Review TRENDS in Cognitive Sciences Vol.8 No.9 September 2004 Box 3.Growing complex networks:local rules and global design Local spatial affect fun oTgrglebalcioertgkhurtfaistogoneratemutiolecusters,as spired m on k en andglobal network design nore the fact that cortical netw esta to t n distr is s in Fig ure 1.This traint.as enath both the cortical and s rts with an are ra nstitut nt compete men 175] ced 600 200 100 dista ces among area freenetw orks [11].Such topological algorithms ogically realistic and do not repre ha combines cro be 1m8 Box )Alternative developmental algorithms were form nodes of proposed recently that acknowledge spatial constraints graph,and their temporal correlation matrix forms the D important challenge to refine these com atational models defining as 'connected'those voxels that are func th of da that is correlated beyond certai hreshold r.. and play in network growt ed in the densit teristic and robust pr rties (Figure 1c).Their degre ary substar ially bet this s [491.It is lity of finding a ity do aches that couple equivalent random networks),although the clustering logy to the dynamical evolution 58 order e larger perIntuitively plausible growth mechanisms have been proposed for the large classes of small-world [10] and scale-free networks [11]. Such topological algorithms, however, are not biologically realistic and do not represent good models for the development of cortical networks (Box 3). Alternative developmental algorithms were proposed recently that acknowledge spatial constraints in biological systems, while also yielding different types of scale-free and small-world networks [46,47]. It will be an important challenge to refine these computational models by drawing on the wealth of data available from studies in developmental neurobiology [48], to reproduce the specific organization of cortical networks. Especially intriguing is the role that experience might play in network growth. Although the same complement of connections appears to exist in different individuals of a species, the density of specific cortical fiber pathways can vary substantially between individual brains [49]. It is currently not clear whether this variability is partly attributable to activity-dependent processes. If so, it might be described by recent approaches that couple changes in connection topology to the dynamical evolution of connection weights [50]. Functional networks and neural dynamics Scale-free functional brain networks Dodel [51] developed a deterministic clustering method that combines cross-correlations between fMRI signal time courses, and elements of graph theory to reveal brain functional connectivity. Image voxels form nodes of a graph, and their temporal correlation matrix forms the weight matrix of the edges between the nodes. Thus a network can be implemented based entirely on fMRI data, defining as ‘connected’ those voxels that are functionally linked, that is correlated beyond a certain threshold rc. A set of experiments examined the resulting functional brain networks [52], obtained from human visual and motor cortex during a finger-tapping task. Over a wide range of threshold values rc the functional correlation matrix resulted in clearly defined networks with charac￾teristic and robust properties (Figure 1c). Their degree distribution (Figure 1d) and the probability of finding a link versus metric distance both decay as a power law. Their CHARACTERISTIC PATH LENGTH is short (similar to that of equivalent random networks), although the clustering coefficient is several orders of magnitude larger. Scaling and small-world properties persisted across different Box 3. Growing complex networks: local rules and global design Local spatial growth rules. To understand how various developmental factors affect functional specializations of brain networks, it is helpful to consider biologically inspired models based on known constraints of neural development. Previous algorithms for the generation of random and scale-free networks (see Box 1) constitute unlikely growth algorithms, as they ignore the fact that cortical networks develop in space. Preferential attachment [2], for instance, would establish links to hubs indepen￾dent of their distance. In biological networks, however, long-distance connections are rare, in part because the concentration of diffusible signaling and growth factors decays with distance. Accounting for this constraint, a spatial growth model was presented [46] in which growth starts with two nodes, and a new node is added at each step. The establishment of connections from a new node u to one of the existing nodes v depends on the distance d(u,v) between nodes, that is, Pðu; vÞZb eKa dðu;vÞ : This spatial growth mechanism can lead to networks with similar clustering coefficients and charac￾teristic path lengths as in cortical networks when growth limits are present, such as extrinsic limits imposed by volume constraints. Lower clustering results if the developing model network does not reach the spatial borders and path lengths among areas increase [47]. By comparison, a preferential attachment model might yield similar global properties, but fails to generate multiple clusters, as found in cortical networks. .and global network design Can local spatial growth rules yield the known corticocortical topology? In addition to similar global properties, defined by clustering coefficient and characteristic path length, the generated networks also exhibit wiring properties similar to the macaque cortex, whose network and wiring distribution is shown in Figure I. This supports the idea that the likelihood of long-range connections among cortical areas of the macaque decreases with distance [47]. Total wiring length both in the cortical and spatially grown networks lies between benchmark networks in which connections are randomly chosen, and in which only the shortest-possible connections are established. The (few) long-range connections existing in the biological networks might constitute shortcuts, ensuring short average paths with only few intermediate nodes. Thus, the minimiz￾ation of this property might compete with global wiring length minimization. As a by-product, the short-distance preference for inter-area connections during spatial growth can lead to optimal component placement [75] without the need of a posteriori optimization. 0 10 20 30 40 50 60 0 100 200 300 400 500 600 Approx. fiber length (mm) Occurrences (a) (b) Figure I. Connectivity and idealized wiring lengths of the macaque cortex. (a) Cortex with associated long-range connectivity among areas (based on Ref. [23]). The connection matrix represents data of three different studies obtained from the CoCoMac database (http://cocomac.org/home.htm). Node positions were calculated by surface coordinates using corresponding parcellation schemes within the Caret software (http://brainmap.wustl.edu/caret/). (b) Distribution of approximate fiber length as calculated by the direct Euclidian distance between the average spatial positions of brain areas. Note that the layout of areas on the cortical sheet might impose limits on the distribution of distances among areas. Nevertheless, the figure indicates that some cortical projections can reach considerable length. (Redrawn from [47].) 422 Review TRENDS in Cognitive Sciences Vol.8 No.9 September 2004 www.sciencedirect.com