Motifs in brain networks

The Minimal k-Core Problem for Modeling k-Assemblies

The concept of cell assembly was introduced by Hebb and formalized mathematically by Palm in the framework of graph theory. In the study of associative memory, a cell assembly is a group of neurons that are strongly connected and represent a "concept" of our knowledge. This group is wired in a specific manner such that only a fraction of its neurons will excite the entire assembly. We link the concept of cell assembly to the closure of a minimal k-core and study a particular type of cell assembly called k-assembly. The goal of this paper is to find all substructures within a network that must be excited in order to activate a k-assembly. Through numerical experiments, we confirm that fractions of these important subgroups overlap. To explore the problem, we present a backtracking algorithm to find all minimal k-cores of a given undirected graph, which belongs to the class of NP-hard problems. The proposed method is a modification of the Bron and Kerbosch algorithm for finding all cliques of an undirected graph. The results in the tested graphs offer insight in analyzing graph structure and help better understand how concepts are stored.

How Robust Is Your Project? From Local Failures to Global Catastrophes: A Complex Networks Approach to Project Systemic Risk

Current societal requirements necessitate the effective delivery of complex projects that can do more while using less. Yet, recent large-scale project failures suggest that our ability to successfully deliver them is still at its infancy. Such failures can be seen to arise through various failure mechanisms; this work focuses on one such mechanism. Specifically, it examines the likelihood of a project sustaining a large-scale catastrophe, as triggered by single task failure and delivered via a cascading process. To do so, an analytical model was developed and tested on an empirical dataset by the means of numerical simulation. This paper makes three main contributions. First, it provides a methodology to identify the tasks most capable of impacting a project. In doing so, it is noted that a significant number of tasks induce no cascades, while a handful are capable of triggering surprisingly large ones. Secondly, it illustrates that crude task characteristics cannot aid in identifying them, highlighting the complexity of the underlying process and the utility of this approach. Thirdly, it draws parallels with systems encountered within the natural sciences by noting the emergence of self-organised criticality, commonly found within natural systems. These findings strengthen the need to account for structural intricacies of a project’s underlying task precedence structure as they can provide the conditions upon which large-scale catastrophes materialise.

Event Networks and the Identification of Crime Pattern Motifs

In this paper we demonstrate the use of network analysis to characterise patterns of clustering in spatio-temporal events. Such clustering is of both theoretical and practical importance in the study of crime, and forms the basis for a number of preventative strategies. However, existing analytical methods show only that clustering is present in data, while offering little insight into the nature of the patterns present. Here, we show how the classification of pairs of events as close in space and time can be used to define a network, thereby generalising previous approaches. The application of graph-theoretic techniques to these networks can then offer significantly deeper insight into the structure of the data than previously possible. In particular, we focus on the identification of network motifs, which have clear interpretation in terms of spatio-temporal behaviour. Statistical analysis is complicated by the nature of the underlying data, and we provide a method by which appropriate randomised graphs can be generated. Two datasets are used as case studies: maritime piracy at the global scale, and residential burglary in an urban area. In both cases, the same significant 3-vertex motif is found; this result suggests that incidents tend to occur not just in pairs, but in fact in larger groups within a restricted spatio-temporal domain. In the 4-vertex case, different motifs are found to be significant in each case, suggesting that this technique is capable of discriminating between clustering patterns at a finer granularity than previously possible.

Communicability angles reveal critical edges for network consensus dynamics

We consider the question of determining how the topological structure influences a consensus dynamical processes taking place on a network. By considering a large data set of real-world networks we first determine that the removal of edges according to their communicability angle, an angle between position vectors of the nodes in an Euclidean communicability space, increases the average time of consensus by a factor of 5.68 in real-world networks. The edge betweenness centrality also identifies, in a smaller proportion, those critical edges for the consensus dynamics; i.e., its removal increases the time of consensus by a factor of 3.70. We justify theoretically these findings on the basis of the role played by the algebraic connectivity and the isoperimetric number of networks on the dynamical process studied and their connections with the properties mentioned before. Finally, we study the role played by global topological parameters of networks on the consensus dynamics. We determine that the network density and the average distance-sum, which is analogous of the node degree for shortest-path distances, account for more than 80% of the variance of the average time of consensus in the real-world networks studied.

Analysing Local Sparseness in the Macaque Brain Network

Understanding the network structure of long distance pathways in the brain is a necessary step towards developing an insight into the brain's function, organization and evolution. Dense global subnetworks of these pathways have often been studied, primarily due to their functional implications. Instead we study sparse local subnetworks of the pathways to establish the role of a brain area in enabling shortest path communication between its non-adjacent topological neighbours. We propose a novel metric to measure the topological communication load on a vertex due to its immediate neighbourhood, and show that in terms of distribution of this local communication load, a network of Macaque long distance pathways is substantially different from other real world networks and random graph models. Macaque network contains the entire range of local subnetworks, from star-like networks to clique-like networks, while other networks tend to contain a relatively small range of subnetworks. Further, sparse local subnetworks in the Macaque network are not only found across topographical super-areas, e.g., lobes, but also within a super-area, arguing that there is conservation of even relatively short-distance pathways. To establish the communication role of a vertex we borrow the concept of brokerage from social science, and present the different types of brokerage roles that brain areas play, highlighting that not only the thalamus, but also cingulate gyrus and insula often act as "relays" for areas in the neocortex. These and other analysis of communication load and roles of the sparse subnetworks of the Macaque brain provide new insights into the organisation of its pathways.

