal responses in the visual system, the role of neural synchrony in cortical networks has been expanded to provide a general mechanism for the coordination of distributed neural activity patterns. In the current paper, we present an update of the status of this hypothesis through summarizing recent results from our laboratory that suggest important new insights regarding the mechanisms, function and relevance of this phenomenon. In the first part, we present recent results derived from animal experiments and mathematical simulations that provide novel explanations and mechanisms for zero and nero-zero phase lag synchronization. In the second part, we shall discuss the role of neural synchrony for expectancy during perceptual organization and its role in conscious experience. This will be followed by evidence that indicates that in addition to supporting conscious cognition, neural synchrony is abnormal in major brain disorders, such as schizophrenia and autism spectrum disorders. We conclude this paper with suggestions for further research as well as with critical issues that need to be addressed in future studies.

Reverse engineering and verification of gene networks: principles, assumptions, and limitations of present methods and future perspectives

Reverse engineering of gene networks aims at revealing the structure of the gene regulation network in a biological system by reasoning backward directly from experimental data. Many methods have recently been proposed for reverse engineering of gene networks by using gene transcript expression data measured by microarray. Whereas the potentials of the methods have been well demonstrated, the assumptions and limitations behind them are often not clearly stated or not well understood. In this review, we first briefly explain the principles of the major methods, identify the assumptions behind them and pinpoint the limitations and possible pitfalls in applying them to real biological questions. With regard to applications, we then discuss challenges in the experimental verification of gene networks generated from reverse engineering methods. We further propose an optimal experimental design for allocating sampling schedule and possible strategies for reducing the limitations of some of the current reverse engineering methods. Finally, we examine the perspectives for the development of reverse engineering and urge the need to move from revealing network structure to the dynamics of biological systems.

Order-based representation in random networks of cortical neurons

The wide range of time scales involved in neural excitability and synaptic transmission might lead to ongoing change in the temporal structure of responses to recurring stimulus presentations on a trial-to-trial basis. This is probably the most severe biophysical constraint on putative time-based primitives of stimulus representation in neuronal networks. Here we show that in spontaneously developing large-scale random networks of cortical neurons in vitro the order in which neurons are recruited following each stimulus is a naturally emerging representation primitive that is invariant to significant temporal changes in spike times. With a relatively small number of randomly sampled neurons, the information about stimulus position is fully retrievable from the recruitment order. The effective connectivity that makes order-based representation invariant to time warping is characterized by the existence of stations through which activity is required to pass in order to propagate further into the network. This study uncovers a simple invariant in a noisy biological network in vitro; its applicability under in vivo constraints remains to be seen.

Spontaneous neuronal burst discharges as dependent and independent variables in the maturation of cerebral cortex tissue cultured in vitro: a review of activity-dependent studies in live 'model' systems for the development of intrinsically generated bioelectric slow-wave sleep patterns

A survey is presented of recent experiments which utilize spontaneous neuronal spike trains as dependent and/or independent variables in developing cerebral cortex cultures when synaptic transmission is interfered with for varying periods of time. Special attention is given to current difficulties in selecting suitable preparations for carrying out biologically relevant developmental studies, and in applying spike-train analysis methods with sufficient resolution to detect activity-dependent age and treatment effects. A hierarchy of synchronized nested burst discharges which approximate early slow-wave sleep patterns in the intact organism is established as a stable basis for isolated cortex function. The complexity of reported long- and short-term homeostatic responses to experimental interference with synaptic transmission is reviewed, and the crucial role played by intrinsically generated bioelectric activity in the maturation of cortical networks is emphasized.

Consciousness related neural events viewed as brain state space transitions

This theoretical and speculative essay addresses a categorical distinction between neural events of sensory-motor cognition and those presumably associated with consciousness. It proposes to view this distinction in the framework of the branch of Statistical Physics currently referred to as Modern Critical Theory (Stanley, Introduction to phase transitions and critical phenomena, 1987; Marro and Dickman, Nonequilibrium phase transitions in lattice, 1999). Based on established landmarks of brain dynamics, network configurations and their role for conveying oscillatory activity of certain frequencies bands, the question is examined: what kind of state space transitions can systems with these properties undergo, and could the relation between neural processes of sensory-motor cognition and those of events in consciousness be of the same category as is characterized by state transitions in non-equilibrium physical systems? Approaches for empirical validation of this view by suitably designed brain imaging studies, and for computational simulations of the proposed principle are discussed.

