Oscillatory Dynamics and Oscillation Death in Complex Networks Consisting of Both Excitatory and Inhibitory Nodes

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1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 6, June 15, 2012 Oscillatory Dynamics and Oscillation Death in Complex Networks Consisting of Both Excitatory and Inhibitory Nodes ZHANG Li-Sheng ( ), LIAO Xu-Hong ( Ê ), MI Yuan-Yuan ( ), GU Wei-Feng ( åþ), and HU Gang ( ) Department of Physics, Beijing Normal University, Beijing , China (Received March 12, 2012) Abstract In neural networks, both excitatory and inhibitory cells play important roles in determining the functions of systems. Various dynamical networks have been proposed as artificial neural networks to study the properties of biological systems where the influences of excitatory nodes have been extensively investigated while those of inhibitory nodes have been studied much less. In this paper, we consider a model of oscillatory networks of excitable Boolean maps consisting of both excitatory and inhibitory nodes, focusing on the roles of inhibitory nodes. We find that inhibitory nodes in sparse networks (small average connection degree) play decisive roles in weakening oscillations, and oscillation death occurs after continual weakening of oscillation for sufficiently high inhibitory node density. In the sharp contrast, increasing inhibitory nodes in dense networks may result in the increase of oscillation amplitude and sudden oscillation death at much lower inhibitory node density and the nearly highest excitation activities. Mechanism under these peculiar behaviors of dense networks is explained by the competition of the duplex effects of inhibitory nodes. PACS numbers: b, Fb, Kd Key words: excitatory node, inhibitory node, oscillation death, cluster synchronization 1 Introduction Neural networks play central roles in controlling various functions of biological systems. Since the pioneering work of McCulloch and Hopfield, [1 3] artificial neural networks have been proposed and applied to study the structures and functions of neural systems and to investigate the dynamical features of neural processes. [4 8] In doing so both ordinary differential equations (ODEs) [5,9] and Boolean maps (BMs) [10 12] have been widely ultilized. BMs has been also called cellular automata in some literatures, [13 15] which can not only considerably simplify analytical and numerical computations but also keep much of the essence of dynamical systems. Thus they have been often used as protypes of the study. [2 3,16] Most of neural cells can excite other cells by their interactions, called excitatory cells, and there also exist other small amount of cells (about 20% of the total cells in actual biological neural systems) playing role of inhibiting their neighbor cells, which are called inhibitory cells. The functions of excitatory cells have been extensively investigated while those of inhibitory cells have not been studied much, [4,17 18] though the roles played by this type of cells are important as well in neural processes. [19 20] In the present paper, we will systematically study their roles by using oscillatory networks of BMs, including both excitatory and inhibitory nodes. An interesting and unexpected discovery in the present paper is that the influences of inhibitory nodes (INs) on the network oscillations are essentially different in sparse (small average connection degree) and dense (large average connection degree) networks. In the former case increasing the number of INs can effectively reduce the activity (i.e., reduce the oscillation amplitude) of the networks while in the latter case increasing INs can apparently enhance this activity before the death of oscillations. By increasing INs the oscillations of both sparse and dense networks must die for sufficiently high IN density. However, the behaviors of oscillation death in both cases are considerably different. For sparse networks the oscillation amplitude decreases as the increase of INs, and then dies at the lowest amplitudes; while for dense networks the death occurs at almost the largest amplitude (the highest neural activities), and the death occurs truly in sudden. All the above features are observed by numerical simulations. The mechanisms underlying these characteristics can be explained by the competing duplex effects of INs and applying the dominant phase advanced driving (DPAD) method proposed in Ref. [21]. The paper is organized as follows. In the next section, we propose the model used throughout the paper. The dynamics of the model is based on BMs of unidirectionally and randomly connected networks consisting of excitable nodes. These excitable nodes include both excitatory and inhibitory nodes. In Sec. 3, we numerically compute the dynamical BM networks and find a number of surprising and significant phenomena. Section 4 is de- Supported by the National Natural Science Foundation of China under Grant Nos and , the Fundamental Research Funds for the Central Universities ganghu@bnu.