Motifs in Brain Networks Olaf Sporns1*, Rolf Kotter2*�� 1 Department of Psychology and Programs in Cognitive and Neural Science, Indiana University, Bloomington, Indiana, United States of America, 2 C. and O. Vogt Brain Research Institute and Institute of Anatomy II, Heinrich Heine University, Dusseldorf, �� Germany Complex brains have evolved a highly efficient network architecture whose structural connectivity is capable of generating a large repertoire of functional states. We detect characteristic network building blocks (structural and functional motifs) in neuroanatomical data sets and identify a small set of structural motifs that occur in significantly increased numbers. Our analysis suggests the hypothesis that brain networks maximize both the number and the diversity of functional motifs, while the repertoire of structural motifs remains small. Using functional motif number as a cost function in an optimization algorithm, we obtain network topologies that resemble real brain networks across a broad spectrum of structural measures, including small-world attributes. These results are consistent with the hypothesis that highly evolved neural architectures are organized to maximize functional repertoires and to support highly efficient integration of information. Citation: Sporns O, Kotter �� R (2004) Motifs in brain networks. PLoS Biol 2(11): e369. Introduction The complex vertebrate brain has evolved from simpler networks of neurons over a time span of many millions of years. Brain networks have increased in size and complexity (Jerison 1973 Butler and Hodos 1996 Kaas 2000 Krubitzer 2000), as have the flexibility of interactions with the environment and the range of potential behaviors that can be generated (Changizi 2003). Most of the rules governing the evolutionary process toward more complex brains are still unknown, although the central roles of modularization (Kaas 2000), conservation of wiring length (Cherniak 1994 Chklov- skii et al. 2002), and of the elaboration of network connectivity (Laughlin and Sejnowski 2003) are becoming increasingly evident. Systematic investigations of neuronal connectivity in the nematode Caenorhabditis elegans (White et al. 1986) and of large-scale interregional pathways in the mammalian cerebral cortex of rat (Burns et al. 2000), cat (Scannell et al. 1995 Scannell et al. 1999 Hilgetag et al. 2000 Kotter �� and Sommer 2000), and macaque monkey (Felleman et al. 1991 Young 1993 Hilgetag et al. 2000 Stephan et al. 2000) have demonstrated that the topology of these networks is neither entirely random nor entirely regular. Instead, analysis of structural and functional data has shown (Hilgetag et al. 2000 Sporns et al. 2000 Stephan et al. 2000 Sporns and Zwi 2004) that these networks can be characterized by a high degree of clustering, with short path lengths linking individual compo- nents, thus exhibiting small-world properties (Watts and Strogatz 1998 Watts 1999) as do many other complex networks (Strogatz 2001 Albert and Barabasi 2002). These structural attributes are instrumental in generating func- tional specialization (Zeki 1978 Passingham et al. 2002) and functional integration (Bressler 1995 Tononi et al. 1998 McIntosh 2000 Varela et al., 2001 Friston 2002), and they support a large repertoire of complex and metastable dynamical states (Bressler and Kelso 2001 Sporns and Tononi 2002 Sporns 2004). Fluctuating and distributed patterns of dynamical interactions among functionally specialized areas result in rapid switches in functional and effective connec- tivity (McIntosh et al. 1999 Buchel �� and Friston 2000 McIntosh et al., 2003 Brovelli et al. 2004). The structural and functional anatomy of brain networks reflects the dual challenges of extracting specialized information and integrat- ing the information in real time (Tononi and Sporns 2003). What rules underlie the organization of the particular types of networks that we see in complex brains? It is likely that, as networks become more complex, already existing simpler networks are largely preserved, extended, and combined, while it is less likely that complex structures are generated entirely de novo. One hypothesis states that complex and highly evolved networks arise from the addition of network elements in positions where they maximize the overall processing power of the neural architecture. This could be achieved by increasing the number of existing processing configurations or by introducing new processing configurations that add to the robustness or range of cognitive and behavioral repertoires. We may gain insight into the rules governing the structure of complex networks by investigating their composition from smaller network build- ing blocks. Those building blocks are called ������motifs������ (in analogy to driving elements that are elaborated in a musical theme or composition), and they have been examined in the context of gene regulatory, metabolic, and other biological and artificial networks (Milo et al. 2002 Milo et al. 2004). Motifs occur in distinct motif classes that can be distin- guished according to the size (M) of the motif, equal to the number of nodes (vertices), and the number and pattern of interconnections. For a more formal definition of motifs and related concepts, see Materials and Methods. While the most common definition of network motifs is based on their structural characteristics (Milo et al. 2002), Received April 14, 2004 Accepted August 26, 2004 Published October 26, 2004 DOI: 10.1371/journal.pbio.0020369 Copyright: �� 2004 Sporns and Kotter. �� This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abbreviations: ID, motif identity number K, number of edges M, motif size N, number of vertices Academic Editor: Karl J. Friston, University College London *To whom correspondence should be addressed. E-mail: osporns@indiana.edu, rk@hirn.uni-duessseldorf.de PLoS Biology | www.plosbiology.org November 2004 | Volume 2 | Issue 11 | e369 1910 Open access, freely available online PLoS BIOLOGY

