## Relating Connectivity and Dynamics

How do the different dimensions of brain connectivity relate to one another? Answering this question demands the combined manipulation and analysis of structural and dynamic attributes. In this final section of the chapter, we briefly review several recent lines of research that have attempted to bridge structural, functional and effective connectivity with the use of computational modeling.

A crucial first step in linking structure to function involves the identification of functionally integrated and structurally connected networks that are potential building blocks of cognitive architectures. Effective integration of neural activity requires causal interactions, which must operate through physical connections. In fact, structural connectivity provides a first approach towards determining potential functional units, by revealing connectedness and modularity within graphs, as defined above. A next step is the identification of functional clusters, using informational measures (Tononi et al., 1998b; Sporns et al., 2000a) or through the application of standard clustering techniques to functional connectivity patterns. Finally, effective connectivity measures such as information integration (Tononi and Sporns, 2003) can aid in the delineation of causally linked neural clusters and complexes. Mapping such clusters in the course of cognitive function would help to identify brain regions that are generating specific cognitive states and discriminate them from others that are activated, but not causally engaged.

Computational approaches allow the systematic study of how neural dynamics is shaped by the structure of connection patterns linking individual elements (Fig. 7). This has been investigated in detailed computer simulations of cortical networks with heterogeneous (Jirsa and Kelso, 2000; Jirsa 2004; Assisi et al., 2005) and spatially patterned (Sporns, 2004) connection topologies. It was found that different connection topologies generated different modes of neuronal dynamics, and some systematic tendencies could be identified. For example, connectivity patterns containing locally clustered connections with a small admixture of long-range connections were shown to exhibit robust small-world attributes (Sporns and Zwi, 2004; Sporns, 2004; Kaiser and Hilgetag, 2006), while conserving wiring length. These connectivity patterns also gave rise to functional connectivity of high complexity with heterogeneous spatially and temporally highly organized patterns. These computational studies suggest the hypothesis that only specific classes of connectivity patterns (which turn out to be structurally similar to cortical networks) simultaneously support short wiring, small-world attributes, clustered architectures (all structural features), and high complexity (a global property of functional connectivity).

The discovery of small-world connectivity patterns in functional connectivity patterns derived from fMRI, EEG and MEG studies (Stam, 2004; Salvador et al., 2005b; Achard et al., 2006; Bassett and Bullmore 2006; Salvador et al., 2007) raises the question of how closely functional connections map onto structural connections. The state- and task-dependence of functional connectivity suggests that a one-to-one mapping of structural to functional connections does not exist. However, it is likely that at least some structural characteristics of individual nodes are reflected in their functional interactions - for example, hub regions should maintain larger numbers of functional relations. A variety of neuro-computational models have suggested that small-world connectivity imposes specific constraints on neural dynamics at the large scale. Numerous studies suggest that small-world attributes facilitate synchronization and sustained activity, irrespective of the details of the node dynamics that are employed in the model (Nishikawa et al., 2003; Buszaki et al., 2004; Masuda and Aihara, 2004; Netoff et al., 2004; Roxin et al., 2004). Synchronization-based rewiring rules promote the emergence of small-world architectures from random topologies (Gong and van Leeuwen, 2004), underscoring the reciprocal "symbiotic" relationship between neural dynamics and underlying brain architectures (Breakspear et al., 2006). Plasticity rules shape structural connectivity, resulting in neural dynamics that in turn shapes plasticity.

Yet another interesting connection between structural connectivity and global dynamics is based on the idea that the continual integration and redistribution of neuronal impulses represents a critical branching process (Beggs anatomy dynamics

