Motifs in brain networks

Abstract

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.

Figure 1. Definition of Structural and Functional Motifs, and Motif Detection

(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 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.

Figure 2. Comparison of Structural Motif Frequency Spectra for Macaque Visual Cortex and C. elegans

(A) Spectra for structural motifs of size M = 3. (B) Spectra for structural motifs of size M = 4.

Figure 3. Structural Motifs that Occurred in Significantly Increased Numbers at Motif Sizes M = 3 and M = 4

(A) Structural motifs found in all three large-scale cortical networks analyzed in this study (see Table 2). (B) Structural motifs found in networks optimized for functional motif number (see Table 4). Numbers refer to the motif's ID.

Figure 4. Motif Fingerprints for Motif size M = 3 in Macaque Visual Cortex

(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 M = 3 (see Figure 1). Note that, despite differences in the absolute motif participation numbers, areas V1, V3, V4, MSTd and DP show highly similar motif fingerprints. (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 Kötter and Stephan [2003]). Areas with more similar motif fingerprints are linked at smaller distances. The five areas with significantly increased motif ID = 9 are indicated in bold typeface. (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.

Figure 5. Properties of Networks (n = 10) Optimized for Structural and Functional Motif Number

(A) Maximization of functional motif number (N = 30, K = 311). Each maximization starts from different random initial conditions, including a different set of 10 random networks. From left to right, each graph shows plots of functional motif number, structural motif number, motif frequency spectrum (M = 3) of optimized networks, and clustering coefficient. (B) Maximization of structural motif number (N = 30, K = 311). Graphs are as in (A). Compare the motif frequency spectrum in (A) with the corresponding plot for the macaque visual cortex in Figure 2A (first row, left bar graph). Initially, random networks in generation 1 exhibited frequency spectra identical to those for random networks in Figure 2A (first row, middle panel).