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 1024 Communications in Theoretical Physics Vol. 57 voted to analyzing and understanding these phenomena by the competition mechanism of the duplex effects of INs and with the DPAD method. In the last section, we present a discussion on the obtained results and anticipate possible developments in the direction of the work. 2 Model We consider unidirectional networks, one of which is schematically shown in Fig. 1(a). For simplicity, each node receives homogeneously k input connections from other nodes randomly chosen (at most one connection from any single node). Therefore, the output connections of any node obeys poisson distribution. In addition, we assure the network strongly connected. The dynamics of the network is determined by the following coupled BMs S(t + 1) = H(S(t)), S(t) = (s 1 (t), s 2 (t),..., s N (t)), { 1 for excited state, s i = (1) 0 for refractory or excitable states, where map H is made in such way that if the i th node is in the excited state (E 1,2 ) or refractory state (R 1,2,3 ) at time t s i (t + 1) follows its own flow independent of interactions, as shown in Fig. 1(b). When the i th node is in one of the excitable rest state G 1,2,... (all these states are simply called G state), we have { 1 if Ii (t) > 0, s i (t + 1) = 0 if I i (t) 0. I i (t) is defined as I i (t) = (2) N M ij s j (t), (3) j=1 where matrix M is generated as 1 if j is i s excitatory input connection, M ij = 0 if j is not i s input connection, if j is i s inhibitory input connection. In biological neural networks, inhibitory interactions are often stronger than excitatory interaction (i.e., a single inhibitory interaction can balance giant excitatory interactions of multiple cells). [13,18,22 24] We therefore adopt approximation of strong inhibitions the absolute inhibition in Eq. (3). And the absolute inhibition approximation has been widely used by scientists as the famous artificial neural network model, McCulloch Pitts model. [1,25 28] Our model constructs complex networks of excitable nodes including refractory states and inhibitory interactions simultaneously in the BM dynamics. Very few previous works considered this type of model. [13,24] We expect that some novel and, perhaps, interesting phenomena may appear in this information-rich model. Fig. 1 An excitable unidirectional network and the dynamical diagram of excited nodes. (a) A schematic example of excitable directed networks with N = 100, k in = 3. Arrowed solid lines indicate directions of interactions and disks denote excitable nodes. (b) A schematic evolution figure of an arbitrary excited node state. E 1 and E 2 represent the 1 st and 2 nd excited phases; R 1, R 2 and R 3 denote the 1 st, 2 nd and 3 rd refractory phases; and G including G 1, G 2, G 3,..., is the ground excitable rest state. Any node evolving from E 1 has to follow the sequence of (b) E 1 E 2 R 1 R 2 R 3 G, independent of any interaction from neighbor nodes. And phase G of a node can be excited to E 1 at time t + 1 if this node receives a total input interaction I i(t) > 0 at time t. And it remains the G phase otherwise. 3 Influences of Inhibitory Nodes on System Dynamics of Sparse and Dense Networks In this section, we focus on numerical simulations of system Eqs. (1) (3). In Fig. 2, we take N = 500 and compute the dynamical networks for k in = 3 (sparse networks) and k in = 50 (dense ones). We start from a network randomly constructed with entirely excitatory nodes (ENs) and an initial condition randomly chosen. After the network oscillation reaches its asymptotic state, an arbitrary EN is changed into inhibitory and a new round of computation continues to another asymptotic. Then another EN will be changed. Such process is repeated by increasing INs one by one, until the network oscillation dies by adding one more IN. Then the last oscillation state will be called critical state, and any quantity, say quantity Q, at the critical state is denoted by Q(C) in Figs. 2(c) 2(f). We compute 100 samples by different networks and different initial conditions with the same N and k in. And then we measure and average these 100 experimental data. The final results are plotted in Figs. 2(a) and 2(b).