Dynamics of Time Delay-Induced Multiple Synchronous Behaviors in Inhibitory Coupled Neurons

The inhibitory synapse can induce synchronous behaviors different from the anti-phase synchronous behaviors, which have been reported in recent studies. In the present paper, synchronous behaviors are investigated in the motif model composed of reciprocal inhibitory coupled neurons with endogenous bursting and time delay. When coupling strength is weak, synchronous behavior appears at a single interval of time delay within a bursting period. When coupling strength is strong, multiple synchronous behaviors appear at different intervals of time delay within a bursting period. The different bursting patterns of synchronous behaviors, and time delays and coupling strengths that can induce the synchronous bursting patterns can be well interpreted by the dynamics of the endogenous bursting pattern of isolated neuron, which is acquired by the fast-slow dissection method, combined with the inhibitory coupling current. For an isolated neuron, when a negative impulsive current with suitable strength is applied at different phases of the bursting, multiple different bursting patterns can be induced. For a neuron in the motif, the inhibitory coupling current, of which the application time and strength is modulated by time delay and coupling strength, can cause single or multiple synchronous firing patterns like the negative impulsive current when time delay and coupling strength is suitable. The difference compared to the previously reported multiple synchronous behaviors that appear at time delays wider than a period of the endogenous firing is discussed. The results present novel examples of synchronous behaviors in the neuronal network with inhibitory synapses and provide a reasonable explanation.

Unsupervised learning of electrocorticography motifs with binary descriptors of wavelet features and hierarchical clustering

We describe a novel method for data mining spectro-spatiotemporal network motifs from electrocorticographic (ECoG) data. The method utilizes wavelet feature extraction from ECoG data, generation of compact binary vectors from these features, and binary vector hierarchical clustering. The potential utility of this method in the discovery of recurring neural patterns is demonstrated in an example showing clustering of ictal and post-ictal gamma activity patterns. The method allows for the efficient and scalable retrieval and clustering of neural motifs occurring in massive amounts of neural data, such as in prolonged EEG/ECoG recordings and in brain computer interfaces.

Brain asymmetry in the white matter making and globularity

Recent studies from the field of language genetics and evolutionary anthropology have put forward the hypothesis that the emergence of our species-specific brain is to be understood not in terms of size, but in light of developmental changes that gave rise to a more globular braincase configuration after the split from Neanderthals-Denisovans. On the grounds that (i) white matter myelination is delayed relative to other brain structures and, in humans, is protracted compared with other primates and that (ii) neural connectivity is linked genetically to our brain/skull morphology and language-ready brain, I argue that one significant evolutionary change in Homo sapiens' lineage is the interhemispheric connectivity mediated by the Corpus Callosum. The size, myelination and fiber caliber of the Corpus Callosum present an anterior-to-posterior increase, in a way that inter-hemispheric connectivity is more prominent in the sensory motor areas, whereas "high- order" areas are more intra-hemispherically connected. Building on evidence from language-processing studies that account for this asymmetry ('lateralization') in terms of brain rhythms, I present an evo-devo hypothesis according to which the myelination of the Corpus Callosum, Brain Asymmetry, and Globularity are conjectured to make up the angles of a co-evolutionary triangle that gave rise to our language-ready brain.

The neuroaesthetics of prose fiction: pitfalls, parameters and prospects

There is a paucity of neuroaesthetic studies on prose fiction. This is in contrast to the very many impressive studies that have been conducted in recent times on the neuroaesthetics of sister arts such as painting, music and dance. Why might this be the case, what are its causes and, of greatest importance, how can it best be resolved? In this article, the pitfalls, parameters and prospects of a neuroaesthetics of prose fiction will be explored. The article itself is part critical review, part methodological proposal and part opinion paper. Its aim is simple: to stimulate, excite and energize thinking in the discipline as to how prose fiction might be fully integrated in the canon of neuroaesthetics and to point to opportunities where neuroimaging studies on literary discourse processing might be conducted in collaborative work bringing humanists and scientists together.

Motif analysis in directed ordered networks and applications to food webs

The analysis of small recurrent substructures, so called network motifs, has become a standard tool of complex network science to unveil the design principles underlying the structure of empirical networks. In many natural systems network nodes are associated with an intrinsic property according to which they can be ordered and compared against each other. Here, we expand standard motif analysis to be able to capture the hierarchical structure in such ordered networks. Our new approach is based on the identification of all ordered 3-node substructures and the visualization of their significance profile. We present a technique to calculate the fine grained motif spectrum by resolving the individual members of isomorphism classes (sets of substructures formed by permuting node-order). We apply this technique to computer generated ensembles of ordered networks and to empirical food web data, demonstrating the importance of considering node order for food-web analysis. Our approach may not only be helpful to identify hierarchical patterns in empirical food webs and other natural networks, it may also provide the base for extending motif analysis to other types of multi-layered networks.

Dwelling quietly in the rich club: brain network determinants of slow cortical fluctuations

For more than a century, cerebral cartography has been driven by investigations of structural and morphological properties of the brain across spatial scales and the temporal/functional phenomena that emerge from these underlying features. The next era of brain mapping will be driven by studies that consider both of these components of brain organization simultaneously--elucidating their interactions and dependencies. Using this guiding principle, we explored the origin of slowly fluctuating patterns of synchronization within the topological core of brain regions known as the rich club, implicated in the regulation of mood and introspection. We find that a constellation of densely interconnected regions that constitute the rich club (including the anterior insula, amygdala and precuneus) play a central role in promoting a stable, dynamical core of spontaneous activity in the primate cortex. The slow timescales are well matched to the regulation of internal visceral states, corresponding to the somatic correlates of mood and anxiety. In contrast, the topology of the surrounding 'feeder' cortical regions shows unstable, rapidly fluctuating dynamics likely to be crucial for fast perceptual processes. We discuss these findings in relation to psychiatric disorders and the future of connectomics.

