Scientific Understanding of Consciousness
Consciousness as an Emergent Property of Thalamocortical Activity

Small-World Networks for Cortical Architecture


Science, November 2013: Vol. 342 no. 6158

Cortical High-Density Counterstream Architectures

Nikola T. Markov, Mária Ercsey-Rava, David C. Van Essen, Kenneth Knoblauch, Zoltán Toroczkai, Henry Kennedy

1Stem cell and Brain Research Institute, INSERM U846, 18 Avenue Doyen Lépine, 69500 Bron, France.

2Université de Lyon, Université Lyon I, 69003 Lyon, France.

3Yale University, Department of Neurobiology, New Haven, CT 06520, USA.

4Faculty of Physics, Babeş-Bolyai University, Cluj-Napoca, 400084 Romania.

5Department of Anatomy and Neurobiology, Washington University School of Medicine, St. Louis, MO 63110–1093, USA.

6Department of Physics and Interdisciplinary Center for Network Science and Applications, University of Notre Dame, Notre Dame, IN 46556, USA.

7Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany.


Small-world networks provide an appealing description of cortical architecture owing to their capacity for integration and segregation combined with an economy of connectivity. Previous reports of low-density interareal graphs and apparent small-world properties are challenged by data that reveal high-density cortical graphs in which economy of connections is achieved by weight heterogeneity and distance-weight correlations. These properties define a model that predicts many binary and weighted features of the cortical network including a core-periphery, a typical feature of self-organizing information processing systems. Feedback and feedforward pathways between areas exhibit a dual counterstream organization, and their integration into local circuits constrains cortical computation. Here, we propose a bow-tie representation of interareal architecture derived from the hierarchical laminar weights of pathways between the high-efficiency dense core and periphery.

Because the concepts of localization of function and parcellation into cortical areas are closely intertwined, elucidating the global pattern of areal interactions is central to understanding higher brain functions. Cerebral cortex in the macaque monkey is subdivided into a mosaic of ~100 cortical areas, each displaying characteristic features, including cytoarchitecture. Each area has a characteristic connectivity profile thought to contribute to determining its functional properties. Here, we review how interareal connectivity at the single-cell level, revealed by quantitative anatomical tract tracing, is relevant to our understanding of large-scale cortical networks and their hierarchical organization.

The circuitry of cerebral cortex is dominated by local (within-area) connections, and interareal connections constitute only about 20% of total cortical connectivity. Hence, the dozens of long-distance projections to areas beyond the immediate neighboring areas account for ~ 5%.    Local networks conform in many ways to a canonical microcircuit that spans all cortical layers and includes recurrent excitation presumed to shape and amplify the sparse input from subcortical and distant cortical sources.

Research studies have showed that interareal connectivity obeys hierarchical constraints rooted in the strong anatomical regularities of feedforward and feedback pathways. In this way, multiple distributed cortical hierarchies form a large-scale model of the cortex that reflects the laminar integration of interareal connectivity into local circuits and is relevant to sensory, motor and cognitive systems. The structural features of interareal interactions may provide important insights into the observed dynamics of large-scale interareal networks controlling information flow through the cortex.

Graph theory provides a powerful framework for investigating complex networks such as those found in the brain. Many insights into the functional processes supported by such networks have been gleaned from analysis at the binary level (i.e., connections present or absent). One important class of models that has received much attention is that of small-world (SW) networks, distinguished by high clustering coupled with a short average path length (also called characteristic path length) across the graph. The relevance of the SW property to understanding the cortex comes from its proposed capacity to optimize essential cortical features, including functional integration and segregation.

Hierarchical Organization

More than 70% of all the projections to a given locus on the cortical sheet arise from within 1.5 to 2.5 mm, so that cortical connectivity is dominated by short-distance, local connections that conform to a canonical microcircuit optimized to amplify and shape weaker long-distance cortical inputs. Therefore, when considering long-distance interareal pathways, it is important to consider not only the strength and the specificity of the connections, but also their pronounced laminar asymmetry determined by the direction of the connection. Hence, feedforward (FF) pathways (mostly directed rostrally) originate principally from supragranular layers and terminate in layer 4 in higher areas, whereas feedback (FB) pathways (mostly directed caudally) originate mainly from infragranular layers in higher areas and avoid layer 4 in lower areas. Pairwise comparison of the connections has been used to reveal cortical hierarchies. While the FVE model is indeterminate, it can be partially resolved by using a continuous scale such as hierarchical distance based on the fraction of supragranular layer neurons (SLNs). The SLN index quantifies an order relation between areas as defined by their laminar profiles of connectivity and thereby allows estimation of the hierarchical distance separating them. Further analysis shows that connections between neighboring areas have the highest weight, defined as the fraction of labeled neurons (FLN) whereas long-distance FF and FB connections have lower weights.

FF and FB processes are distinct physiologically. A useful, albeit oversimplified characterization is that FF connections are “driving” and FB connections are “modulatory”. Further, FF and FB pathways engage different glutamate receptor subtypes. Not only do FF and FB constitute distinct populations, but they also form two segregated streams, consistent with earlier observations. The supra- and the infragranular layers each have a counterstream organization, most pronounced in the supragranular layers. Further, the supragranular counterstream showed a point-to-point (i.e., topographical) precise connectivity in both FF and FB directions, whereas the infragranular counterstream has a more diffuse topography showing high divergence and convergence in both directions.