Fig. 7. Relation of structural connectivity and functional dynamics in intermediate-scale cortical networks. The model consists of a single map of 40 x 40 nonlinear neuronal units, each modeled as a single reciprocally coupled Wilson-Cowan excitatory and inhibitory unit, interconnected in different patterns. Three patterns of structural connectivity are shown: "sparse" (intra-map connections absent), "uniform" (intra-map connections are assigned at random with uniform probability across the map), and "clustered" (most intra-map connections are generated within a local neighborhood, with a small admixture of longer-range connections). The "clustered" pattern is most like the one found in cortex. Panels at the left show connection patterns of 40 randomly chosen units, middle panels show a single frame of the unfolding neural dynamics (mpeg movies are available at http://www.indiana.edu/~cortex/complexity.html), and rightmost panels show spatially averaged activity traces obtained from near the center of the map (circled area). The values for the characteristic path length A(G), clustering coefficient 7(G), complexity C(X), and total wiring length (lwire) were: A = 0, 7 = 0, C(X) = 0.143, Iwire = 0 ("sparse"); A = 3.1310, 7 = 0.0076, C(X) = 0.289, Wire = 10, 807 ("uniform"); A = 5.6878, 7 = 0.2637, C(X) = 0.579, Wire = 1,509 ("clustered"), all means of 5 runs. Note that C(X) is highest for the "clustered" network, which shows a rich set of spatiotemporal patterns including waves and spirals. This network also exhibits small-world attributes (low A(G), high 7(G)) and short wiring length. Modified after Sporns (2004)

and Plenz, 2003; Haldeman and Beggs, 2005; see also Beggs et al., 2007). In neural architectures, critical branching processes give rise to sequences of propagating spikes that form neuronal avalanches. In the critical regime, the branching parameter expressing the ratio of descendant spikes from ancestor spikes is found to be near unity, such that a triggering event causes a long chain of spikes that neither dies out quickly (subcriticality) nor grows explosively (supercriticality). Slice preparations of rat cortex operate at or near criticality, generating neuronal avalanches with a size distribution following a power law (Beggs and Plenz, 2003). Criticality is found to be associated with maximal information transfer and thus high efficacy of neuronal information processing, as well as with a maximal number of metastable dynamical states. These results point to additional important links between structural connectivity patterns and the informational processes carried out within them.

The association of certain kinds of dynamics with particular features of structural connectivity opens up a new computational approach. If we fix key aspects of the dynamics (for example, by enforcing a high value of integration or complexity, or of information integration) and then search for connection patterns that are compatible with this type of dynamics, what relationship, if any, do we find? For example, what kinds of structural connection patterns are associated with high values for integration, complexity or information integration? We used complexity (and other information theoretical measures of functional connectivity, such as entropy or integration) as cost functions in simulations designed to optimize network architectures and found that networks that are optimized for high complexity develop structural motifs that are very similar to those observed in real cortical connection matrices (Sporns et al., 2000a; 2000b; Sporns and Tononi, 2002). Specifically, such networks exhibit an abundance of reciprocal (reentrant) connections, a strong tendency to form clusters and they have short characteristic path lengths. Other measures (entropy or integration) produce networks with strikingly different structural characteristics. While it is computationally expensive to employ most types of nonlinear dynamics in the context of such optimizations, a closer examination of specific connection topologies (sparse, uniform and clustered, or cortexlike) that are simulated as nonlinear systems has shown that the association of small-world attributes and complex functional dynamics can hold for more realistic models of cortical architectures as well (Sporns, 2004; Fig. 7). Thus, high complexity, a measure of global statistical features and of functional connectivity, appears to be strongly and uniquely associated with the emergence of small-world networks (Sporns et al., 2004; Sporns, 2006).

Evolutionary algorithms for growing connectivity patterns have been used in evolving motor controllers (Psujek et al., 2006), networks for path integration (Vickerstaff and DiPaolo, 2005), or in the context of sensorimotor coordination (Seth and Edelman, 2004; Seth, 2005). While many current applications, for example those used in evolutionary robotics, rely on small networks with limited connectivity patterns (due to constraints requiring the convergence of evolutionary algorithms in finite computational time), the gap to larger, more brain-like networks is rapidly closing. An exciting future avenue for computational research in this area involves the evolution of behaviorally capable architectures that incorporate features of biological organization. Results from this research may ultimately contribute to resolving long-standing controversies such as whether biological evolution inherently tends towards biological structures of greater and greater complexity. Initial studies of evolving connectivity patterns embedded in simulated creatures within a computational ecology (Yaeger and Sporns, 2006) suggest that as econiches become more demanding neural architectures evolve towards greater structural elaboration, elevated levels of plasticity, and with functional activity patterns of higher neural complexity.

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