The frustrated brain: from dynamics on motifs to communities and networks

Cognitive function depends on an adaptive balance between flexible dynamics and integrative processes in distributed cortical networks. Patterns of zero-lag synchrony likely underpin numerous perceptual and cognitive functions. Synchronization fulfils integration by reducing entropy, while adaptive function mandates that a broad variety of stable states be readily accessible. Here, we elucidate two complementary influences on patterns of zero-lag synchrony that derive from basic properties of brain networks. First, mutually coupled pairs of neuronal subsystems-resonance pairs-promote stable zero-lag synchrony among the small motifs in which they are embedded, and whose effects can propagate along connected chains. Second, frustrated closed-loop motifs disrupt synchronous dynamics, enabling metastable configurations of zero-lag synchrony to coexist. We document these two complementary influences in small motifs and illustrate how these effects underpin stable versus metastable phase-synchronization patterns in prototypical modular networks and in large-scale cortical networks of the macaque (CoCoMac). We find that the variability of synchronization patterns depends on the inter-node time delay, increases with the network size and is maximized for intermediate coupling strengths. We hypothesize that the dialectic influences of resonance versus frustration may form a dynamic substrate for flexible neuronal integration, an essential platform across diverse cognitive processes.

Using Graph Components Derived from an Associative Concept Dictionary to Predict fMRI Neural Activation Patterns that Represent the Meaning of Nouns

In this study, we introduce an original distance definition for graphs, called the Markov-inverse-F measure (MiF). This measure enables the integration of classical graph theory indices with new knowledge pertaining to structural feature extraction from semantic networks. MiF improves the conventional Jaccard and/or Simpson indices, and reconciles both the geodesic information (random walk) and co-occurrence adjustment (degree balance and distribution). We measure the effectiveness of graph-based coefficients through the application of linguistic graph information for a neural activity recorded during conceptual processing in the human brain. Specifically, the MiF distance is computed between each of the nouns used in a previous neural experiment and each of the in-between words in a subgraph derived from the Edinburgh Word Association Thesaurus of English. From the MiF-based information matrix, a machine learning model can accurately obtain a scalar parameter that specifies the degree to which each voxel in (the MRI image of) the brain is activated by each word or each principal component of the intermediate semantic features. Furthermore, correlating the voxel information with the MiF-based principal components, a new computational neurolinguistics model with a network connectivity paradigm is created. This allows two dimensions of context space to be incorporated with both semantic and neural distributional representations.

Directed network motifs in Alzheimer's disease and mild cognitive impairment

Directed network motifs are the building blocks of complex networks, such as human brain networks, and capture deep connectivity information that is not contained in standard network measures. In this paper we present the first application of directed network motifs in vivo to human brain networks, utilizing recently developed directed progression networks which are built upon rates of cortical thickness changes between brain regions. This is in contrast to previous studies which have relied on simulations and in vitro analysis of non-human brains. We show that frequencies of specific directed network motifs can be used to distinguish between patients with Alzheimer’s disease (AD) and normal control (NC) subjects. Especially interesting from a clinical standpoint, these motif frequencies can also distinguish between subjects with mild cognitive impairment who remained stable over three years (MCI) and those who converted to AD (CONV). Furthermore, we find that the entropy of the distribution of directed network motifs increased from MCI to CONV to AD, implying that the distribution of pathology is more structured in MCI but becomes less so as it progresses to CONV and further to AD. Thus, directed network motifs frequencies and distributional properties provide new insights into the progression of Alzheimer’s disease as well as new imaging markers for distinguishing between normal controls, stable mild cognitive impairment, MCI converters and Alzheimer’s disease.

Encoding of both analog- and digital-like behavioral outputs by one C. elegans interneuron

Model organisms usually possess a small nervous system but nevertheless execute a large array of complex behaviors, suggesting that some neurons are likely multifunctional and may encode multiple behavioral outputs. Here, we show that the C. elegans interneuron AIY regulates two distinct behavioral outputs: locomotion speed and direction-switch by recruiting two different circuits. The "speed" circuit is excitatory with a wide dynamic range, which is well suited to encode speed, an analog-like output. The "direction-switch" circuit is inhibitory with a narrow dynamic range, which is ideal for encoding direction-switch, a digital-like output. Both circuits employ the neurotransmitter ACh but utilize distinct postsynaptic ACh receptors, whose distinct biophysical properties contribute to the distinct dynamic ranges of the two circuits. This mechanism enables graded C. elegans synapses to encode both analog- and digital-like outputs. Our studies illustrate how an interneuron in a simple organism encodes multiple behavioral outputs at the circuit, synaptic, and molecular levels.

Intrinsic connectivity in the human brain does not reveal networks for 'basic' emotions

We tested two competing models for the brain basis of emotion, the basic emotion theory and the conceptual act theory of emotion, using resting-state functional connectivity magnetic resonance imaging (rs-fcMRI). The basic emotion view hypothesizes that anger, sadness, fear, disgust and happiness each arise from a brain network that is innate, anatomically constrained and homologous in other animals. The conceptual act theory of emotion hypothesizes that an instance of emotion is a brain state constructed from the interaction of domain-general, core systems within the brain such as the salience, default mode and frontoparietal control networks. Using peak coordinates derived from a meta-analysis of task-evoked emotion fMRI studies, we generated a set of whole-brain rs-fcMRI 'discovery' maps for each emotion category and examined the spatial overlap in their conjunctions. Instead of discovering a specific network for each emotion category, variance in the discovery maps was accounted for by the known domain-general network. Furthermore, the salience network is observed as part of every emotion category. These results indicate that specific networks for each emotion do not exist within the intrinsic architecture of the human brain and instead support the conceptual act theory of emotion.

When less is more: non-monotonic spike sequence processing in neurons

Fundamental response properties of neurons centrally underly the computational capabilities of both individual nerve cells and neural networks. Most studies on neuronal input-output relations have focused on continuous-time inputs such as constant or noisy sinusoidal currents. Yet, most neurons communicate via exchanging action potentials (spikes) at discrete times. Here, we systematically analyze the stationary spiking response to regular spiking inputs and reveal that it is generically non-monotonic. Our theoretical analysis shows that the underlying mechanism relies solely on a combination of the discrete nature of the communication by spikes, the capability of locking output to input spikes and limited resources required for spike processing. Numerical simulations of mathematically idealized and biophysically detailed models, as well as neurophysiological experiments confirm and illustrate our theoretical predictions.

