# Synchronization of Neuronal Networks via  Control Rank Pinning Scheme

Authors: Qingying Miao; W. K. Wong; Dan Shan
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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.


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## Body

## 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 [3–6]. 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 [7–12]. 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 [15–25]. 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 R and the maximum imaginary part σ 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 R was minimized and σ was regarded as a constraint [22–24]. 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 Section2, 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 𝒢=[𝒱,ℰ], where 𝒱={1,…,N} denotes the vertex set and ℰ={e(i,j)} denotes the edge set.We consider a general complex network model composed ofN identical nodes with diffusively coupling. The model is described as follows:
(1)x˙i=f(xi)-c∑j=1N‍gijΓxj,i∈𝒱,
where xi=(xi1,xi2,…,xin)T is the state variable of the ith node; f(xi) is a continuous vector function; the positive constant c>0 is the overall coupling strength; Γ=diag(r1,r2,…,rn) is the inner coupling matrix and reveals the coupling terms of two connected nodes. Connection matrix G=(gij)∈ℝN×N is related to the topology of the network and represents the coupling configuration of the network. If there is a directed link from node j to i(i≠j) and the weight of the link is wij, then gij=wij, otherwise gij=0. Let OD(i) and ID(i) denote the output degree (outdegree) and input degree (indegree) of the ith node, respectively. The outdegree OD(i)=∑j=1,j≠iN‍gji,i∈𝒱 of a node i is the number of efferent connections that it projects to other nodes, and its indegree ID(i)=∑j=1,j≠iN‍gij,i∈𝒱, is the number of the afferent connections it receives. In this paper, the matrix G is asymmetric, and it has complex eigenvalues. Assume, the eigenvalues set of matrix G is μi=μir+jμim(j=-1),i∈𝒱, and they are ordered in such a way that μ1r≤μ2r≤⋯≤μNr. The Laplacian matrix of the directed network (1) can be defined as L=-G, Lii=∑j=1,j≠iN‍Lij,i∈𝒱. Thus, the matrix has ∑j=1N‍Lij=0,i∈𝒱.Assume,s(t) is the desired target trajectory and satisfies(2)s˙(t)=f(s(t)).Applying pinning control to (1) to control the network to desired reference evolution s(t), we get the following expression:
(3)x˙i=f(xi)-c∑j=1N‍gijΓ(xj)+ui,i=1,2,…,l,x˙i=f(xi)-c∑j=1N‍gijΓ(xj),i=l+1,l+2,…,N,
where ui=-kiΓ(xi-s),ki>0,i=1,2,…,l. ki are control gains and assumed to be constant here, for example, ki=k,i=1,2,…,l. Let p denote the ratio of the pinned nodes over all nodes of the network (0≤p≤1), then the pinned nodes set I={i1,i2,…,il}, where l=⌊p×N⌋. Here, assume that for any node i(i∈[1,N]), we can always find a node j in the pinned nodes set I such that there is a directed path from the pinned node j to node i(i≠j).Assume thats(t) is a virtual system and by setting yi(t)=xi(t),i∈𝒱, and yN+1(t)=s(t), we obtain a network with N+1 dynamical systems as follows [21]:
(4)y˙i=f(yi)-c∑j=1N+1‍MijΓ(yj),i=1,2,…,N+1,
where M={Mij} is an (N+1)-dimensional squared matrix defined by(5)M=(L11+δ1k1L12…L1N-δ1k1L21L22+δ2k2…L2N-δ2k2⋮⋮⋱…⋮LN1LN2…LNN+δNkN-δNkN00…00),where if i∈I, δi=1, else δi=0. Through the above transformation from (1) to (6), the problem of controllability of complex networks is converted into synchronizability of networks. Let λir+jλim(j=-1) be the eigenvalues of the matrix M and in the order of 0=λ1r≤λ2r≤⋯≤λN+1r. Similar to the analysis method of synchronizability of the networks, the bounded systems’ controllability can be evaluated by eigenvalues ratio(6)R=λN+1rλ2r,σ=max{λim},i=1,2,…,N+1.
The smaller the R and σ are, the easier the network is controllable. It is evident that the values of control gains k and the number of pinned nodes l will influence the eigenvalues of matrix M.