3 No. 6 Communications in Theoretical Physics In Figs. 2(a) and 2(b), we plot the death tolls of nodes ND vs. the number of INs NI for kin = 3 (a) and 50 (b), respectively, for all samples. The average data are plotted in the small frames. From the figures, we observe that ND increases gradually to N in Fig. 2(a), or say the system almost continuously go to death where all nodes of the network reach the rest state and remain this state forever. And the death firstly occurs around NI 250 in sparse networks, which takes up about 50% of the total number of the network nodes. However, the behavior of the dense network shown in Fig. 2(b) is rather different. The oscillation death of networks firstly appears around NI 50, and ND directly jumps from small numbers to ND = 500 (i.e., reaches the total oscillation death). It is interesting to compare the oscillation death behaviors of Fig. 2(a) for sparse networks with Fig. 2(b) for dense networks. In the former case, network oscillation death occurs at a large number of INs (NI (c) N & 50% in our case), and before the network death the number of dead nodes increases gradually with increasing NI. And near the critical NI (c), the number of total dead nodes ND becomes very large, and then only a very small number of living nodes support the network oscillation. Therefore, 1025 the oscillation death has a clear premonition of weakening oscillation. Nevertheless, in the case of dense networks, oscillation death occurs much suddenly without any anticipating clue. Namely, this death happens after rather small NI of the critical state where most of nodes are alive in oscillation. At the critical state the network oscillation has been much sensible to one or few more increased INs. This phenomenon may be important for biological neural networks some of which are dense indeed. In Figs. 2(c) and 2(d) we plot the distributions of critical NI (c) for the network oscillation death in sparse ((c), kin = 3) and dense ((d), kin = 50) networks. It is observed that the distribution in Fig. 2(d) is much sharper than that in Fig. 2(c), and the peak of Fig. 2(d) locates at much smaller NI, compared with Fig. 2(c). In Figs. 2(e) and 2(f) we plot the distributions of ND at the critical NI (c). We find that the peak in Fig. 2(e) locates at very large ND > 450 for sparse networks (where a major portion of nodes have already died right before the oscillation death). Obviously, in dense networks ND s are sharply distributed around a rather small ND number (ND 60) and the whole network collapses suddenly to full death by increasing a single IN. Fig. 2 Dynamical behaviors of the asymptotic states of networks by increasingly changing ENs into INs. N = 500, kin = 3 for (a), (c), (e) and kin = 50 for (b), (d), (f). (a) The number of dead nodes ND vs. the number of INs NI for kin = 3. Starting from NI = 0 and an arbitrary initial condition, we compute the asymptotic state and count ND. Then continue the computation by changing one EN to inhibitory and use the ending state of the previous round as the initial state of the current round. And we repeat these processes by increasing NI one by one. The results are given in (a). The scattering plots are given by 100 different randomly chosen networks and initial conditions. The small frame plots the averages of these 100 samples. (b) The same as (a) with kin = 50. Sudden oscillation death at relatively small NI is observed. (c) and (d) Distributions P (NI (c)) vs. NI (c) with NI (c) being the number of INs at the critical states. (c) kin = 3, (d) kin = 50. (e) and (f) Distributions P (ND (c)) vs. ND (c) with ND (c) being the numbers of dead nodes in the networks at the critical states. (e) kin = 3, (f) kin = 50.

4 1026 Communications in Theoretical Physics Vol. 57 In Figs. 3(a) and 3(b), we plot the average oscillation amplitude A vs. N I for the sparse and dense networks, respectively, following the same numerical computation procedure as Fig. 2 to empty circle curve. The amplitude A is defined as N A = S max S min, S(t) = s i (t), (4) i=1 where S max and S min are the largest and smallest S(t) in the asymptotic states. Very interestingly, we observe essentially different tendencies of A against N I in both cases: for sparse networks A almost monotonously decreases as N I increases while we observe just opposite situation for the dense ones. Fig. 3 Amplitude A vs. N I. A is defined as the oscillation amplitude in Eq. (4). (a) k in = 3, (b) k in = 50. The empty circles are obtained by increasing N I from N I = 0, and the black disks are obtained by decreasing N I from 380 for k in = 3 and 75 for k in = 50. Reversibility of (a) in sparse networks k in = 3 and irreversibility of (b) in dense networks k in = 50 are clear shown. The black disks in Fig. 3 are plotted by inverting the process of the empty circles. Namely, we start from the states with N I increasing to a given value in the previous circle computations (N I = 380 for k in = 3 in Fig. 3(a) and N I = 75 for k in = 50 in Fig. 3(b)), and do exactly the same numerical experiments as empty circles by randomly changing IN back to EN one by one for each step. Therefore, the data of black disks are obtained by the inverse computation process against the data of empty circles. At N I = 0 the states of black disk and empty circle are produced by the same topological structure of networks consisting of pure ENs. It is interesting to see that for sparse networks the black disks practically reverse the path of the empty circles, and almost return back to the original state at N D = 0, where the forward process start. While for