Graph analysis of the anatomical network organization of the hippocampal formation and parahippocampal region in the rat

Graph theory was used to analyze the anatomical network of the rat hippocampal formation and the parahippocampal region (van Strien et al., Nat Rev Neurosci 10(4):272-282, 2009). For this analysis, the full network was decomposed along the three anatomical axes, resulting in three networks that describe the connectivity within the rostrocaudal, dorsoventral and laminar dimensions. The rostrocaudal network had a connection density of 12% and a path length of 2.4. The dorsoventral network had a high cluster coefficient (0.53), a relatively high path length (1.62) and a rich club was identified. The modularity analysis revealed three modules in the dorsoventral network. The laminar network contained most information. The laminar dimension revealed a network with high clustering coefficient (0.47), a relatively high path length (2.11) and four significantly increased characteristic network building blocks (structural motifs). Thirteen rich club nodes were identified, almost all of them situated in the parahippocampal region. Six connector hubs were detected and all of them were located in the entorhinal cortex. Three large modules were revealed, indicating a close relationship between the perirhinal and postrhinal cortex as well as between the lateral and medial entorhinal cortex. These results confirmed the central position of the entorhinal cortex in the (para)hippocampal network and this possibly explains why pathology in this region has such profound impact on cognitive function, as seen in several brain diseases. The results also have implications for the idea of strict separation of the "spatial" and the "non-spatial" information stream into the hippocampus. This two-stream memory model suggests that the information influx from, respectively, the postrhinal-medial entorhinal cortex and the perirhinal-lateral entorhinal cortex is separate, but the current analysis shows that this apparent separation is not determined by anatomical constraints.

Nonlinear network dynamics under perturbations of the underlying graph

Many natural systems are organized as networks, in which the nodes (be they cells, individuals or populations) interact in a time-dependent fashion. The dynamic behavior of these networks depends on how these nodes are connected, which can be understood in terms of an adjacency matrix and connection strengths. The object of our study is to relate connectivity to temporal behavior in networks of coupled nonlinear oscillators. We investigate the relationship between classes of system architectures and classes of their possible dynamics, when the nodes are coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We illustrate how the phase space dynamics and bifurcations of the system change when perturbing the underlying adjacency graph. We differentiate between the effects on dynamics of the following operations that directly modulate network connectivity: (1) increasing/decreasing edge weights, (2) increasing/decreasing edge density, (3) altering edge configuration by adding, deleting, or moving edges. We discuss the significance of our results in the context of real life networks. Some interpretations lead us to draw conclusions that may apply to brain networks, synaptic restructuring, and neural dynamics.

Motif-role-fingerprints: the building-blocks of motifs, clustering-coefficients and transitivities in directed networks

Complex networks are frequently characterized by metrics for which particular subgraphs are counted. One statistic from this category, which we refer to as motif-role fingerprints, differs from global subgraph counts in that the number of subgraphs in which each node participates is counted. As with global subgraph counts, it can be important to distinguish between motif-role fingerprints that are 'structural' (induced subgraphs) and 'functional' (partial subgraphs). Here we show mathematically that a vector of all functional motif-role fingerprints can readily be obtained from an arbitrary directed adjacency matrix, and then converted to structural motif-role fingerprints by multiplying that vector by a specific invertible conversion matrix. This result demonstrates that a unique structural motif-role fingerprint exists for any given functional motif-role fingerprint. We demonstrate a similar result for the cases of functional and structural motif-fingerprints without node roles, and global subgraph counts that form the basis of standard motif analysis. We also explicitly highlight that motif-role fingerprints are elemental to several popular metrics for quantifying the subgraph structure of directed complex networks, including motif distributions, directed clustering coefficient, and transitivity. The relationships between each of these metrics and motif-role fingerprints also suggest new subtypes of directed clustering coefficients and transitivities. Our results have potential utility in analyzing directed synaptic networks constructed from neuronal connectome data, such as in terms of centrality. Other potential applications include anomaly detection in networks, identification of similar networks and identification of similar nodes within networks. Matlab code for calculating all stated metrics following calculation of functional motif-role fingerprints is provided as S1 Matlab File.

Zero-lag synchronization despite inhomogeneities in a relay system

A novel proposal for the zero-lag synchronization of the delayed coupled neurons, is to connect them indirectly via a third relay neuron. In this study, we develop a Poincaré map to investigate the robustness of the synchrony in such a relay system against inhomogeneity in the neurons and synaptic parameters. We show that when the inhomogeneity does not violate the symmetry of the system, synchrony is maintained and in some cases inhomogeneity enhances synchrony. On the other hand if the inhomogeneity breaks the symmetry of the system, zero lag synchrony can not be preserved. In this case we give analytical results for the phase lag of the spiking of the neurons in the stable state.