## 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 G.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[CR0]N×1 is a row vector composed of initial CR values of each node. After k times iteration, the final stable CR values CRk are calculated as follows:
(7)CRk=CRk-1×ID-1×L,
where ID=diag(ID1,ID2,…,IDN). At each iteration in (7), vertex j will contribute with CRj×(1/IDj) to each of its indegree neighbors, which is in direct proportion to 1/IDj. If there is a path from vertex p to j through vertex i, j will contribute with CRj×(1/IDi)×(1/IDj) to the CR value of vertex p. Hence, the contribution of CR value from vertex j to vertex p attenuates with increment of the path length from p to j. 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 Table1. 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.Table 1
CR value of areas of the cortical network with sizeN=53.
CR Order
Name
CR 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
FrontolimbicFrom 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 Figures1(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 s(t). 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.Time evolution of pinning control errorE for the cortical network with N=53. The coupling strength c=0.3, and the feedback gains k=30. (a) p=0.2. Red line is used in the case of pinning the Hubs, and blue line is used in the case of descending CR value pinning scheme. (b) p=0.03. Red line is used in the case that Ig  and 36 are pinned, and blue line is used in the case that CGa and PFCMd  are pinned.
(a)
(b)


## 5. Analysis of CR Pinning Scheme

In this section, we analyze the descending CR pinning scheme based on the cortical network described in Section4. The dynamics of each area are modeled by Rössler’s system. The Rössler system is given by the following differential equations:
(8)x˙1=-x2-x3,x˙2=x1+ax2,x˙3=b+x3(x1-c),
where a, b, and c are constants. In this study, a=0.165, b=0.2, and c=10 are used. The output function Γ = ((1 0 0), (0 1 0), (0 0 1)) indicates that the systems are coupled through the variables xi1, xi2, and xi3. Here we evaluate the effects of n pinned nodes with feedback gains k1=k2=⋯=kn on the control error E. The control error is denoted by E=(1/(ΔT)N)∑i=1N‍∫TT+ΔT‍∥xi(t)-s(t)∥dt, in which ∥xi∥=|xi1|+|xi2|+|xi3|.Next, we present numerical results of the descending CR pinning scheme under variations of the coupling strengthc and the pinning probability p on the cortical network.Control errorE has been calculated in Figures 2(a) and 2(b) on networks with c=0.1,0.2,0.3, and 0.5 (pinning ratio p=0.1 and p=0.3, resp.). A strong coupling strength will be conducive to control of the network, which is profound when the pinning ratio is large such as p=0.3 and above. Moreover, we take a further research on the network with coupling c=0.1 and c=0.3. 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 c=0.3 that the network is stabilized on orbit s(t) when p=0.4 and above. In Figure 3(a), especially a large number of pinned nodes are needed to stabilize the network with c=0.1 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.Time evolution of pinning control errorE for the cortical network with N=53, feedback gains k=30. With pinning ratio (a) p=0.1 and (b) p=0.3.
(a)
(b)Control errorE as a function of time t with different pinning ratios p=0.1,0.4,0.7,1.0. For the cortical network with N=53, the feedback gains k=30, and the coupling strength (a) c=0.1 and (b) c=0.3.
(a)
(b)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∑i=1N‍∥xi(t)-s(t)∥<0.01, in which ∥xi∥=|xi1|+|xi2|+|xi3|. 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.Table 2
Pinned nodes percentage in communities withp=0.25.
Pinned nodes number
Pinned nodes percentage in community
Community
6
43%
Frontolimbic
5
31%
Somato-Motor
2
13%
Visual
0
0
AuditoryThe CR pinning scheme of the cortical network withN=53, the coupling strength c=0.3, the pinning ratio p=0.25, and the feedback gains k=30. (a) Control error E as a function of time t. (b) The nodes synchronized to the orbit S(t) with function of time t. In all the plots, the legend is as follows: red (blue, green, black, and pink) lines are used in the case of the cortical network (visual community, auditory community, somatomotor community, and frontolimbic community).
(a)
(b)


## 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.



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*Source: 835037-2013-03-18.xml*