structural motifs of neuronal networks form the physical substrate for a repertoire of distinct functional modes of information processing. In brain networks, a structural motif may consist of a set of brain areas and pathways that can potentially engage in different patterns of interactions depending on their degree of activation, the surrounding neural context or the behavioral state of the organism. Thus, we propose a distinction between structural and functional motifs. Structural motifs quantify anatomical building blocks, whereas functional motifs represent elementary processing modes of a network (Figure 1). In this paper, functional motifs refer to specific combinations of nodes and con- nections (contained within structural motifs) that may be selectively recruited or activated in the course of neural information processing. Sorting all possible structural motifs within a network as a function of motif class yields a motif frequency spectrum that records the number of distinct motifs in each structural motif class. Given the motif frequency spectrum, one can easily obtain the motif number, defined as the total number of distinct occurrences of any motif of size M, and the motif diversity, defined as the number of classes that are represented within the network by at least one example. Clearly, the number of vertices (N) and edges (K) within a large network has a strong effect on the motif number and diversity of its constituent structural and functional motifs. But even if N and K are held constant, different connection patterns will result in different repertoires of such network motifs, expressed in terms of both number and diversity. These considerations lead us to formulate hypotheses concerning the rules for brain network organization in terms of network motifs. We hypothesize that neuronal networks have evolved such that their repertoire of potential func- tional interactions (functional motifs) is both large and highly diverse, while their physical architecture is constructed from structural motifs that are less numerous and less diverse. A large functional repertoire facilitates flexible and dynamic processing, while a small structural repertoire promotes efficient encoding and assembly. We investigate this hypothesis first by performing an analysis of structural and functional motifs in various brain networks. We compare the motif properties of real brain networks with random networks and with networks that follow specific connection rules such as neighborhood connectivity (lattice networks). We identify some motif classes that occur more frequently in real brain networks, as compared to random or lattice topologies. Second, by rewiring random networks and imposing a cost function that maximizes functional motif number, network topologies are generated that resemble real brain networks across a broad spectrum of structural measures, including small-world attributes. The results of our analyses are consistent with the hypothesis that complex brain networks maximize func- tional motif number and diversity while maintaining rela- tively low structural motif number and diversity. Results Motif Frequency Analysis We obtained complete structural motif frequency spectra for large-scale connection matrices of macaque visual cortex, macaque cortex, and cat cortex, for motifs sizes of M = 2, 3, 4, and 5 (estimations). In addition, we obtained motif frequency spectra for the matrix of interneuronal connec- tions (������chemical synapses������) of C. elegans, for motif sizes M = 2, 3, and 4 (estimations). For each neural connectivity matrix we generated equivalent (N, K) random and lattice matrices, preserving degree distributions (n = 100 see Materials and Methods), and we obtained their structural motif frequency spectra for comparison. Thus, statistical significance of a motif can only be reached if it occurs in significantly increased proportions with respect to both random and lattice reference cases. Table 1 summarizes the data for structural and functional motif number. Large-scale connection matrices exhibit a consistent statistical trend. Their structural motif number is relatively low, and their functional motif number is relatively high, with both measures approaching the corresponding values of lattice networks. All of these brain networks contain Figure 1. Definition of Structural and Functional Motifs, and Motif Detection (A) From a network, we select a subset of three vertices and their interconnec- tions, representing a candidate structur- al motif. (B) The candidate motif is matched to the 13 motif classes for motif size M = 3. Numbers refer to the ID. The candidate motif is detected as a motif with ID = 13. In detecting structural motifs, only exact matches of candidate motif and motif class are counted. (C) A single instance of a structural motif contains many instances of functional motifs. Here, a structural motif (M = 3, ID = 13) is shown to contain, for example, two distinct instances of the functional motif ID = 9, one motif ID = 2, and one motif ID = 7. Many other distinct instances of functional motifs are present that are not shown in the figure. Note that, in order to be counted as a functional motif of size M = 3, all three vertices of the original structural motif must participate. For a very similar distinction between structural and functional motifs (������interlaced circuits������) and an illustration see Ashby (1960), p. 53. DOI: 10.1371/journal.pbio.0020369.g001 PLoS Biology | www.plosbiology.org November 2004 | Volume 2 | Issue 11 | e369 1911 Motifs in Brain Networks