Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2013 / Article
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Networked Systems with Complexities and Their Applications to Engineering

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Volume 2013 |Article ID 835037 |

Qingying Miao, W. K. Wong, Dan Shan, "Synchronization of Neuronal Networks via Control Rank Pinning Scheme", Mathematical Problems in Engineering, vol. 2013, Article ID 835037, 7 pages, 2013.

Synchronization of Neuronal Networks via Control Rank Pinning Scheme

Academic Editor: Yang Tang
Received11 Jan 2013
Accepted26 Jan 2013
Published18 Mar 2013


Recent studies have proposed the controlling regions in the corticocortical network of cats' brain at different scales. Here we study pinning control using a simple model of coupled oscillators assigned to cortical areas in the corticocortical network of cats' brain. We analyze control rank (CR) values of areas in the cortical network. It is found that most of the hubs have bigger CR values than other nodes in the same functional community. Moreover, we analyze the synchrony state of the functional communities in the cortical network, revealing that a community with a larger percentage of pinned nodes inside will be conducive to synchronization.

1. Introduction

Over the last years, the complex networks approach has been widely applied in the field of neuroscience. The study changes from the topological structure of the global neuronal networks to the segregation and the integration of different subnetworks. The analysis of the anatomical connectivity of the mammalian cortex has uncovered that large-scale neuronal networks display high clustering and short path length [1, 2]. The cortical network also shows a hierarchy of complex connectivity and a hierarchy of interacting elements on different levels: the microscopic level of interacting neurons, the mesoscopic level of local neural circuits, and the macroscopic level of large-scale organization of the brain areas [36]. The understanding of the topological structure in different levels of neural networks helpf in studying the dynamical process on such networks.

Synchronization is widely observed in many fields such as coupled nonlinear systems and neural networks, which closely relates to neural information processing [712]. It is found that the synchronization phenomena, over a wide range of spatial and temporal scales, reflect the relevance for cognitive dysfunctions and pathophysiology [13]. The process of synchronization is from partial nodes to the global network, and it also reflects the underlying topology of interactions [14]. Pinning control is an effective method to attain synchronization of networks [1525]. By injecting feedback control action on part of the nodes, the network is synchronized to equilibrium state or nonlinear dynamical orbits. Utilizing control theory and master stability function (MSF) method, the concept of controllability was proposed to evaluate how easily a network is controllable [21]. The Eigenratio and the maximum imaginary part of an extended matrix were proposed to evaluate the network’s controllability. Both the number of pinned nodes and control gains influence the network’s synchronization. The detection of controlling regions was converted into a constrained optimization problem, where the objective function was minimized and was regarded as a constraint [2224]. By utilizing control theory and assuming linear dynamics in each vertex, the results of structure controllability?±?revealed that, in both model and real systems, the driver nodes tend to avoid the high-degree nodes [25, 26]. The concept of control rank (CR) was defined and used to control a scale-free (SF) directed dynamical network to an equilibrium state [27].

In this work, we study synchronization in the cortical network of cats’ brain. Each cortical area is modeled by a nonlinear dynamical system with different initial states. The coupling between areas is modeled using the inner state coupling of the systems’ state variables. Based on the network model, we analyze CR values of each area in the cortical network. Then we study CR pinning scheme in terms of the number and the distributions of the pinned nodes. Moreover, we analyze global synchronization and community synchronization in the cortical network.

The remainder of this paper is organized as follows. In Section 2, the topic of pinning control is introduced. Then, we introduce CR pinning scheme in Section 3. In Section 4, we present CR values of each area in the cortical network. In Section 5, we analyze synchronization under CR pinning scheme. Conclusions are given in Section 6.

2. Overview of Pinning Control

Notations. Throughout this paper, define a graph by , where denotes the vertex set and denotes the edge set.

We consider a general complex network model composed of identical nodes with diffusively coupling. The model is described as follows: where is the state variable of the th node; is a continuous vector function; the positive constant is the overall coupling strength; is the inner coupling matrix and reveals the coupling terms of two connected nodes. Connection matrix is related to the topology of the network and represents the coupling configuration of the network. If there is a directed link from node to and the weight of the link is , then , otherwise . Let and denote the output degree (outdegree) and input degree (indegree) of the th node, respectively. The outdegree of a node is the number of efferent connections that it projects to other nodes, and its indegree , is the number of the afferent connections it receives. In this paper, the matrix is asymmetric, and it has complex eigenvalues. Assume, the eigenvalues set of matrix is , and they are ordered in such a way that . The Laplacian matrix of the directed network (1) can be defined as , . Thus, the matrix has .

