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Network motifs, overrepresented small local connection patterns, are assumed to act as functional meaningful building blocks of a network and, therefore, received considerable attention for being useful for understanding design principles and functioning of networks. We present an extension of the original approach to network motif detection in single, directed networks without vertex labeling to the case of a sample of directed networks with pairwise different vertex labels. A characteristic feature of this approach to network motif detection is that subnetwork counts are derived from the whole sample and the statistical tests are adjusted accordingly to assign significance to the counts. The associated computations are efficient since no simulations of random networks are involved. The motifs obtained by this approach also comprise the vertex labeling and its associated information and are characteristic of the sample. Finally, we apply this approach to describe the intricate topology of a sample of vertex-labeled networks which originate from a previous EEG study, where the processing of painful intracutaneous electrical stimuli and directed interactions within the neuromatrix of pain in patients with major depression and healthy controls was investigated. We demonstrate that the presented approach yields characteristic patterns of directed interactions while preserving their important topological information and omitting less relevant interactions.

Many processes and systems have a network structure that consists of interacting units which can be represented as a graph. Accordingly, analysis from a graph theory perspective has recently become a focus of research as unique insights are obtained into the working and organization of various complex systems. For example, in the study of cellular signaling pathways associated with cancer it was revealed that the activity of p53, a central tumor suppressor that regulates many different genes, can only be understood by considering associated tangled signaling networks in their entirety and the position of p53 integration within these networks, instead of considering interactions of p53 with single network components [

Network motifs, the subject of this publication, constitute an exceptional influential measure of local network topology that enables a detailed description of overrepresented local patterns of interconnections [

The original network motif detection approach attempts to find significant frequent subnetworks in one single-directed network with (usually many) unlabeled vertices that are indistinguishable from each other. Original network motif detection basically consists of three computationally expensive subtasks.

Exhaustively enumerating [

The second subtask in network motif detection encompasses determining graph isomorphism for grouping found subnetworks into equivalence classes. It is believed that graph isomorphism cannot be solved in polynomial time. Several algorithms for solving graph isomorphism with miscellaneous performance in practice have been presented [

The last subtask is assessing statistical significance of subnetwork occurrences. Subnetworks that occur in significantly large numbers in the input network as compared to their occurrence in a large set of null model random networks are accepted to be motifs. The comparison of a network with a set of associated random networks should reveal deviations of network properties such as the number of subnetwork occurrences from randomness. Therefore, the underlying random graph model has to be chosen carefully, because it is this model that specifies the notion of randomness. Hence, it has to strike a balance between preserving functional constraints and characteristics of the input network while at the same time comprising random edge patterns so that at best no subnetwork appearance is being favored [

We present a novel approach to network motif detection that differs from the original approach and avoids some of its limitations. In this approach, we intend to detect subnetworks that are motifs for a sample of directed networks where each network possesses the same pairwise different vertex labels. Furthermore, we intend to not discard these vertex labels but rather preserve the functional important topological information associated with them. Topological information has already been used in previous studies to visualize spatio-temporal connectivity structures [

We denote the networks from the samples we investigate in the present study by effective connectivity networks (ECNs). The network data originates from a previous study in which we used effective connectivity analysis to investigate the processing of moderately painful intracutaneous electrical stimuli and directed interactions within the neuromatrix of pain in both groups, MD and HCs, by means of frequency selective generalized partial directed coherence (gPDC) [

The present study directly follows up on the EEG experiments and the connectivity analysis published in [

Eighteen patients (10 women, 8 men) with major depression (mean age ± standard deviation:

All subjects were electrically stimulated intracutaneously at the tip of the middle fingers of both the right and the left hand. The intensity level was adjusted between 10

In this study, we refine the view of this effective connectivity data by examining effective connectivity from a different perspective: we model each participant’s directed interactions, which are given by significant gPDC values, as effective connectivity networks (ECNs). The topology of ECNs consequently represents a valuable source of information about the relationship between pain and depression, which is incompletely understood. Subsequently, we apply our network motif detection approach to group-specific samples of ECNs to find patterns of directed interactions that may be considered as a characteristic of the group of patients or the group of controls, respectively. These characteristic patterns may shed light on the basic neural activity which occurs during the processing of painful stimuli in patients with major depression and in the healthy controls.