The integration of interareal connections into the canonical microcircuit of a target area presumably determines how top-down and bottom-up streams affect processing within the target area. The concept of hierarchical processing has influenced theories of cortical computation, predictive coding, and emerging concepts of inference in cortical function. Recent progress in elucidating interareal communication includes the demonstration of gamma-band phase coherence in the supragranular and beta-band coherence in the infragranular layers. These differences in coherence reflect differences in interareal synchronization, which are thought to facilitate effective communication. Recently beta-gamma asymmetries have been shown to correlate with the aforementioned SLN fraction, suggesting a match between anatomically and functionally defined hierarchies.

Rich-Club and Bow-Tie Structure

Previous studies based on sparse cortical networks have suggested an important structural heterogeneity in the cortex, where hub areas are statistically more interconnected than expected, forming a so-called rich club or central core.

A Cortical Distance Rule as Cost-of-Wiring Principle: The EDR Model

The above observations suggest a general picture of the interareal network: It is a dense network, but with high binary specificity ensured by long-distance connections and characterized by the existence of a core-periphery structure organized into a bow tie via FF/FB pathways.

A clue to the importance of weight-distance relations for understanding the properties of the cortical network comes from the observation that the FLN weights are highly heterogeneous, following a log-normal distribution varying over five orders of magnitude. The log-normal distribution may directly reflect the interplay between metabolic costs associated with projection lengths and a geometrical or spatial property of areal locations. Axonal projections out to a distance d through the white matter come at an energy (metabolic) cost, irrespective of the areas involved. This is suggested by the exponential decay of the number of labeled neurons as a function of projection distance d: p(d) = cexp(–λd), corresponding to an exponential distance rule (EDR). The spatial decay constant λ = 0.188 mm–1 expresses the growth rate of the metabolic cost with distance. We therefore expect that the FLN between two areas separated by a distance d is determined to a first approximation by this cost, independently of areal identity. A relevant spatial property, expressed by the fraction of area pairs separated by a distance d, is well approximated by a truncated Gaussian. Hence, combining the EDR with the Gaussian distribution of interareal distances, we find that the distribution of area pairs with a given FLN obeys a log-normal distribution.

Efficiency of Information Transfer

A simple measure of bandwidth for information transfer in complex networks can be defined via the average conductance between all source-target pairs in the network called global efficiency, or (Eg). Conductance here is interpreted as in physics, by the inverse resistance of the directed path of minimal total path resistance through the network from the source node to the target node. The path resistance can be interpreted as the negative logarithm of the probability that activity in the source node will generate activity in the target node. Here, we equate bandwidth with axon number as reflected by FLN. Thus, a sequence of edges having large FLN values (or “high bandwidth” edges) directed from source to target would form a path of low resistance (high conductance), providing a high-bandwidth pathway for information transmission. To obtain a graded measure of global efficiency within the structure of G29x29 and to understand the role of weak projections, we computed Eg on the remaining network after the sequential removal of the weakest link (smallest FLN) and plotted it as function of the network density. The global efficiency of the network does not change before 76% of the weakest links are removed, indicating the existence of a high global efficiency (high bandwidth) backbone formed by short-range paths. Indeed, the average length of the remaining edges at 24% remaining density is 16 mm compared to the 27-mm average length of the removed edges. This suggests that the cortical network is organized in such a manner as to be independent of the activity along the weak projections for high-bandwidth information transfer. We speculate that these long-range pathways, which we have shown to have a high binary specificity, may contribute to interareal synchronization between cooperating areas.

Concluding Remarks

In summary, the interareal network achieves economy of connectivity and communication efficiency by means of a distribution of weights, spatial organization and a core-periphery structure in the form of a bow tie with a dense core. Interareal connections integrate across the local circuits via dual counter-streams located in the supra- and infragranular layers, which have distinct physiological properties.

The broadband connections involving large number of cells that form the global efficiency backbone and that are expected to shape receptive fields in target areas are short range and strongly cluster areas of a given modality. The considerably more numerous long-distance connections are sparse and exhibit high binary specificity; we speculate that these connections serve to promote cortical communication by controlling oscillatory coherence, possibly via contraction dynamics. During primate evolution the increase in numbers of areas and their spatially restricted broadband connections raises the possibility that increases in brain size would lead to increased isolation of clustered communities. We hypothesize that weak long-distance connections appeared during recent evolution and may facilitate cooperation between distant areas. Assuming that a high-density connectivity is a characteristic feature of the interareal cortical graph, long-distance weak links may allow preservation of the connectedness of the cortical network following cortical expansion. Further, long-distance connections allow accommodation of novel information processing, thereby building over evolutionary time upon the existing repertoire of dynamic functions. There are a number of advantages to be gained by developing a high efficiency central core that we along with others have observed. Long-distance connections particularly for the higher association areas play a special role in forming the cortical core. The high incidence of long-distance projections involving prefrontal areas suggests a particular importance of this cortical region in core-periphery integration, indicating the cortical core as an important component of higher order interareal architectures.

Self-organization in cortical development is well established. The findings summarized above suggest interesting parallels with other, highly functional, self-organizing information-processing networks. The high density, yet highly specific nature, of G29x29 suggests a heterogeneous, expandable, and cost-efficient information-processing network subject to evolutionary constraints. In particular, we find (i) high connectivity, (ii) high global accessibility, and (iii) large path diversity. Additional evidence supports several further constraints: (iv) a high bisection bandwidth; (v) resilience to connectivity failures (global conductance measurements show that during sequential removal of the weakest links, a substantial decline in the network’s global efficiency is not reached until a density of 24%); and (vi) incremental expandability that allows addition of new areas to the network during evolution, without a substantial wiring overhaul to maintain or improve performance.

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