dense networks black disks follow a path with considerably higher A than the circle path, and reach a state at N D = 0 which can not be found by random initial condition. This type of irreversibility can be observed in all cases of dense networks. For comparing the reversibility of sparse networks and irreversibility of dense ones in more detail, in Fig. 4 we plot amplitude distributions of various networks for k in = 3 and 50 at N I = 0 (states of empty circles); N I = N I (c) and N I = 0 (states of black disks), respectively. The figures for dense networks are the most interesting where the distribution of Fig. 4(f) at N I = 0 is similar to that in Fig. 4(d) at N I (c) 0, while does not resemble that in Fig. 4(b). Although the sets of networks in Figs. 4(b) and 4(f) have exactly the same topological structures, the behaviors of amplitude distributions in Figs. 4(f) and 4(b) differ from each other greatly. So far we have studied the dynamical behaviors of excitable BM networks with both excitatory and inhibitory nodes for some particular k in s (k in = 3 and k in = 50), and found dramatic differences of inhibitory influences between sparse and dense networks. Summarizing the observations of numerical simulations we obtain the following interesting results. (i) Sparse and dense networks have opposite tendencies in oscillation intensity, i.e., the oscillation amplitude A decreases (increases) in sparse (dense) networks when increasing the IN density. (ii) The network oscillation deaths in sparse and dense networks occur in essentially different manners. In sparse systems the death is reached by continuously weakening the oscillation with increase of INs, while in dense systems we observe sudden death of oscillation at a large oscillation amplitude. (iii) By successively changing INs back into ENs from the state with high IN density, the backward path of sparse networks practically reverses the forward path obtained by successively increasing INs. However, the path of dense networks is irreversible, and the oscillation amplitude in backward path keeps much higher than that of the forward path. In all above discussion, we adopted absolute inhibition where INs have infinite inhibitory capacity. In most of realistic neural systems inhibitory neurons have strong while finite inhibition. Therefore, whether the phenomena found in the present paper are also valid in the case of finite inhibition becomes very important. Thus we performed extensive numerical simulations for the case. We compute the networks of Eq. (1) (3) with the interaction matrix replaced by

5 No. 6 Communications in Theoretical Physics 1 Mij = 0 J if j is i s excitatory input connection, if j is not i s input connection, ( < J < 1) if j is i s inhibitory input connection. Fig. 4 Distributions P (A) vs. A for different states, and different kin s realizations are used for plotting each distributions. (a), (c), (d) kin = 3. (b), (d), (f) kin = 50. (a) (b) P (A) and A are measured in the circle states of Fig. 3 at NI = 0. (c) (d) The same as (a) (b), with the critical states at NI (c)s measured. (e) (f) The same as (a) (b) at NI = 0 with the disk states of Fig. 3 measured. It is interesting to observe that the distribution in (f) (at NI = 0) is similar to that of (d) (at NI 6= 0), but completely different from (b) (at the same NI = 0). Fig. 5 Dynamical behaviors of networks in finite inhibition (J = 6 if j is the inhibitory input of i in Eq. (5). N = 500, kin = 3 for (a) and (c), kin = 50 for (b) and (d). (a) (b) and (c) (d) are the exactly same as Figs. 2(a), 2(b) and Figs. 3(a), 3(b), respectively, with absolutes inhibition replaced by strong while finite inhibition of J = 6. (a) The number of dead nodes gradually increase with NI increasing for sparse systems, and oscillation death occurs at large NI and NI. (b) Sudden oscillation death occurs at the small number of INs for dense networks. (c) Decrease of oscillation amplitude with increasing NI and approximately reversible property in the forward and backward process are observed for sparse systems. (d) For dense networks, increase of oscillation amplitude with increasing NI and the strongly irreversible property in the inverse process, are found. All the features found in Figs. 2(a), 2(b) and Figs. 3(a), 3(b) for absolute inhibition are observed here in (a) (b) and (c) (d), respectively, for strong while finite inhibition (5)

6 1028 Communications in Theoretical Physics Vol. 57 In Fig. 5, we use J = 6 in Eq. (5) for inhibition (close to realistic neural system [13,18,24] and find that all the above results (i), (ii), (iii) can be observed (our model needs J 3 for the above obvious phenomena). In Fig. 5(a), we do exactly the same as Fig. 2(a) for sparse networks k in = 3. We do observe the number of dead nodes gradually increase to the system death with N I increasing and oscillation death occurs at large N I and N D. In Fig. 5(b) compared with Fig. 2(b), the same phenomenon of sudden oscillation death at small numbers of INs is observed for dense networks k in = 50. In Fig. 5(c), we adopt the identical operation as Fig. 3(a) for sparse networks k in = 3 and identify the decrease of oscillation amplitude with N I increasing and the almost reversible property in the inverse process. In Fig. 5(d), we adopt the identical operation as Fig. 3(b) for dense networks k in = 50 and identify the increase of oscillation amplitude with increasing N I and the strongly irreversible property in the inverse process. Therefore, all the remarkable features explored for infinite inhibition are justified for strong while finite inhibition. All the above three features are interesting, and the underlying mechanisms should be further studied. 