Elsevier

Clinical Neurophysiology

Volume 118, Issue 11, November 2007, Pages 2317-2331
Clinical Neurophysiology

Invited review
The application of graph theoretical analysis to complex networks in the brain

https://doi.org/10.1016/j.clinph.2007.08.010Get rights and content

Abstract

Considering the brain as a complex network of interacting dynamical systems offers new insights into higher level brain processes such as memory, planning, and abstract reasoning as well as various types of brain pathophysiology. This viewpoint provides the opportunity to apply new insights in network sciences, such as the discovery of small world and scale free networks, to data on anatomical and functional connectivity in the brain. In this review we start with some background knowledge on the history and recent advances in network theories in general. We emphasize the correlation between the structural properties of networks and the dynamics of these networks. We subsequently demonstrate through evidence from computational studies, in vivo experiments, and functional MRI, EEG and MEG studies in humans, that both the functional and anatomical connectivity of the healthy brain have many features of a small world network, but only to a limited extent of a scale free network. The small world structure of neural networks is hypothesized to reflect an optimal configuration associated with rapid synchronization and information transfer, minimal wiring costs, resilience to certain types of damage, as well as a balance between local processing and global integration. Eventually, we review the current knowledge on the effects of focal and diffuse brain disease on neural network characteristics, and demonstrate increasing evidence that both cognitive and psychiatric disturbances, as well as risk of epileptic seizures, are correlated with (changes in) functional network architectural features.

Introduction

Traditionally, neuroscientists correlate ‘focal’ brain lesions, for instance brain tumors, with ‘focal’ clinical deficits. This approach gave important insights into the localization of brain functions; a classical example is the identification of the motor speech center in the lower left frontal cortex by the French neurologist Paul Broca at the end of the 19th century. Particularly during the last decades of the 20th century, this essentially reductionistic program led to significant progress in neuroscience in terms of molecular and genetic mechanisms.

Despite the impressive increase of knowledge in neuroscience, however, progress in true understanding of higher level brain processes has been disappointing. Evidence has accumulated that functional networks throughout the brain are necessary, particularly for higher cognitive functions such as memory, planning, and abstract reasoning. It is more and more acknowledged that the brain should be conceived as a complex network of dynamical systems, consisting of numerous functional interactions between closely related as well as more remote brain areas (Varela et al., 2001).

Evaluation of the strength and temporal and spatial patterns of interactions in the brain and the characteristics of the underlying functional and anatomical networks may contribute substantially to the understanding of brain function and dysfunction. A major advantage of this approach is that a lot can be learned from other fields of science, particularly the social sciences, that are also devoted to the study of complex systems. In the last decade of the 20th century, considerable progress has been made in the study of complex systems consisting of large numbers of weakly interacting elements. The modern theory of networks, which is derived from graph theory, has proven to be particularly valuable for this purpose (Amaral and Ottino, 2004, Boccaletti et al., 2006).

The aim of this paper is to review applications of network theories to neuroscience in general, and clinical neurophysiology in particular (a more technically oriented review can be found in Stam and Reijneveld (Stam and Reijneveld, 2007)). After a brief historical introduction, we will summarize the basic characteristics of networks in general, and some important results on the relation between network properties and dynamical processes in these networks. Subsequently we will discuss applications of network theories in experimental neuroscience, both neuro-anatomical and neurophysiological. In the last section of this review, we will determine the effect of interventions or brain disease on neural network properties, in the light of patients with disturbed brain function, e.g. cognitive disturbances, epilepsy, and psychiatric disorders.

Section snippets

Historical background

The modern theory of networks has its roots both in mathematics and sociology. In 1736 the mathematician Leonard Euler solved the problem of ‘the bridges of Konigsberg’. This problem involved the question whether it was possible to make a walk crossing exactly one time each of the seven bridges connecting the two islands in the river Pregel and its shores. Euler proved that this is not possible by representing the problem as an abstract network: a ‘graph’. This is often considered as the first

Basics of modern network theory

The discovery of small world networks and scale free networks set off a large body of theoretical and experimental research, which has led to increasing knowledge on various aspects of network properties in the last decade. Before we move on to the application of network theories to experimental neural networks, and healthy and diseased brain, we will provide some basic knowledge on several aspects of network properties. As mentioned before, more detailed mathematical descriptions can be found

Do biological neural networks display small world and scale free properties?

An important question is to what extent the abovementioned features are relevant for networks of neuron-like elements. It is thought that generally speaking the brain is faced with two opposing requirements: (i) segregation, which means local specialization for specific tasks and (ii) integration, combining all the information at a global level (Sporns et al., 2000a, Sporns et al., 2000b, McIntosh, 2000). The first key question is which kind of anatomical and functional architecture allows both

What is the effect of damage on network properties?

Considering functional connectivity and network properties to be a physiological substrate for segregated and distributed information processing (Salvador et al., 2005a, Salvador et al., 2005b, Achard et al., 2006, Achard and Bullmore, 2007), intervention, whether on purpose (e.g. medication) or accidentally (e.g. brain disease), would lead to changes in these parameters. In the next section, we will summarize existing evidence on the effect of intentional manipulation or disease-related

Conclusions and future prospects

In this review, we demonstrate that the modern network theories provide a very useful framework for the study of complex networks in the brain. They offer powerful realistic models and an increasing number of measures to study complex networks in the brain, thereby enabling better understanding of the correlation between network structure and the processes taking place in these networks, in particular synchronization processes, and providing scenarios how complex networks might respond to

Acknowledgement

Thanks to Els van Deventer who helped to retrieve many of the papers used in this review.

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