The connectome of the basal ganglia

The basal ganglia of the laboratory rat consist of a few core regions that are specifically interconnected by efferents and afferents of the central nervous system. In nearly 800 reports of tract-tracing investigations the connectivity of the basal ganglia is documented. The readout of connectivity data and the collation of all the connections of these reports in a database allows to generate a connectome. The collation, curation and analysis of such a huge amount of connectivity data is a great challenge and has not been performed before (Bohland et al. PloS One 4:e7200, 2009) in large connectomics projects based on meta-analysis of tract-tracing studies. Here, the basal ganglia connectome of the rat has been generated and analyzed using the consistent cross-platform and generic framework neuroVIISAS. Several advances of this connectome meta-study have been made: the collation of laterality data, the network-analysis of connectivity strengths and the assignment of regions to a hierarchically organized terminology. The basal ganglia connectome offers differences in contralateral connectivity of motoric regions in contrast to other regions. A modularity analysis of the weighted and directed connectome produced a specific grouping of regions. This result indicates a correlation of structural and functional subsystems. As a new finding, significant reciprocal connections of specific network motifs in this connectome were detected. All three principal basal ganglia pathways (direct, indirect, hyperdirect) could be determined in the connectome. By identifying these pathways it was found that there exist many further equivalent pathways possessing the same length and mean connectivity weight as the principal pathways. Based on the connectome data it is unknown why an excitation pattern may prefer principal rather than other equivalent pathways. In addition to these new findings the local graph-theoretical features of regions of the connectome have been determined. By performing graph theoretical analyses it turns out that beside the caudate putamen further regions like the mesencephalic reticular formation, amygdaloid complex and ventral tegmental area are important nodes in the basal ganglia connectome. The connectome data of this meta-study of tract-tracing reports of the basal ganglia are available for further network studies, the integration into neocortical connectomes and further extensive investigations of the basal ganglia dynamics in population simulations.

Generation of dense statistical connectomes from sparse morphological data

Sensory-evoked signal flow, at cellular and network levels, is primarily determined by the synaptic wiring of the underlying neuronal circuitry. Measurements of synaptic innervation, connection probabilities and subcellular organization of synaptic inputs are thus among the most active fields of research in contemporary neuroscience. Methods to measure these quantities range from electrophysiological recordings over reconstructions of dendrite-axon overlap at light-microscopic levels to dense circuit reconstructions of small volumes at electron-microscopic resolution. However, quantitative and complete measurements at subcellular resolution and mesoscopic scales to obtain all local and long-range synaptic in/outputs for any neuron within an entire brain region are beyond present methodological limits. Here, we present a novel concept, implemented within an interactive software environment called NeuroNet, which allows (i) integration of sparsely sampled (sub)cellular morphological data into an accurate anatomical reference frame of the brain region(s) of interest, (ii) up-scaling to generate an average dense model of the neuronal circuitry within the respective brain region(s) and (iii) statistical measurements of synaptic innervation between all neurons within the model. We illustrate our approach by generating a dense average model of the entire rat vibrissal cortex, providing the required anatomical data, and illustrate how to measure synaptic innervation statistically. Comparing our results with data from paired recordings in vitro and in vivo, as well as with reconstructions of synaptic contact sites at light- and electron-microscopic levels, we find that our in silico measurements are in line with previous results.

Meeting the brain on its own terms

In contemporary human brain mapping, it is commonly assumed that the "mind is what the brain does". Based on that assumption, task-based imaging studies of the last three decades measured differences in brain activity that are thought to reflect the exercise of human mental capacities (e.g., perception, attention, memory). With the advancement of resting state studies, tractography and graph theory in the last decade, however, it became possible to study human brain connectivity without relying on cognitive tasks or constructs. It therefore is currently an open question whether the assumption that "the mind is what the brain does" is an indispensable working hypothesis in human brain mapping. This paper argues that the hypothesis is, in fact, dispensable. If it is dropped, researchers can "meet the brain on its own terms" by searching for new, more adequate concepts to describe human brain organization. Neuroscientists can establish such concepts by conducting exploratory experiments that do not test particular cognitive hypotheses. The paper provides a systematic account of exploratory neuroscientific research that would allow researchers to form new concepts and formulate general principles of brain connectivity, and to combine connectivity studies with manipulation methods to identify neural entities in the brain. These research strategies would be most fruitful if applied to the mesoscopic scale of neuronal assemblies, since the organizational principles at this scale are currently largely unknown. This could help researchers to link microscopic and macroscopic evidence to provide a more comprehensive understanding of the human brain. The paper concludes by comparing this account of exploratory neuroscientific experiments to recent proposals for large-scale, discovery-based studies of human brain connectivity.

Functional brain networks: great expectations, hard times and the big leap forward

Many physical and biological systems can be studied using complex network theory, a new statistical physics understanding of graph theory. The recent application of complex network theory to the study of functional brain networks has generated great enthusiasm as it allows addressing hitherto non-standard issues in the field, such as efficiency of brain functioning or vulnerability to damage. However, in spite of its high degree of generality, the theory was originally designed to describe systems profoundly different from the brain. We discuss some important caveats in the wholesale application of existing tools and concepts to a field they were not originally designed to describe. At the same time, we argue that complex network theory has not yet been taken full advantage of, as many of its important aspects are yet to make their appearance in the neuroscience literature. Finally, we propose that, rather than simply borrowing from an existing theory, functional neural networks can inspire a fundamental reformulation of complex network theory, to account for its exquisitely complex functioning mode.

Highlighting the structure-function relationship of the brain with the Ising model and graph theory

With the advent of neuroimaging techniques, it becomes feasible to explore the structure-function relationships in the brain. When the brain is not involved in any cognitive task or stimulated by any external output, it preserves important activities which follow well-defined spatial distribution patterns. Understanding the self-organization of the brain from its anatomical structure, it has been recently suggested to model the observed functional pattern from the structure of white matter fiber bundles. Different models which study synchronization (e.g., the Kuramoto model) or global dynamics (e.g., the Ising model) have shown success in capturing fundamental properties of the brain. In particular, these models can explain the competition between modularity and specialization and the need for integration in the brain. Graphing the functional and structural brain organization supports the model and can also highlight the strategy used to process and organize large amount of information traveling between the different modules. How the flow of information can be prevented or partially destroyed in pathological states, like in severe brain injured patients with disorders of consciousness or by pharmacological induction like in anaesthesia, will also help us to better understand how global or integrated behavior can emerge from local and modular interactions.

A statistical mechanical problem?