Due to the nature of the underlying EEG experiment, eight samples of ECNs may be considered. They are defined by all combinations of the group assignment (MD—patients suffering from major depression versus HC—healthy control subjects), the stimulated side (left versus right), as well as the time window with respect to the stimulus conditions (pre—time window before noxious stimulation versus post—time window directly following the stimulation, i.e., including the processing of the noxious stimulus). The nomenclature is MD-pre-left, MD-pre-right, MD-post-left, MD-post-right, HC-pre-left, and so forth. The sample size for the MD-post-right sample is fifteen, where the sample size equals sixteen for all other samples.

As directed graphs, effective connectivity networks consist of a nonempty finite set

Examples of effective connectivity networks (ECNs). ECNs consist of nine vertices corresponding to EEG-electrodes that are pairwise differently labeled by associated EEG-electrode identifiers and directed edges corresponding to directed interactions between EEG-electrodes as indicated by significant gPDC values. In their wiring patterns, samples of ECNs contain intrinsic information about the processing of painful electrical stimuli in a group of patients suffering from major depression and a group of healthy controls.

Dealing with network samples of directed networks with identical pairwise different vertex labeling instead of single networks without such labeling imposes certain constraints on the approach to network motif detection and also on the definition of a network motif. The most important constraint is that each subnetwork can occur at most only once in a single network, which affects the statistical analysis of subnetwork occurrences. It is not possible to assign significance to subnetwork counts in one network or in very small samples of networks. Therefore, motif detection in a sufficiently large sample of networks constitutes not only a novel approach to reveal common topological characteristics of all sample elements but is also a necessity. Given the pairwise different vertex labeling, two subnetworks are identical if and only if they share the same set of edges, that is, they have identical adjacency matrices. Therefore, isomorphic subnetworks do not exist and consequently it is unnecessary to address the problem of determining graph isomorphism for subnetworks. It is completely different for networks without vertex labeling. In the unlabeled case, different topological equivalence classes of subnetworks exist, also called motif classes or identities and each of them might consist of isomorphic subnetworks. For example, there are 13 equivalence classes of 3-subnetworks without vertex labeling comprising a total of 54 isomorphic subnetworks [

In order to keep the constraints given by the vertex labeling, one has to extend the original notion of network motifs [

Let

In order to identify subnetworks that occur significantly more often than expected in random networks, a suitable model for such random networks is required. Such a model is called null model. Due to pairwise different vertex labels, each subnetwork can occur at most once in a network. Thus, the usual

Let

We applied our approach to detect network motifs in eight group-specific samples of ECNs that were obtained from our effective connectivity data [

The mean number of all 2-motifs in the eight samples of effective connectivity networks (ECNs). The 2-motifs represent important interactions before and during the processing of painful electrical stimuli. The samples originate from all combinations of the group assignment. MD—patients suffering from major depression, HC—healthy control subjects, left and right—stimulated side, pre and post—time window with respect to the stimulus condition.

(1) | 13.0 | 11.0 | 13.0 | — | — | — | — | — | |

(2) | 11.0 | 11.0 | 14.0 | 11.0 | 12.0 | 13.0 | 11.0 | 12.0 | |

(3) | — | — | 11.0 | — | — | — | — | — | |

(4) | — | — | 13.0 | — | 11.9 | — | — | 11.0 | |

(5) | — | — | 11.0 | — | — | — | — | — | |

(6) | — | 11.0 | 11.0 | — | — | — | — | 11.0 | |

(7) | — | — | — | — | 11.0 | — | — | — | |

(8) | — | — | — | — | — | — | 11.0 | — | |

(9) | 12.2 | 12.8 | 12.2 | 11.2 | 11.0 | — | 13.0 | 14.0 | |

(10) | — | — | 12.0 | — | 11.0 | — | 10.0 | — | |

(11) | — | — | 12.0 | — | — | — | — | — | |

(12) | — | — | 11.3 | — | — | — | — | — | |

(13) | — | — | 11.9 | — | — | — | — | — | |

(14) | — | 14.0 | 11.0 | — | — | 12.0 | — | — | |

(15) | — | — | — | — | — | 13.3 | — | 13.2 |

The mean number of 3-motifs with interesting similarities and differences in the eight samples of effective connectivity networks (ECNs). The 3-motifs represent important patterns of interactions before and during the processing of painful electrical stimuli. The samples originate from all combinations of the group assignment. MD—patients suffering from major depression, HC—healthy control subjects, left and right—stimulated side, pre and post—time window with respect to the stimulus condition.