4 Mechanisms Underlying Characteristics of Numerical Observations 4.1 Why the Oscillation Amplitude of the Dense (Sparse) Network can be Increased (Decreased) by Increasing INs The first interesting point to be explained is why the oscillations of sparse networks are weakened while dense networks strengthen their oscillations by the same operations of increasing N I. These opposite tendencies are due to the competition of two facts induced by increasing INs: (i) Increasing N I can increase the number of dead nodes in networks due to the inhibitory function of INs described in Eq. (3) and Fig. 2. This fact exists in both sparse and dense networks and causes oscillation weakening. (ii) Increasing N I can increase cluster synchronization in networks. This is due to the fact that excited INs can in the next time steps inhibit the excitation of their neighbors in rest state. Therefore, all living neighbors of an IN prefer to stay at the phase three steps latter than the phase of the given IN (for the model Fig. 1(b) all nodes have excitation duration of two time steps). For any IN, this tendency leads to inhomogeneous phase distribution of living neighbors in respect with the phase of the IN, and therefore, induces cluster synchronization of nodes. This tendency leads to increase of oscillation amplitude by increasing INs. The total gain of the oscillation amplitude by changing N I is due to the competition of these two tendencies. In sparse systems, the synchronization tendency is very weak because the average connection degree is small. Then the first negative tendency dominates the system behavior, we can only observe decrease of A with increasing N I. However, in dense systems, the synchronization tendency becomes much stronger than the decreasing tendency of node death, because in average each IN can control and synchronize a large number of nodes. Thus the competition causes increase of A with N I, shown in the circle curve of Fig. 3(b). In Figs. 6(a) and 6(b), we show the node distribution in all phases for the empty circle states at N I = 0 with k in = 3 (k in = 50). The distributions are practically homogeneous in all phases for both k in = 3 and 50. In Figs. 6(c) and 6(d), we do exactly the same as (a) and (b), respectively, with the distributions at critical N I (c)s plotted. Now considerably different distributions are observed. In the sparse network Fig. 6(c), the distribution is still practically homogeneous (some large fluctuations exist due to the small number of living nodes). On the contrast, in the dense network Fig. 6(d), the distribution is strongly heterogeneous. Most of nodes (both excitatory and inhibitory nodes) locate on two phases whose distance is three time steps away. This cluster synchronization yields large amplitude A (though some nodes are already dead), and produces the positive slope of the empty circle curve in Fig. 3(b). In Fig. 6 we show the distributions only for N I = 0 and N I = N I (c). In order to study the dependence of the synchronization intensity on variation of N I, we use a quantity of information entropy T E Inf = p i lnp i, (6) i=1 where p i is the proportion of nodes in the i th phase, and T is the total time steps in an oscillation period (note, here we only consider periodic oscillations). In Fig. 6(e), we plot E Inf vs. N I for k in = 3 with E Inf averaged with 1000 samples, where E Inf is practically independent of N I. The small negative slope for large N I is caused by the large fluctuations of phase distributions with very small living nodes. In Fig. 6(f), we do the same as Fig. 6(e) with k in = 50. Now we find clear decreasing of E Inf with N I, showing cluster synchronization with N I increasing. 4.2 Different Mechanisms for Oscillation Death In Figs. 2(a) and 2(b), we observe two different death phenomena by increasing N I : In sparse networks, increasing N I can increase dead nodes in the networks, and finally results in the oscillation death of the entire networks as the number of living nodes in the critical state becomes very small. In dense networks, we see oscillation death suddenly occurring after the critical state in which most of nodes are alive. The mechanism underlying the oscillation death of sparse networks through gradual weakening in Figs. 2(a) and 3(a) can be well explained by Fig. 7(a) where we plot all existing unidirectional loops in the phase advanced driving (PAD) pattern [29] of a critical state for k in = 3, N I (c) = 327. The phase-advanced driving of node i to node j means that i interacts j topologically and i plays dynamic role driving j at the G state. In Fig. 7(a), all

7 No. 6 Communications in Theoretical Physics arrows represent the topological interactions dynamically functioning in driving the driven nodes for oscillations. It is well known that any self-sustained oscillation in networks of nonoscillatory nodes must be supported by oscillation source loops.