The problem of deriving the processes of perception and cognition or the modes of behavior from states of the brain appears to be unsolvable in view of the huge numbers of elements involved. However, neural activities are not random, nor independent, but constrained to form spatio-temporal patterns, and thanks to these restrictions, which in turn are due to connections among neurons, the problem can at least be approached. The situation is similar to what happens in large physical ensembles, where global behaviors are derived by microscopic properties. Despite the obvious differences between neural and physical systems a statistical mechanics approach is almost inescapable, since dynamics of the brain as a whole are clearly determined by the outputs of single neurons. In this paper it will be shown how, starting from very simple systems, connectivity engenders levels of increasing complexity in the functions of the brain depending on specific constraints. Correspondingly levels of explanations must take into account the fundamental role of constraints and assign at each level proper model structures and variables, that, on one hand, emerge from outputs of the lower levels, and yet are specific, in that they ignore irrelevant details.

Identification of important nodes in directed biological networks: a network motif approach

Identification of important nodes in complex networks has attracted an increasing attention over the last decade. Various measures have been proposed to characterize the importance of nodes in complex networks, such as the degree, betweenness and PageRank. Different measures consider different aspects of complex networks. Although there are numerous results reported on undirected complex networks, few results have been reported on directed biological networks. Based on network motifs and principal component analysis (PCA), this paper aims at introducing a new measure to characterize node importance in directed biological networks. Investigations on five real-world biological networks indicate that the proposed method can robustly identify actually important nodes in different networks, such as finding command interneurons, global regulators and non-hub but evolutionary conserved actually important nodes in biological networks. Receiver Operating Characteristic (ROC) curves for the five networks indicate remarkable prediction accuracy of the proposed measure. The proposed index provides an alternative complex network metric. Potential implications of the related investigations include identifying network control and regulation targets, biological networks modeling and analysis, as well as networked medicine.

Prediction of essential proteins based on overlapping essential modules

Many computational methods have been proposed to identify essential proteins by using the topological features of interactome networks. However, the precision of essential protein discovery still needs to be improved. Researches show that majority of hubs (essential proteins) in the yeast interactome network are essential due to their involvement in essential complex biological modules and hubs can be classified into two categories: date hubs and party hubs. In this study, combining with gene expression profiles, we propose a new method to predict essential proteins based on overlapping essential modules, named POEM. In POEM, the original protein interactome network is partitioned into many overlapping essential modules. The frequencies and weighted degrees of proteins in these modules are employed to decide which categories does a protein belong to? The comparative results show that POEM outperforms the classical centrality measures: Degree Centrality (DC), Information Centrality (IC), Eigenvector Centrality (EC), Subgraph Centrality (SC), Betweenness Centrality (BC), Closeness Centrality (CC), Edge Clustering Coefficient Centrality (NC), and two newly proposed essential proteins prediction methods: PeC and CoEWC. Experimental results indicate that the precision of predicting essential proteins can be improved by considering the modularity of proteins and integrating gene expression profiles with network topological features.

Aspects of randomness in neural graph structures

In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the presence of serious uncertainties, such as major undersampling of the experimental data. In the specific case of neural systems, however, a few moderately robust experimental reconstructions have been reported, and these have long served as fundamental prototypes for studying connectivity patterns in the nervous system. In this paper, we provide a comparative analysis of these "historical" graphs, both in their directed (original) and symmetrized (a common preprocessing step) forms, and provide a set of measures that can be consistently applied across graphs (directed or undirected, with or without self-loops). We focus on simple structural characterizations of network connectivity and find that in many measures, the networks studied are captured by simple random graph models. In a few key measures, however, we observe a marked departure from the random graph prediction. Our results suggest that the mechanism of graph formation in the networks studied is not well captured by existing abstract graph models in their first- and second-order connectivity.

Robust design of polyrhythmic neural circuits

Neural circuit motifs producing coexistent rhythmic patterns are treated as building blocks of multifunctional neuronal networks. We study the robustness of such a motif of inhibitory model neurons to reliably sustain bursting polyrhythms under random perturbations. Without noise, the exponential stability of each of the coexisting rhythms increases with strengthened synaptic coupling, thus indicating an increased robustness. Conversely, after adding noise we find that noise-induced rhythm switching intensifies if the coupling strength is increased beyond a critical value, indicating a decreased robustness. We analyze this stochastic arrhythmia and develop a generic description of its dynamic mechanism. Based on our mechanistic insight, we show how physiological parameters of neuronal dynamics and network coupling can be balanced to enhance rhythm robustness against noise. Our findings are applicable to a broad class of relaxation-oscillator networks, including Fitzhugh-Nagumo and other Hodgkin-Huxley-type networks.

Auditory-prefrontal axonal connectivity in the macaque cortex: quantitative assessment of processing streams

Primate sensory systems subserve complex neurocomputational functions. Consequently, these systems are organised anatomically in a distributed fashion, commonly linking areas to form specialised processing streams. Each stream is related to a specific function, as evidenced from studies of the visual cortex, which features rather prominent segregation into spatial and non-spatial domains. It has been hypothesised that other sensory systems, including auditory, are organised in a similar way on the cortical level. Recent studies offer rich qualitative evidence for the dual stream hypothesis. Here we provide a new paradigm to quantitatively uncover these patterns in the auditory system, based on an analysis of multiple anatomical studies using multivariate techniques. As a test case, we also apply our assessment techniques to more ubiquitously-explored visual system. Importantly, the introduced framework opens the possibility for these techniques to be applied to other neural systems featuring a dichotomised organisation, such as language or music perception.