(1) | 4.0 | 7.0 | 9.0 | 5.0 | 7.0 | — | 5.0 | 8.0 | |

(2) | 5.0 | — | 4.0 | — | 5.0 | — | — | — | |

(3) | 4.5 | 5.0 | 7.0 | 4.0 | 6.0 | — | — | 5.0 | |

(4) | 4.2 | — | 6.3 | — | — | — | 4.2 | — | |

(5) | 4.5 | 6.0 | 6.0 | 6.0 | — | — | — | 4.0 | |

(6) | 4.0 | — | 4.3 | — | 5.0 | — | 4.2 | — | |

(7) | 5.5 | 4.5 | 4.0 | — | 4.4 | 5.0 | 5.9 | — | |

(8) | — | — | 7.0 | — | 7.0 | — | 5.0 | — | |

(9) | — | — | 5.0 | 5.0 | — | — | 3.0 | 5.0 | |

(10) | — | — | 7.0 | 6.0 | 4.0 | — | 4.0 | 5.0 | |

(11) | — | 4.5 | 8.5 | 7.0 | 5.6 | — | 5.5 | 6.0 | |

(12) | — | — | 5.0 | 6.0 | 4.0 | — | 4.0 | 4.0 | |

(13) | — | — | — | — | 4.0 | — | 3.0 | 5.0 | |

(14) | — | — | — | 4.0 | — | 4.1 | — | 5.0 | |

(15) | — | — | 4.3 | — | 4.3 | — | 4.0 | — | |

(16) | — | — | 4.5 | 7.3 | — | — | — | 4.3 |

Table

Motifs of size 3 (Table

These results offer a number of intriguing insights into various patterns of directed interactions associated with the processing of painful and, therefore, salient stimuli, characteristic of both groups over the course of time during the experiment. However, the concept of motif detection remains controversial and questions remain. First, it is clear that motif detection misses any functional meaningful subnetworks that appear only infrequently. Conversely, subnetworks that appear with significant frequency are not necessarily important for the functioning of their network. Another criticism refers to the claim that the occurrence of specific motifs is characteristic for a certain network or a type of networks. It seems that some motifs of a network’s motif distribution might occur due to contingencies in the network structure and due to topological effects known as spatial clustering (closeness of vertices in topological space or in attribute space) [

The potential ambiguity of the structure-function relationship of subnetworks and the influence of selection pressure versus variability operators on network topology, as well as the role of entanglement of subnetworks with the rest of the network, has also been debated [

Yet although these critiques underline potential limitations and pitfalls in assigning functionality to motifs (which is the reason for detecting them), it surely does not invalidate the concept of furthering the understanding of a network’s functionality and uncovering its design principles by first analyzing local functional substructures, and then combining this information to infer network behavior at a global level. Moreover, this criticism in its entirety does not hold for our use and interpretation of network motifs, because we are primarily interested in obtaining patterns of interactions that are overrepresented in a sample of networks. It is solely this overrepresentation that allows for an interpretation of these network motifs as a characteristic of this sample. At the same time, we ignore those patterns of interactions that are not overrepresented. In this respect, our approach might be seen as a tool that simplifies the intricate topology of each member in a sample of networks by thinning out interactions that are less important for the sample of networks. Finally, after this simplification, we are able to compare different samples of networks, for example, samples of ECNs. Currently, given the outlined criticisms and lack of neurophysiological knowledge on pain processing, an understanding of the information processing roles network motifs carry out in ECNs is not yet attainable.

We have applied network motif detection to unipolar data with a linked-ears reference. It has been shown previously that the reference might affect the results of such analyses. Specifically for coherence estimates, it is not possible to accurately predict reference effects without an accurate volume conductor model and prior knowledge of all source locations [

The underlying gPDC analysis has been performed on the basis of SEPs, where a multitrial estimator was applied to estimate the autoregressive model parameters [

The design of a suitable null model defines the notion of randomness and is crucial for distinguishing regular topological effects from true topological contingencies in the sample of ECNs and thus is crucial for obtaining valid results [

We have presented an approach to analytically detect network motifs in a sample of directed networks with pairwise different vertex labels. The importance of choosing an appropriate null model random network that contrasts topological regularities of the input networks with topological contingency is outlined. Clearly, a refinement of our analytical null model, which accounts for the average number of edges in a sample of networks, is desirable. Such a refinement is currently under investigation. Nonetheless, we have demonstrated that our approach to network motif detection is suitable to act as a filter to reveal locatable patterns of directed interactions that might be interpreted as characteristic for each of several group-specific samples of ECNs. These networks originate from effective connectivity data obtained in our previous study that investigated cortical activity before and after painful stimulation of patients with major depression and healthy control subjects [

This work was supported by the Grants WI 1166/9-2 and NI 369/7 of the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG). The authors thank Elizabeth Ahrens-Kley for grammatical aid.