2003 Special IssueNetwork participation indices: characterizing component roles for information processing in neural networks
Introduction
Organisation and mechanisms of neural networks in living organisms are poorly understood. The only large-scale network that has been completely reconstructed is the nervous system of the nematode worm Caenorhabditis elegans with 302 neurons in the hermaphrodite individual (White et al., 1986, Chen et al., 2002). Larger networks are nearly impossible to reconstruct in detail for the sheer number of neurons: already the nervous system of the fruitfly Drosophila melanogaster consists of about 100,000 neurons, the cerebral cortex of the mouse comprises 2×107 neurons, and the human neocortex is estimated to contain about 4×1010 neurons, which approximates the number of stars in our galaxy (Braitenberg and Schüz, 1991, Cragg, 1975, Williams, 2000). Therefore, anatomical reconstructions of these nervous systems are either limited to the detailed investigation of a very small local network or based on global estimates extrapolating from local samples (Braendgaard et al., 1990, Cragg, 1975, Braitenberg and Schüz, 1991, Murre and Sturdy, 1995).
Over the last decade much effort has been made to reconstruct large-scale biological networks at the coarser level of brain regions in rats, cats and macaques (Burns and Young, 2000, Felleman and Van Essen, 1991, Kamper et al., 2002, Scannell et al., 1995, Scannell et al., 1999, Stephan et al., 2001, Young, 1993, Young et al., 1994). Extensive analysis of these networks has provided insights into e.g. the hierarchical arrangement, the cluster structure, and the likely spread of activity to produce topographically specific activation patterns (Felleman and Van Essen, 1991, Hilgetag et al., 1996, Hilgetag et al., 2000, Kötter and Sommer, 2000, Kötter et al., 2001a, Kötter et al., 2002, Petroni et al., 2001, Sporns et al., 2000, Stephan et al., 2000).
Interestingly, the reduced network models that have been created on the basis of neuronal geometry differ in a profound way from the networks that are defined by inter-regional connectivity patterns: The former tend to describe nervous systems as either systematically or randomly connected networks, whose functional specialization is assumed to arise from superimposed learning processes. Correspondingly, much research effort is directed at the investigation of the principles of synaptic modification. The latter networks, by contrast, have been found to be neither regular nor random, but somewhere in between the two extremes, expressing a characteristic ‘small world’ connectivity pattern: the combination of dense local clusters of network components combined with short path lengths between them (Watts and Strogatz, 1998, Hilgetag et al., 2000, Stephan et al., 2000). A local differentiation defined by the empirical connectivity patterns is intrinsically present, most likely as a result of phylogenetic and ontogenetic selection mechanisms. Although the level of resolution of this intrinsic differentiation is coarse compared to what could be gained from knowing the detailed neuronal connectivity patterns, it offers a formidable opportunity to explore the modular processing structure of real brains.
How to identify candidate processing structures is a major question. Inspiration has been taken from general network measures in graph theory (Hilgetag et al., 2002, Sporns, 2002); alternatively, a brute force method has been used to identify at least some simple network motifs (Milo et al., 2002). The criterion for what feature qualifies as interesting is defined usually by statistical comparison of the frequency of its occurrence with randomly or somehow regularly connected networks. A more principled approach, however, is highly desirable since (1) there is no a priori reason to assume that only unusual features are important and (2) even the unusual frequency of occurrence does not tell us what the feature is good for, i.e. it does not yet create understanding.
Our endeavor here is to make progress with relating specific connectional patterns in real neural networks to certain types of interactions among brain regions. The advances made are of four kinds: (1) we consider explicitly the case of missing information, which is a common problem in the analysis of biological networks with substantial influence on the results; (2) we present a new classification of network nodes based on some simple characteristics of their participation in the network, i.e. the density, convergence/divergence, and symmetry of a node's afferent and efferent connections; (3) we propose a relationship between certain types of network participation and specific modes of information transfer that leads to a functional characterization of nodes using terms such as ‘sender’, ‘receiver’, ‘relay’, etc. (4) we apply this approach to the real neural network architecture of the brain and identify several regions that show interesting types of network participation with functional relevance. The key message of this paper is that the network organization as viewed from the point of an individual node or brain region and characterized through our network participation indices (NPIs), provides clues as to the possible roles that this node can take in information processing and helps to understand how segregated and integrated representations co-exist in cortical networks.
Section snippets
Network participation indices
We provide formal descriptions of three simple structural characteristics of neural networks that are computed for each node individually: (i) connection density, (ii) relative proportions between efferent and afferent connections, and (iii) symmetry of connections. An important difference to conventional graph-theoretical approaches towards describing these features is that, for real-world neural connectivity data beyond C. elegans, not all information about the presence or absence of a
Results
We restricted our analyses to data sets that had already been investigated in the past with other analytical methods; this facilitates the assessment of insights gained from the NPIs relative to established techniques.
Prefrontal cortex
The comparison of the analytic results for connections between Walker's (1940) areas in prefrontal cortex presented in Fig. 2 illustrates that the assumptions made about unknown connectivity information can have profound effects on the interpretation of connectivity data. Note that even for area 8B, which has the largest proportion of missing data, 64% of the matrix entries were known. This is considerably more than the average for most published large-scale connectivity matrices, including
Acknowledgements
Supported by the DFG (LIS 4-554 95 (2) Düsseldorf and Graduate School 320). We thank Claus Hilgetag for supplying the connectivity matrix underlying Fig. 6.
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