Understanding Structural-Functional Relationships in the Human Brain

Relating the brain’s structural connectivity (SC) to its functional connectivity (FC) is a fundamental goal in neuroscience because it is capable of aiding our understanding of how the relatively fixed SC architecture underlies human cognition and diverse behaviors. With the aid of current noninvasive imaging technologies (e.g., structural MRI, diffusion MRI, and functional MRI) and graph theory methods, researchers have modeled the human brain as a complex network of interacting neuronal elements and characterized the underlying structural and functional connectivity patterns that support diverse cognitive functions. Specifically, research has demonstrated a tight SC-FC coupling, not only in interregional connectivity strength but also in network topologic organizations, such as community, rich-club, and motifs. Moreover, this SC-FC coupling exhibits significant changes in normal development and neuropsychiatric disorders, such as schizophrenia and epilepsy. This review summarizes recent progress regarding the SC-FC relationship of the human brain and emphasizes the important role of large-scale brain networks in the understanding of structural-functional associations. Future research directions related to this topic are also proposed.

Mapping small-world properties through development in the human brain: disruption in schizophrenia

Evidence from imaging studies suggests that the human brain has a small-world network topology that might be disrupted in certain brain disorders. However, current methodology is based on global graph theory measures, such as clustering, C, characteristic path length, L, and small-worldness, S, that lack spatial specificity and are insufficient to identify regional brain abnormalities. Here we propose novel ultra-fast methodology for mapping local properties of brain network topology such as local C, L and S (lC, lL and lS) in the human brain at 3-mm isotropic resolution from ‘resting-state’ magnetic resonance imaging data. Test-retest datasets from 40 healthy children/adolescents were used to demonstrate the overall good reliability of the measures across sessions and computational parameters (intraclass correlation > 0.5 for lC and lL) and their low variability across subjects (< 29%). Whereas regions with high local functional connectivity density (lFCD; local degree) in posterior parietal and occipital cortices demonstrated high lC and short lL, subcortical regions (globus pallidus, thalamus, hippocampus and amygdala), cerebellum (lobes and vermis), cingulum and temporal cortex also had high, lS, demonstrating stronger small-world topology than other hubs. Children/adolescents had stronger lFCD, higher lC and longer lL in most cortical regions and thalamus than 74 healthy adults, consistent with pruning of functional connectivity during maturation. In contrast, lFCD, lC and lL were weaker in thalamus and midbrain, and lL was shorter in frontal cortical regions and cerebellum for 69 schizophrenia patients than for 74 healthy controls, suggesting exaggerated pruning of connectivity in schizophrenia. Follow up correlation analyses for seeds in thalamus and midbrain uncovered lower positive connectivity of these regions in thalamus, putamen, cerebellum and frontal cortex (cingulum, orbitofrontal, inferior frontal) and lower negative connectivity in auditory, visual, motor, premotor and somatosensory cortices for schizophrenia patients than for controls, consistent with prior findings of thalamic disconnection in schizophrenia.

Mechanisms of zero-lag synchronization in cortical motifs

Zero-lag synchronization between distant cortical areas has been observed in a diversity of experimental data sets and between many different regions of the brain. Several computational mechanisms have been proposed to account for such isochronous synchronization in the presence of long conduction delays: Of these, the phenomenon of “dynamical relaying” – a mechanism that relies on a specific network motif – has proven to be the most robust with respect to parameter mismatch and system noise. Surprisingly, despite a contrary belief in the community, the common driving motif is an unreliable means of establishing zero-lag synchrony. Although dynamical relaying has been validated in empirical and computational studies, the deeper dynamical mechanisms and comparison to dynamics on other motifs is lacking. By systematically comparing synchronization on a variety of small motifs, we establish that the presence of a single reciprocally connected pair – a “resonance pair” – plays a crucial role in disambiguating those motifs that foster zero-lag synchrony in the presence of conduction delays (such as dynamical relaying) from those that do not (such as the common driving triad). Remarkably, minor structural changes to the common driving motif that incorporate a reciprocal pair recover robust zero-lag synchrony. The findings are observed in computational models of spiking neurons, populations of spiking neurons and neural mass models, and arise whether the oscillatory systems are periodic, chaotic, noise-free or driven by stochastic inputs. The influence of the resonance pair is also robust to parameter mismatch and asymmetrical time delays amongst the elements of the motif. We call this manner of facilitating zero-lag synchrony resonance-induced synchronization, outline the conditions for its occurrence, and propose that it may be a general mechanism to promote zero-lag synchrony in the brain.

Understanding large-scale neuronal dynamics – and how they relate to the cortical anatomy – is one of the key areas of neuroscience research. Despite a wealth of recent research, the key principles of this relationship have yet to be established. Here we employ computational modeling to study neuronal dynamics on small subgraphs – or motifs – across a hierarchy of spatial scales. We establish a novel organizing principle that we term a “resonance pair” (two mutually coupled nodes), which promotes stable, zero-lag synchrony amongst motif nodes. The bidirectional coupling between a resonance pair acts to mutually adjust their dynamics onto a common and relatively stable synchronized regime, which then propagates and stabilizes the synchronization of other nodes within the motif. Remarkably, we find that this effect can propagate along chains of coupled nodes and hence holds the potential to promote stable zero-lag synchrony in larger sub-networks of cortical systems. Our findings hence suggest a potential unifying account of the existence of zero-lag synchrony, an important phenomenon that may underlie crucial cognitive processes in the brain. Moreover, such pairs of mutually coupled oscillators are found in a wide variety of physical and biological systems suggesting a new, broadly relevant and unifying principle.