[21,29 31] At NI = 0, networks have a large number of such loops. By increasing NI, more and more nodes die and thus more and more PAD loops are broken. At the critical states the living nodes of sparse networks become much smaller than the original states of oscillating networks, and most probably, there remain 1029 only one or few source loops coupled with each other, one of such structures is shown in Fig. 7(a). In our numerical experiment the 328th excitatory-inhibitory change accidently occurs at node 188, then all the remaining PAD source loops no longer function and thus the oscillation of the network must die quickly. The evolution of the dead process is shown in Fig. 7(b) where S(t) decreases as time (with some fluctuations) and goes to zero within a duration of a cycle period. Fig. 6 Analysis of cluster synchronization for different kin s and NI s. (a) (c) (e) kin = 3; (b) (d) (f) kin = 50. (a) (b) (c) (d) various snapshots of node distributions in different phases for an arbitrarily chosen sample. Disk nodes are excitatory and square nodes inhibitory. (a) (b) Distributions of the circle states in Fig. 3 for NI = 0. The letter on each column labels the phase of this column, the number on each column represents the number of nodes in the given phase. (c) (d) The same distributions as (a) (b), respectively, at the critical states NI (c). The numbers outside (inside) the brackets indicate the total (the inhibitory) number of nodes in the given phases. Random and practically homogeneous distributions in different phases are observed in all (a) (b) (c) while strongly inhomogeneous distribution inducing cluster synchronization is clearly seen in (d). (e) (f) Information entropies EInf defined in Eq. (6) and averaged with 1000 samples plotted vs. NI. While heinf i keeps a high level in (e) for sparse networks, heinf i in (f) decreases apparently for increasing NI, indicating enhancement of synchronization with increasing NI. In dense networks, the mechanism of oscillation death becomes completely different. In Fig. 8, we consider the case of the critical state of a network with kin = 50 for NI = NI (c) = 66. In this case, most of nodes are alive in self-sustained oscillation and there remain greatly many PAD loops supporting the network oscillation, so that breaks of any one or few loops can not cause the oscillation death. For simplifying the pattern of driving pathes we draw in Fig. 8(a) dominant phase-advanced driving (DPAD) pattern[29] instead of the PAD pattern in Fig. 7(a). A node may be driven by a number of PADs, among which DPAD chooses the drivings exciting the given node first. Moreover, if there are a number of drivings exciting the given node first, we randomly choose one of them as DPAD. Therefore, in DPAD pattern each node is driven by one and only one DPAD. In Fig. 8(a) a significant feature, not appearing in the PAD pattern of the sparse network Fig. 7(a), is the strong clustering synchronization. This feature shows the dynamical aspect of the phase distribution of Fig. 6(d). Changing node 199 of Fig. 8(a) from excitatory to inhibitory, the order parameter S(t) undergoes a widely varying and long transient in Fig. 8(b) which is ended by a sudden death after a final rapid raise of S(t) at t = 63. During the death process most of dead nodes in critical state can be revived from time to time. This is in sharp contrast with the short and smooth decreasing of S(t) in Fig. 7(b) where almost no dead nodes can be revived during the transient. Instead of the mechanism of oscillation death in sparse networks by breaking the single or few remaining source

8 1030 Communications in Theoretical Physics Vol. 57 loops of the critical state with an added IN, the mechanism in dense networks is to break a large number of source loops by enhancing the clustering synchronization. In Fig. 8(c), we do the same as Figs. 6(a) 6(d) with the state at the moment t = 63 of (b), one step before oscillation death, plotted. In this figure, we observe indeed: after the clustering accumulation in the transient, the network occasionally reaches such a high cluster synchronization of phase distribution that the final S(t) eruptively break all PAD loops. At t = 63 it just happens that the revived or migrated INs into the E 2 phase suppress all the ENs in phase G (only few nodes), and sustained oscillation can thus no longer be maintained. Oscillation death occurs at its high excitation activity. Fig. 7 Critical state and evolution of oscillation death in a sparse network. (a) All phase-advanced driving (PAD) loops (defined in the context) at the critical state. Different colors represent different phases. The three PAD loops serve as the oscillation sources. Solid arrowed lines mean that the driving nodes of arrowed line advance over the driven node with one phase step difference, and dotted arrowed lines denote differences of two phase steps. By changing node 188 from excitatory to inhibitory all the PAD source loops break, leading to oscillation death. (b) Evolution of S(t) defined by Eq. (4) after the excitatory-inhibitory change of node 188 at t = 1. Fig. 8 (a) Partial DPAD pattern of the critical state of a dense network with N = 500, k in = 50. Disk nodes are excitatory, and square ones inhibitory. Solid and dotted arrowed lines have the same meanings as Fig. 7(a). Dash arrowed lines represent other DPAD branches. The DPAD pattern has clustering structure and high intensity of synchronization. (b) Transient S(t) evolution maintains high levels of synchronization with large fluctuation. And sudden deaths always occur after the last bursts of excitations in which the enhancements of synchronization occasionally break all PAD loops. (c) The phase distribution of nodes one time step before the oscillation death, indicated by an arrowed line in (b) at t = 63. The letters and numbers of all columns have the same meaning as Fig. 6(d). Now the nodes in E 2 state can no longer excite any of their excitatory neighbors in G state. And all driving loops are broken, leading to oscillation death in the next time step. 4.3 Reversibility and Irreversibility in Manipulating Excitatory and Inhibitory Nodes Based on the