Assume, is the desired target trajectory and satisfies

Applying pinning control to (1) to control the network to desired reference evolution , we get the following expression: where . are control gains and assumed to be constant here, for example, . Let denote the ratio of the pinned nodes over all nodes of the network , then the pinned nodes set , where . Here, assume that for any node , we can always find a node in the pinned nodes set such that there is a directed path from the pinned node to node .

Assume that is a virtual system and by setting , and , we obtain a network with dynamical systems as follows [21]: where is an -dimensional squared matrix defined bywhere if , , else . Through the above transformation from (1) to (6), the problem of controllability of complex networks is converted into synchronizability of networks. Let be the eigenvalues of the matrix and in the order of . Similar to the analysis method of synchronizability of the networks, the bounded systems’ controllability can be evaluated by eigenvalues ratio The smaller the and are, the easier the network is controllable. It is evident that the values of control gains and the number of pinned nodes will influence the eigenvalues of matrix .

3. CR Pinning Scheme

Usually, two kinds of pinning control strategies are considered on general network model. (i) Random pinning: the pinned nodes are randomly selected with uniform probability. (ii) Selective pinning: the nodes are sorted according to a certain property of the nodes, for instance, the nodes’ degree, betweenness, or closeness centrality, then the pinned nodes are chosen according to the order. It is mentioned that random pinning scheme cannot guarantee synchronization of the directed networks [20]. It is verified that synchronization can be achieved if and only if forcing is applied to the roots of trees in a spanning-directed forest of the interaction graph of connection matrix .

Supposing that each node with indegree has contribution to the nodes point to it, so the nodes which have more children and grandchildren and such as this will have larger CR values than others. According to CR values of each node, the concept of CR pinning scheme was defined and utilized to deal with pinning control of the directed networks [27].

Supposing that is a row vector composed of initial CR values of each node. After times iteration, the final stable CR values are calculated as follows: where . At each iteration in (7), vertex will contribute with to each of its indegree neighbors, which is in direct proportion to . If there is a path from vertex to through vertex , will contribute with to the CR value of vertex . Hence, the contribution of CR value from vertex to vertex attenuates with increment of the path length from to . The CR values have an obvious positive correlation with the nodes’ outdegree. It is also related to the largest distance from it to all the other vertices. Considering both the property of outdegree and the distance from the pinned nodes to the others in the network, pinning a vertex according to its CR value will be an effective strategy in dealing with control of directed networks.

4. The Cortical Network of Cats’ Brain

The cortical network of cats’ brain can be divided into 53 cortical areas, linked by about 830 fibres of different densities into a weighted and directed complex network. Here we use a data set from [28]. The cerebral cortex consists of four topological interconnected communities that agree with four functional cortical subdivisions: visual cortex (V, 16 areas), Auditory (A, 7 areas), somatomotor (SM, 16 areas), and frontolimbic (FL, 14 areas). The auditory community is sparsely connected, while the visual, somatomotor, and frontolimbic are densely connected among each other.

In the cortical network, the hub areas (Hubs) are densely interconnected at the top of the cortical hierarchy. Here we consider the case that the Hubs comprise 11 nodes from four functional communities, 20a, 7, and AES from visual cortex, EPp from auditory, 6?m and 5AI from somatosensory-motor, and Ia, Ig, CGp, 35, and 36 from frontolimbic [29].

We present the CR values of 53 areas of the cortical network in Table 1. The CR values are sorted in a descending way. The area named CGp in frontolimbic has the biggest CR value. It is impressive that most areas in auditory have smaller CR values. Taking an analysis of the CR values of the Hubs, we find that the Hubs in visual, auditory, somatosensory-motor communities have bigger CR values than the others in the same community. But for frontolimbic, the hub areas are Ig and 36 other than CGa and PFCMd, which have bigger CR values than the two areas mentioned before.