Small modifications to network topology can induce stochastic bistable spiking dynamics in a balanced cortical model

Directed random graph models frequently are used successfully in modeling the population dynamics of networks of cortical neurons connected by chemical synapses. Experimental results consistently reveal that neuronal network topology is complex, however, in the sense that it differs statistically from a random network, and differs for classes of neurons that are physiologically different. This suggests that complex network models whose subnetworks have distinct topological structure may be a useful, and more biologically realistic, alternative to random networks. Here we demonstrate that the balanced excitation and inhibition frequently observed in small cortical regions can transiently disappear in otherwise standard neuronal-scale models of fluctuation-driven dynamics, solely because the random network topology was replaced by a complex clustered one, whilst not changing the in-degree of any neurons. In this network, a small subset of cells whose inhibition comes only from outside their local cluster are the cause of bistable population dynamics, where different clusters of these cells irregularly switch back and forth from a sparsely firing state to a highly active state. Transitions to the highly active state occur when a cluster of these cells spikes sufficiently often to cause strong unbalanced positive feedback to each other. Transitions back to the sparsely firing state rely on occasional large fluctuations in the amount of non-local inhibition received. Neurons in the model are homogeneous in their intrinsic dynamics and in-degrees, but differ in the abundance of various directed feedback motifs in which they participate. Our findings suggest that (i) models and simulations should take into account complex structure that varies for neuron and synapse classes; (ii) differences in the dynamics of neurons with similar intrinsic properties may be caused by their membership in distinctive local networks; (iii) it is important to identify neurons that share physiological properties and location, but differ in their connectivity.

Key bifurcations of bursting polyrhythms in 3-cell central pattern generators

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control.

Influence of wiring cost on the large-scale architecture of human cortical connectivity

In the past two decades some fundamental properties of cortical connectivity have been discovered: small-world structure, pronounced hierarchical and modular organisation, and strong core and rich-club structures. A common assumption when interpreting results of this kind is that the observed structural properties are present to enable the brain's function. However, the brain is also embedded into the limited space of the skull and its wiring has associated developmental and metabolic costs. These basic physical and economic aspects place separate, often conflicting, constraints on the brain's connectivity, which must be characterized in order to understand the true relationship between brain structure and function. To address this challenge, here we ask which, and to what extent, aspects of the structural organisation of the brain are conserved if we preserve specific spatial and topological properties of the brain but otherwise randomise its connectivity. We perform a comparative analysis of a connectivity map of the cortical connectome both on high- and low-resolutions utilising three different types of surrogate networks: spatially unconstrained ('random'), connection length preserving ('spatial'), and connection length optimised ('reduced') surrogates. We find that unconstrained randomisation markedly diminishes all investigated architectural properties of cortical connectivity. By contrast, spatial and reduced surrogates largely preserve most properties and, interestingly, often more so in the reduced surrogates. Specifically, our results suggest that the cortical network is less tightly integrated than its spatial constraints would allow, but more strongly segregated than its spatial constraints would necessitate. We additionally find that hierarchical organisation and rich-club structure of the cortical connectivity are largely preserved in spatial and reduced surrogates and hence may be partially attributable to cortical wiring constraints. In contrast, the high modularity and strong s-core of the high-resolution cortical network are significantly stronger than in the surrogates, underlining their potential functional relevance in the brain.

Attractor dynamics in local neuronal networks

Patterns of synaptic connectivity in various regions of the brain are characterized by the presence of synaptic motifs, defined as unidirectional and bidirectional synaptic contacts that follow a particular configuration and link together small groups of neurons. Recent computational work proposes that a relay network (two populations communicating via a third, relay population of neurons) can generate precise patterns of neural synchronization. Here, we employ two distinct models of neuronal dynamics and show that simulated neural circuits designed in this way are caught in a global attractor of activity that prevents neurons from modulating their response on the basis of incoming stimuli. To circumvent the emergence of a fixed global attractor, we propose a mechanism of selective gain inhibition that promotes flexible responses to external stimuli. We suggest that local neuronal circuits may employ this mechanism to generate precise patterns of neural synchronization whose transient nature delimits the occurrence of a brief stimulus.

Experience-dependent rewiring of specific inhibitory connections in adult neocortex

Optogenetic mapping of connectivity in the primary somatosensory cortex shows that mature cortical circuits adapt to sensory deprivation by selectively altering specific network motifs.

Although neocortical connectivity is remarkably stereotyped, the abundance of some wiring motifs varies greatly between cortical areas. To examine if regional wiring differences represent functional adaptations, we have used optogenetic raster stimulation to map the laminar distribution of GABAergic interneurons providing inhibition to pyramidal cells in layer 2/3 (L2/3) of adult mouse barrel cortex during sensory deprivation and recovery. Whisker trimming caused large, motif-specific changes in inhibitory synaptic connectivity: ascending inhibition from deep layers 4 and 5 was attenuated to 20%–45% of baseline, whereas inhibition from superficial layers remained stable (L2/3) or increased moderately (L1). The principal mechanism of deprivation-induced plasticity was motif-specific changes in inhibitory-to-excitatory connection probabilities; the strengths of extant connections were left unaltered. Whisker regrowth restored the original balance of inhibition from deep and superficial layers. Targeted, reversible modifications of specific inhibitory wiring motifs thus contribute to the adaptive remodeling of cortical circuits.

Many natural and engineered networks contain recurring patterns of local connectivity. Although these so-called network motifs are thought to have functional significance, direct tests of the idea that network topology reflects function remain scarce. We have performed such a test in the area of mammalian neocortex that is devoted to the sensory representation of touch. To this end, we equipped inhibitory interneurons in the mouse with light-activated ion channels that allowed us to stimulate interneuron activity optically and record light-evoked inhibitory currents in their postsynaptic partners, thereby revealing maps of connectivity. We find that excitatory pyramidal cells in layer 2/3 of primary somatosensory cortex receive inhibition from GABAergic interneurons located in different cortical layers, with a characteristic balance of inhibitory connections from deep and superficial layers. Trimming the whiskers to remove sensory input in adult animals alters this balance—inhibitory connections from deep cortical layers are depleted, while inhibitory connections from superficial layers are augmented. These changes revert when the whiskers regrow, restoring the original balance between wiring motifs. This see-saw relationship between deep and superficial inhibition demonstrates that mature cortical circuits adapt to functional change by selectively altering specific network motifs.