analysis in the above subsections, the mechanisms underlying the reversibility (irreversibility) in Figs. 3(a) and 3(b) for sparse (dense) networks can be also well explained. In sparse systems network nodes keep random distributions in various phases of evolutions by changing more and more ENs to INs. Then it can be easily accepted that this type of random distribution is retained in the inverse process of changing INs back to excitatory ones. This explains the reversibility of Fig. 3(a) for empty circle and black disk curves. However, in dense systems, stronger and stronger cluster synchronization is observed through changing more and more ENs to inhibitory ones. In the process of decreasing the number of INs from the critical state with high IN density, the structure of synchronization is retained. In Fig. 9(a), we show the phase distribution of

9 No. 6 Communications in Theoretical Physics 1031 the state after all the INs are turned back to ENs along the curve of black disks of Fig. 3(b). It is found that the phase distribution of nodes is similar to that of Fig. 6(d) at N I = N I (c), but essentially different from Fig. 6(b) representing the starting state of the empty circle curve at N I = 0. In Fig. 9(b), we plot a partial DPAD pattern at N I = 0, corresponding to the ending state of the backward process of Fig. 3(b). Again we find that pattern Fig. 9(b) keeps the cluster synchronization established at N I (c) in Fig. 8(a), while pattern Fig. 9(c) shows similar random driving paths at the beginning of the forward process for the same N I = 0 in the circle curve of Fig. 3(b). Fig. 9 (a) Phase distribution of nodes in an ending state of an inverse process (N D = 0 of the disk curve of Fig. 3(b)). The labels of columns have the same meaning as Figs. 6(a) and 6(b). The phase distribution of nodes in (a) has cluster-synchronous structure similar to Fig. 6(d) while different from that of Fig. 6(b) though the distributions of Fig. 9(a) and Fig. 6(b) are produced by exactly the same network topology N I = 0. (b) The DPAD pattern of the ending state for N I = 0 of the disk curve of Fig. 3(b), which is again similar to that of Fig. 8(a) at N I = N I(c) = 75. The different types of lines have the same meanings as Fig. 8(a). (c) The same as (b) with the starting state of the forward process (empty circle of Fig. 3(b) at N I = 0) plotted. The driving pathes look random, they are essentially different from the pattern of (b) though both Figs. 9(b) and 9(c) have exactly the same network topology. It is emphasized that distribution Fig. 6(b) and pattern Fig. 9(b) are produced in the same network topology as that producing Fig. 6(a) and Fig. 7(a). However, with k in = 50, N I = 0 and random initial conditions we can hardly reach (actually we have never reached with a huge number of tests) the state of cluster synchronization of Figs. 9(a) and 9(b). The method of manipulation in Fig. 3(b) provides a convenient and robust way to produce this highly synchronous states if they are preferred due to some practical reasons. 5 Conclusion and Discussion In conclusion we have studied the influences of INs on the dynamics of oscillatory excitable networks by using a model of coupled BMs. We found essentially different features of these influences in sparse and dense networks. (i) In sparse networks INs play negative role in excitations and oscillations. By increasing the number of INs the intensity of oscillations can be continually weakened. While in the dense cases oscillation intensity can be enhanced by increasing INs. (ii) Oscillation death can occur in both sparse and dense systems for sufficiently high IN density. The death in sparse networks takes place at the smallest oscillation amplitude after continually weakened oscillation by increasing INs to a very large number, while dense networks show sudden oscillation death at much lower IN density and at the almost highest oscillation amplitude. (iii) By continually increasing INs to the nearly critical number and then inversely decreasing them, we observe practically reversible paths in sparse networks while strongly irreversible processes in dense systems. All the seemly peculiar features of dense networks can be well explained by the competition of duplex effects of INs both in dense and sparse networks. The above three results have certain potential significance of applications. With (i) we understand why one can often find the positive role of INs in increasing activation intensity of the neural networks. [32 33] With (ii) we may know that sudden collapse of neuron activities may more easily happen in dense subnetworks of neurons while sparse networks may be immune from this type of dangers. With (iii) it may be expected that some diseases induced by increase of inhibitory neurons may be hardly (easily) recovered in dense (sparse) networks by reducing the inhibition levels. The entire human neural networks may have neural cells, and each has about 1000 interactions in average. These networks are between the sparse and dense networks exampled in the present paper. Since various subnetworks of human being may have diverse interaction densities, some of these subsystems may fall into the ranges of sparse or dense systems investigated here. Moreover, many neural networks of lower animals may have relatively small N and k in and may be easily classified to the category of sparse or dense networks. The results obtained in this presentation may have realistic correspondence or be suggestive in realistic biological neural systems. However, all these points remain unknown and to be further studied.

