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OS and RK conceived and designed the experiments. OS and RK analyzed the data. OS and RK contributed reagents/materials/analysis tools. OS and RK wrote the paper.

The authors have declared that no conflicts of interest exist.

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.

Analysis of characteristic patterns of connectivity in neuroanatomical datasets suggests that nervous systems evolved to maximize functional repertoires and support highly efficient integration of information.

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 (

Systematic investigations of neuronal connectivity in the nematode

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 building 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 (

While the most common definition of network motifs is based on their structural characteristics (

(A) From a network, we select a subset of three vertices and their interconnections, representing a candidate structural motif.

(B) The candidate motif is matched to the 13 motif classes for motif size

(C) A single instance of a structural motif contains many instances of functional motifs. Here, a structural motif (

Clearly, the number of vertices

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 functional motif number and diversity while maintaining relatively low structural motif number and diversity.

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

Numbers are actual values (for real matrices) and mean and standard deviation (in parentheses, for random and lattice matrices,

(A) Spectra for structural motifs of size

(B) Spectra for structural motifs of size

(A) Structural motifs found in all three large-scale cortical networks analyzed in this study (see

(B) Structural motifs found in networks optimized for functional motif number (see

See Figure 3 for displays of the significant motifs (shown with their ID). Note that no significant differences are found for any of the networks at

Compare motif ID with those shown in

All networks were optimized for high functional motif number (

Each vertex (brain area) participates in a subset of the structural motifs that compose the entire network. We asked whether individual brain areas participate in similar or different sets of motifs and whether motif participation might reveal functional relationships. We define the motif fingerprint of a brain area as the number of distinct structural motifs of size

(A) Motif fingerprints for five areas with significantly increased motif ID = 9 (V1, V3, V4, MSTd, DP, names in bold) as well as areas V2, V4t, and PITv. Polar plots display the motif participation number for 13 motif classes with

(B) Hierarchical cluster analysis of motif fingerprints. The Pearson correlation coefficients between all pairs of motif fingerprints were used in a consecutive linking procedure using Euclidean distances based on the farthest members of each cluster (for details see

(C) Hierarchical cluster analysis of single area motif frequency spectra using the same procedures on orthogonal data of (B). Motif classes 9, 12, and 13 covary across the 30 visual areas and form a distinct branch of the cluster tree.

We hypothesized that high functional motif number and diversity represent important ingredients in the global organization of cortical networks, and that a selective advantage for these two properties might contribute to the generation of other significant structural properties. To test this hypothesis we applied an evolutionary algorithm (

(A) Maximization of functional motif number (

(B) Maximization of structural motif number (

To further characterize these networks, we calculated their clustering coefficient and their path length to determine if they exhibited small-world properties (

The importance of a large repertoire of functional circuits for flexible and efficient neural processing has long been recognized (

The functional implications of some network structures—such as reciprocal, convergent, and divergent connections or cycles—have been discussed in the context of network participation indices (

Optimizing functional motif number yields networks that resemble real brain networks across a broad spectrum of structural measures, including several that did not appear to be linked in trivial ways to the optimized measure. Increasing the functional motif number tends to lead to a concomitant decrease in structural motif number, as individual connections become locally dense, thus increasing the abundance of motifs with more local connections and thus greater functional diversity. We note that maximal numbers of functional motifs are not reached in ideal lattices (nearest-neighbor connectivity); rather, optimized networks routinely exhibit functional motif numbers that exceed those of ideal lattices, and they belong to a general class of networks that maintain a mixture of “local” and “long-range” connectivity. Importantly, even though structural and functional motifs are directly related (each structural motif contains a fixed set and spectrum of functional motifs), optimizing structural and functional motif number yielded strikingly different connection topologies.

Optimizing functional (but not structural) motif number produced a tendency toward the emergence of small-world attributes (high clustering coefficient and short path length), a mode of connectivity that promotes functional cooperation, recurrent processing, and efficient information exchange (

All networks and network motifs in this paper are described as graphs of units (called nodes or vertices) with directed (i.e., nonsymmetrical) connections (called edges).

A “motif” is a connected graph or network consisting of ^{2} –

A “structural motif” of size

A structural motif provides the complete anatomical substrate for possible functional interactions among its constituent vertices. However, in real neuronal networks, not all structural connections participate in functional interactions at all times. As different edges or connections become functionally engaged, different “functional motifs” emerge within a single structural motif. The former (functional) refers to “processing modes” or “effective circuits,” while the latter (structural) refers to “anatomical elements” or “building blocks.” The existence of different functional motifs greatly enhances the processing power of any neuronal architecture. We then distinguish structural motifs from functional motifs that form a set of subgraphs of the structural motif. All such functional motifs consist of the original

This definition implies that functional motifs are more naturally applied to networks with vertices that contain multiple neurons or neuronal populations. In the present study, our main focus is on motifs of large-scale connection matrices; data for the single neuron network of

A “connected motif” is a structural motif that forms a strongly connected graph. In a connected motif, all constituent vertices can be reached from all other constituent vertices. Such a motif, in principle, allows all vertices to exert causal effects on each other. For

A “motif frequency spectrum” records the number of occurrences of each motif of a given class for a size

“Motif number” is the total number of all motifs of all classes (for a given size

“Motif diversity” is the total number of all motif classes (for a given size

“Motif participation number” is the number of instances of a given motif class that a particular vertex participates in. For example, if a vertex participates in 12 distinct motifs with

The “motif fingerprint” is the spectrum of motif participation numbers for all motifs of a given size

All datasets used in this study are available in Matlab format at

Currently available datasets are likely to contain errors or missing connections that have not been investigated and do not take into account possible intersubject variability or rank-ordered or graded connection densities or strengths. While these issues have not been addressed systematically, some exploratory analyses suggest that the results reported in this paper are invariant with respect to small variations in connection patterns.

A statistical evaluation of motif frequencies depends on a choice of reference cases (“null hypotheses”).

Random and lattice matrices that preserve the in-degree and out-degree for each vertex are generated from the original anatomical connection matrices by a Markov-chain algorithm (_{1},_{1}) and (_{2},_{2}) is selected for which c_{i}_{1j1} = 1, c_{i}_{2j2} = 1, c_{i}_{1j2} = 0, and c_{i}_{2j1} = 0. Then we set c_{i}_{1j1} = 0, c_{i}_{2j2} = 0, c_{i}_{1j2} = 1, and c_{i}_{2j1} = 1. This is repeated until the connection topology of the original matrix is randomized.

For lattice matrices, the same Markov procedure is employed but swaps are only carried out if the resulting matrix has nonzero entries that are located closer to the main diagonal (thus approximating a lattice or ring topology). This algorithm is implemented as a probabilistic optimization using a weighted cost function.

All graph theory methods used in this paper—including those for calculating clustering coefficients and path lengths (

We thank John Tuley (supported through Indiana University's Science, Technology, and Research Scholar's Program) for help in implementing motif detection algorithms. This work was supported by US government contract NMA201–01-C-0034 to OS, and DFG GRK 320 and Forschungskommission, Medizinische Fakultät, HHU Düsseldorf to RK. The views, opinions, and findings contained in this paper are those of the authors and should not be construed as official positions, policies, or decisions of NGA or the US government.

motif identity number

number of edges

motif size

number of vertices