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. 2008 Sep 26;4(9):e1000190.
doi: 10.1371/journal.pcbi.1000190.

Organization of excitable dynamics in hierarchical biological networks

Mark Müller-Linow  1 , Claus C Hilgetag, Marc-Thorsten Hütt
Affiliations

Organization of excitable dynamics in hierarchical biological networks

Mark Müller-Linow et al. PLoS Comput Biol. .

Abstract

This study investigates the contributions of network topology features to the dynamic behavior of hierarchically organized excitable networks. Representatives of different types of hierarchical networks as well as two biological neural networks are explored with a three-state model of node activation for systematically varying levels of random background network stimulation. The results demonstrate that two principal topological aspects of hierarchical networks, node centrality and network modularity, correlate with the network activity patterns at different levels of spontaneous network activation. The approach also shows that the dynamic behavior of the cerebral cortical systems network in the cat is dominated by the network's modular organization, while the activation behavior of the cellular neuronal network of Caenorhabditis elegans is strongly influenced by hub nodes. These findings indicate the interaction of multiple topological features and dynamic states in the function of complex biological networks.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Basic graph models representing different combinations of both modular and hub characteristics.
The degree of a node (as an example of a hub characteristic) is indicated by its size, while the grouping of the nodes reveals the modular structure. (A) The Erdös-Rényi (ER) random graph lacks both hubs and modules; (B) the scale-free Barabási-Albert (BA) graph displays a center of interlinked hubs only; the (C) random modular graph and the (D) scale-free modular graph consist of planarly linked modules, which are composed of smaller ER graph and BA graphs, respectively. The hubs in the BA graph version are distributed among the modules. The hierarchical graphs in (E) and (F) are featured by modules consisting of modules. In contrast to the hierarchical cluster graph in (E), the hierarchical scale-free graph (F) is additionally characterized by a hierarchical structure of hubs with one hub dominating the center.
Figure 2
Figure 2. Construction of a color-coded topological reference based on the TM of a network (Top), and formation of a dynamic clustering tree on the basis of the dynamic model simulation (Bottom).
(Top) The distance relations between all nodes are converted into a distance matrix L. (The color label encodes the distances between pairs of nodes.) The matrix L is then translated into a topological reference tree via UPGMA clustering (see Methods). The node indices in the graph correspond to the ones in the tree, the circles in the graph representation denote the modules found in the cluster tree after assigning a threshold (dotted line) which separates the downstream branches. Next, color labels are assigned (TM reference). (Bottom) The model produces a space-time pattern of excitations (white lines) which is then converted into a correlation matrix C. (The color labels encode the number of simultaneous excitations.) The matrix C is translated into a clustering tree (from the excitation patterns). The color labels of the leaves are copied from the TM reference.
Figure 3
Figure 3. Construction of a color-coded topological reference which is based on the location of the CN in the network (top row), and computation of the dynamic clustering tree is carried out as described in Figure 2 (bottom row).
(Top row) The central node h (inner circle in the graph representation) displays the highest betweenness centrality B (see Methods: betweenness). It is surrounded by modules of equidistant nodes (from h). The nodes of the resulting distance vector are re-sorted according to their distance to h.
Figure 4
Figure 4. Graph representation of the modular scale-free network.
The nodes are colored according to the dynamic clustering tree (resulting from a simulation with f = 0.01) after assigning a threshold for 5 modules (the number of topological modules). The dynamic clustering agrees with the topological modules almost completely.
Figure 5
Figure 5. Dynamically detected cluster (DDC vectors) for 10−6<f<1 (right) re-sorted and colored according to the TM reference (left), as described in Figure 2.
The region of the image displaying the highest consistency between the TM reference and the DDC vectors (10−3<f<0.1) marks the range, where the dynamics is able to exploit the given topological modules rather precisely. In this range of f the distribution patterns of excitations are dominated by burst regimes (as discussed in . The pattern formation for f>0.1 is strongly influenced by random firing events, while for f<10−3 the modular boundaries are followed only partly by the dynamics, hinting at another form of correlation between dynamics and topology, which acts on a larger topological scale.
Figure 6
Figure 6. Network representation of the BA graph.
The nodes are colored according to the dynamic cluster tree (resulting from a simulation with f = 10−5) after assigning a threshold for 7 modules (the maximal distance to the hub). Most of the dynamically detected clusters are arranged in a ring-like fashion around the central hub highlighted in black.
Figure 7
Figure 7. Dynamically detected clusters (DDC vectors) for 10−6<f<1 (right) re-sorted and colored according to the CN reference (left), as described in Figure 3.
In this reference, the nodes sharing the same color have the same distance d to the central node h (see Figure 3 top row). Up to a value of f = 10−3, the equidistant nodes are almost completely integrated dynamically according to this topological reference. In this f-regime the dynamics is characterized by excitation waves (spikes), which cover the whole system and which emerge from h preferentially and independently of the location of the accidentally excited node. The increasing scattering of colors for higher values of f indicates a change of the dynamic regime, the spike dynamics is increasingly replaced by burst dynamics.
Figure 8
Figure 8. Levels of dynamic organization in different graphs with a hierarchical distribution of modules.
The dynamic modularity Qdyn for both the TM reference (blue ▵) and the CN reference (red ○) is depicted as a function of the rate of spontaneous excitations f. (A) The hierarchical scale-free graph displays properties of both, the modular and the BA graph. Thus, the two levels of dynamic integration are visible within the same network for the respective values of f. The transition between these two levels corresponds to the transition from spike to burst dynamics. (B) The mapped fractal graph from lacks a scale-free degree distribution and, consequently, hubs, which is reflected in low values of formula image. The absence of ring-like excitation patterns also explains the extension of the high-value range of formula image towards low values of f.
Figure 9
Figure 9. Levels of dynamic organization in two different neuronal networks.
The highlighted curves (bigger symbols; top row) correspond to the respective DDC vector results (bottom row). (A) The dominance of modular elements in the cortical network of the cat is reflected by a distinct increase of Qdyn for the TM-dependent results in the high-f regime (blue ▵; top) as well as by the homogeneous clustering of the DDC vectors (TM-dependent results; bottom), while central node effects seem to play only a marginal role (see the slight superelevation in the low-f regime [red ○]; top). (B) By contrast, the cellular network of C. elegans displays a strong dependency on two adjoining central nodes which dominate the dynamics in a wide range of f. The drastic increase of the CN-dependent results for Qdyn in the low-f region (red ○; top) reflects the high order of the DDC vectors (CN-dependent results; bottom) with a conserved distance ranking of the topologically detected node clusters. Even here, there exists a noticeable but comparatively subordinate influence of the module-based excitation patterns (blue ▵; top).
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