CR Order NameCR Value Community

1 CGp 0.04478 Frontolimbic
2 5Al 0.03849 Somato-Motor
3 CGa 0.03760 Frontolimbic
4 Ia 0.03692 Frontolimbic
5 6m 0.03499 Somato-Motor
6 7 0.03319 Visual
7 5Bm 0.03234 Somato-Motor
8 5Am 0.03125 Somato-Motor
9 PFCMd 0.03114 Frontolimbic
10 5Bl 0.02990 Somato-Motor
11 35 0.02973 Frontolimbic
12 PFCL 0.02952 Frontolimbic
13 20a 0.02939 Visual
14 Ig 0.02919 Frontolimbic
15 SSAo 0.02887 Somato-Motor
16 AES 0.02766 Visual
17 4g 0.02667 Somato-Motor
18 6l 0.02564 Somato-Motor
19 EPp 0.02558 Auditory
20 SSSAi 0.02450 Somato-Motor
21 36 0.02136 Frontolimbic
22 SIV 0.02131 Somato-Motor
23 PFCMil 0.02091 Frontolimbic
24 20b 0.01867 Visual
25 SII 0.01847 Somato-Motor
26 4 0.01825 Somato-Motor
27 19 0.01732 Visual
28 RS 0.01680 Frontolimbic
29 Enr 0.01608 Frontolimbic
30 PLLS 0.01329 Visual
31 PMLS 0.01307 Visual
32 3b 0.01297 Somato-Motor
33 PS 0.01292 Visual
34 18 0.01070 Visual
35 P 0.01064 Auditory
36 21a 0.01059 Visual
37 3a 0.01018 Somato-Motor
38 17 0.00997 Visual
39 AII 0.00952 Auditory
40 2 0.00887 Somato-Motor
41 AMLS 0.00881 Visual
42 1 0.00881 Somato-Motor
43 AI 0.00840 Auditory
44 21b 0.00796 Visual
45 VLS 0.00751 Visual
46 ALLS 0.00618 Visual
47 DLS 0.00598 Visual
48 pSb 0.00594 Frontolimbic
49 Sb 0.00550 Frontolimbic
50 VP(ctx) 0.00438 Auditory
51 Tem 0.00430 Auditory
52 AAF 0.00381 Auditory
53 Hipp 0.00318 Frontolimbic

From the viewpoint of a control scheme design, we compare the effect of pinning the Hubs and the nodes based on descending CR values. As shown in Figures 1(a) and 1(b), the pinning scheme according to descending the CR value has an advantage at the beginning of pinning control, then it becomes consistent with pinning the Hubs till all nodes synchronize to . Moreover, a further analysis between CGa, PFCMd, Ig, and 36 in frontolimbic is presented. We study the dynamics via pinning only two nodes of CGa and PFCMd or Ig and 36. From Figure 1(b), it is evident that CGa and PFCMd have better property in controlling the network.

5. Analysis of CR Pinning Scheme

In this section, we analyze the descending CR pinning scheme based on the cortical network described in Section 4. The dynamics of each area are modeled by Rössler’s system. The Rössler system is given by the following differential equations: where , , and are constants. In this study, , , and are used. The output function = ((1 0 0), (0 1 0), (0 0 1)) indicates that the systems are coupled through the variables , , and . Here we evaluate the effects of pinned nodes with feedback gains on the control error . The control error is denoted by , in which .

Next, we present numerical results of the descending CR pinning scheme under variations of the coupling strength and the pinning probability on the cortical network.

Control error has been calculated in Figures 2(a) and 2(b) on networks with , and (pinning ratio and , resp.). A strong coupling strength will be conducive to control of the network, which is profound when the pinning ratio is large such as and above. Moreover, we take a further research on the network with coupling and . From Figures 3(a) and 3(b), it can be seen that increasing the number of pinned nodes will help to synchronize the network. It is very prominent in the network with coupling strength that the network is stabilized on orbit when and above. In Figure 3(a), especially a large number of pinned nodes are needed to stabilize the network with or with even smaller coupling strength. Hence, it is easier to control a strong coupling network, and only a small number of pinned nodes can synchronize the network to target orbit.

In the following, the effect of the descending CR pinning scheme on synchronization of the cortical network and the four communities is analyzed. The nodes which attain synchronization are calculated by , in which . The results are shown in Figures 4(a) and 4(b). Note that somatomotor community is firstly synchronized and then visual community, frontolimbic community, and auditory community, successively. The distributions of the pinned nodes among communities are listed in Table 2. It is found that the distributions of the pinned nodes influence the dynamics of the communities. A community with a larger percentage of pinned nodes in it is easier to synchronize, for example, somatomotor community in the cortical network. Without pinned nodes inside it, the auditory is the last community to attain synchronization.

Pinned nodes number Pinned nodes percentage in community Community


6. Conclusions

In this paper, the problem of pinning control of the weighted and directed cortical network has been studied. Firstly, the CR values of different areas in the cortical network are listed and then applied to analyze the hub areas in the cortical network. It is found that most of the hubs have bigger CR values than the others in the same functional community. Next, we studied the control effect of CR pinning scheme. It is found that the coupling strength and the pinned nodes demonstrate mutual complement in controlling the network to the desired dynamics. Finally, we analyzed the global synchronization and community synchronization from the viewpoint of pinning control. We pointed out that community’s synchronization is positively correlated with the number of pinned nodes inside it.


The authors acknowledge the support from German Academic Exchange Service (DAAD), the Major State Basic Research Development Program of China (973 Program, Grant no. 2010CB731400), the Key Creative Project of Shanghai Education Community (Grant no. 13ZZ050), and, the Key Foundation Project of Shanghai (Grant no. 12JC1400400).


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Copyright © 2013 Qingying Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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