Effects of Hebbian learning on the dynamics and structure of random networks with inhibitory and excitatory neurons

The aim of the present paper is to study the effects of Hebbian learning in random recurrent neural networks with biological connectivity, i.e. sparse connections and separate populations of excitatory and inhibitory neurons. We furthermore consider that the neuron dynamics may occur at a (shorter) time scale than synaptic plasticity and consider the possibility of learning rules with passive forgetting. We show that the application of such Hebbian learning leads to drastic changes in the network dynamics and structure. In particular, the learning rule contracts the norm of the weight matrix and yields a rapid decay of the dynamics complexity and entropy. In other words, the network is rewired by Hebbian learning into a new synaptic structure that emerges with learning on the basis of the correlations that progressively build up between neurons. We also observe that, within this emerging structure, the strongest synapses organize as a small-world network. The second effect of the decay of the weight matrix spectral radius consists in a rapid contraction of the spectral radius of the Jacobian matrix. This drives the system through the "edge of chaos" where sensitivity to the input pattern is maximal. Taken together, this scenario is remarkably predicted by theoretical arguments derived from dynamical systems and graph theory.

A scale-free systems theory of motivation and addiction

Scale-free organizations, characterized by uneven distributions of linkages between nodal elements, describe the structure and function of many life-based complex systems developing under evolutionary pressures. We explore motivated behavior as a scale-free map toward a comprehensive translational theory of addiction. Motivational and behavioral repertoires are reframed as link and nodal element sets, respectively, comprising a scale-free structure. These sets are generated by semi-independent information-processing streams within cortical-striatal circuits that cooperatively provide decision-making and sequential processing functions necessary for traversing maps of motivational links connecting behavioral nodes. Dopamine modulation of cortical-striatal plasticity serves a central-hierarchical mechanism for survival-adaptive sculpting and development of motivational-behavioral repertoires by guiding a scale-free design. Drug-induced dopamine activity promotes drug taking as a highly connected behavioral hub at the expense of natural-adaptive motivational links and behavioral nodes. Conceptualizing addiction as pathological alteration of scale-free motivational-behavioral repertoires unifies neurobiological, neurocomputational and behavioral research while addressing addiction vulnerability in adolescence and psychiatric illness. This model may inform integrative research in defining more effective prevention and treatment strategies for addiction.

Synchronization in weighted uncorrelated complex networks in a noisy environment: optimization and connections with transport efficiency

Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to (kikj)beta with ki and kj being the degrees of the nodes connected by the link. Subject to the constraint that the total edge cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at beta*= -1 . Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of beta*. Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.

Dynamics and effective topology underlying synchronization in networks of cortical neurons

Cognitive processes depend on synchronization and propagation of electrical activity within and between neuronal assemblies. In vivo measurements show that the size of individual assemblies depends on their function and varies considerably, but the timescale of assembly activation is in the range of 0.1-0.2 s and is primarily independent of assembly size. Here we use an in vitro experimental model of cortical assemblies to characterize the process underlying the timescale of synchronization, its relationship to the effective topology of connectivity within an assembly, and its impact on propagation of activity within and between assemblies. We show that the basic mode of assembly activation, "network spike," is a threshold-governed, synchronized population event of 0.1-0.2 s duration and follows the logistics of neuronal recruitment in an effectively scale-free connected network. Accordingly, the sequence of neuronal activation within a network spike is nonrandom and hierarchical; a small subset of neurons is consistently recruited tens of milliseconds before others. Theory predicts that scale-free topology allows for synchronization time that does not increase markedly with network size; our experiments with networks of different densities support this prediction. The activity of early-to-fire neurons reliably forecasts an upcoming network spike and provides means for expedited propagation between assemblies. We demonstrate this capacity by observing the dynamics of two artificially coupled assemblies in vitro, using neuronal activity of one as a trigger for electrical stimulation of the other.

Periodic neural activity induced by network complexity

We study a model for neural activity on the small-world topology of Watts and Strogatz and on the scale-free topology of Barabási and Albert. We find that the topology of the network connections may spontaneously induce periodic neural activity, contrasting with nonperiodic neural activities exhibited by regular topologies. Periodic activity exists only for relatively small networks and occurs with higher probability when the rewiring probability is larger. The average length of the periods increases with the square root of the network size.

From scale-free to Erdos-Rényi networks

We analyze a model that interpolates between scale-free and Erdos-Rényi networks. The model introduced generates a one-parameter family of networks and allows one to analyze the role of structural heterogeneity. Analytical calculations are compared with extensive numerical simulations in order to describe the transition between these two important classes of networks. Finally, an application of the proposed model to the study of the percolation transition is presented.