Introduction

Neural connections of the mammalian cerebral cortex exhibit specific patterns ranging in scale from interconnections linking whole brain regions to intraareal patterns of connections between cell populations or individual cortical neurons (Cajal, 1909; Brodmann, 1909; Zeki, 1993; Salin and Bullier, 1995; Swanson, 2003). Detailed anatomical and physiological studies have revealed many of the basic components and interconnections of cortical microcircuitry (Douglas and Martin, 1991), and of their arrangement into columns and mini-columns (Mountcastle, 1978; 1997). Columns and other localized populations of neurons maintain connections within and between brain regions, constituting large-scale patterns of anatomical connectivity. While the large-scale networks of human cortex remain largely unmapped (Sporns et al., 2005), comprehensive descriptions of anatomical patterns of cortical connectivity have been collated for several other mammalian species (e.g. Felleman and Van Essen, 1991; Scannell et al., 1999). Closer analysis has revealed that these patterns are neither completely regular nor completely random, but combine structural aspects of both of these extremes (reviewed in Sporns et al., 2004). This basic insight has sparked significant interest in characterizing the structure of brain networks, using methods that are also applied in parallel efforts to map and describe other biological networks, e.g. those of cellular metabolism, gene regulation, or ecology. This chapter is intended as an overview of recent quantitative approaches to brain networks (see also Sporns, 2005), with an emphasis on theoretical and computational studies that inform us about the structural features that determine functional brain dynamics.

In neuroscience, the term connectivity has multiple meanings and connotations that are sometimes difficult to define or disentangle (Horwitz, 2003; Lee et al., 2003). A fundamental distinction is that between structural, functional and effective connectivity, and we will adhere to this distinction for the remainder of the chapter. Anatomical connectivity is the set of physical or structural (synaptic) connections linking neurons within the network, as well as their associated structural biophysical attributes encapsulated in parameters such as strength or effectiveness. Anatomical connections range in scale from local circuits to large-scale networks of inter-regional pathways. Their physical pattern may be thought of as relatively static at shorter time scales (seconds to minutes), but may be plastic or dynamic at longer time scales (hours to days), for example during learning or development. Functional connectivity (Friston, 1993; 1994) captures patterns of deviations from statistical independence between distributed and often spatially remote neuronal units, measuring their correlation/covariance, spectral coherence or phase-locking. Functional connectivity is highly time-dependent (on a scale of hundreds of milliseconds) and does not make any explicit reference to causal effects or an underlying structural model. Effective connectivity describes the network of causal effects of one neural system over another (Friston, 1994; Biichel and Friston, 2000), and can be inferred experimentally through perturbations or time series analysis. Unlike functional connectivity, effective connectivity is not "model-free", but usually requires the specification of a causal model including structural parameters.

The relationship between anatomical, functional and effective connectivity in the cortex currently represents one of the most significant challenges to computational cognitive neuroscience. An emerging view suggests that structural connection patterns are major determinants of the functional dynamics of cortical circuits and systems, as captured by functional or effective connectivity. According to this view, structural connections are essential for shaping patterns of activation and co-activation associated with specific cognitive states. Two potential linking principles are those of segregation and integration (Tononi et al., 1994; 1998; Friston, 2002; 2005). Segregation and integration are found throughout a broad range of cortical systems and may represent a set of complementary organizational principles. We will start this review by considering segregation and integration as basic principles, before turning to methods and approaches aimed at quantifying structural connection patterns, global measures of brain dynamics, and their interrelations.

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