10 1032 Communications in Theoretical Physics Vol. 57 References [1] W.S. McCulloch and W. Pitts, Bulletin of Mathematical Biophysics 5 (1943) 115. [2] J.J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79 (1982) [3] J.J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 81 (1984) [4] D.J. Amit and A.Treves, Proc. Natl. Acad. Sci. U.S.A. 86 (1989) [5] A. Hjelmfelt and J. Ross, Proc. Natl. Acad. Sci. U.S.A. 91 (1994) 63. [6] E. Greenfield and H. Lecar, Phys. Rev. E 63 (2001) [7] G. Grinstein and R. Linsker, Proc. Natl. Acad. Sci. U.S.A. 102 (2005) [8] M.D. McDonnell, Phys. Rev. E 79 (2009) [9] R. Huerta and M. Rabinovich, Phys. Rev. Lett. 93 (2004) [10] D.Stauffer, A.Aharony, L.daF. Costa, and J.Adler, Eur. Phys. J. B 32 (2003) 395. [11] O. Kinouchi and M. Copelli, Nature (London) 2 (2006) 289. [12] S. defranciscis, S. Johnson, and J.J. Torres, Phys. Rev. E 83 (2011) [13] A.V. Goltsev, F.V. deabreu, S.N. Dorogovtsev, and J.F.F. Mendes, Phys. Rev. E 81 (2010) [14] F. Rozenblit and M. Copelli, J. Stat. Mech. 1 (2011) P [15] T.J. Lewis and J. Rinzel, Network: Comput. Neural Syst. 11 (2000) 299. [16] J.M. Greenberg and S.P. Hastings, SIAM J. Appl. Math. 34 (1978) 515. [17] G.M. Shim, D. Kim, and M.Y. Choi, Phys. Rev. A 43 (1991) [18] Y. Adini, D. Sagi, and M. Tsodyks, Proc. Natl. Acad. Sci. U.S.A. 94 (1997) [19] J. Buhmann, Phys. Rev. A 40 (1989) [20] MikhailI. Rabinovich, P. Varona, AllenI. Selverston, and Henry D.I. Abarbanel, Rev. Mod. Phys. 78 (2006) [21] X. Liao, Y. Qian, Y. Mi, Q. Xia, X. Huang, and G. Hu, Front. Phys. 6 (2011) 124, arxiv: (2009). [22] N. Brunel, J. Comput. Neurosci. 8 (2000) 183. [23] R. Dodla and C.J. Wilson, Phys. Rev. Lett. 102 (2009) [24] L. Acedo, Mathematical and Computer Modelling 50 (2009) 717. [25] Gall A. Carpenter, Neural Networks 2 (1989) 243. [26] E. Gelenbe and A. Stafylopatis, Appl. Math. Modelling 15 (1991) 534. [27] B. Yegnanarayana, Sādhanā 19(2) (1994) 189. [28] Z. Tang, H. Tamura, M. Kuratu, O. Ishizuka, and K. Tanno, Electronics and Communications in Japan 84(8) (2001) 11. [29] Y.Qian, X. Huang, G. Hu, and X. Liao, Phys. Rev. E 81 (2010) [30] Y. Mi, L. Zhang, X. Huang, Y.Qian, G. Hu, and X. Liao, Europhys. Lett. 95 (2011) [31] X. Liao, Q. Xia, Y.Qian, L. Zhang, G. Hu, and Y. Mi, Phys. Rev. E 83 (2011) [32] C. van Vreeswijk, L.F. Abbott, and G.Bard Ermentrout, J. Comput. Neurosci. 1 (1995) 313. [33] X. Wang and G. Buzsáki, J. Neurosci. 16(20) (1996) 6402.