PHYSICAL REVIEW E 83, 046207 (2011) Delay and diversity-induced synchronization transitions in a small-world neuronal network Jun Tang, 1,* Jun Ma, 2 Ming Yi, 3 Hui Xia, 1 and Xianqing Yang 1 1 College of Science, China University of Mining and Technology, Xuzhou 221008, China 2 Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China 3 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China (Received 3 December 2010; published 13 April 2011) The synchronized behaviors of a noisy small-world neuronal network with delay and diversity is numerically studied by calculating a synchronization measure and plotting firing pattern. We show that delay in the information transmission can induce fruitful synchronization transitions, including transition from phase locking to antiphase synchronization, and transition from antiphase synchronization to complete synchronization. Furthermore, the delay-induced complete synchronization can be changed by diversity, which causes the oscillatory-like transition between antiphase synchronization and complete synchronization. DOI: 10.1103/PhysRevE.83.046207 PACS number(s): 05.45.−a, 05.40.−a, 89.75.Kd I. INTRODUCTION For centuries, synchronization phenomena have been paid much attention due to their apparent ubiquity in many biology, ecology, climatology, and sociology systems [1,2]. Theoretically, many types of synchronization are identified in chaotic or limit-cycle coupled systems [3], such as complete synchronization, phase synchronization, antiphase synchro- nization, phase-lock synchronization, cluster synchronization, lag synchronization, etc. In the study of neuron systems, synchronized behavior of a population of interacting neurons, namely, neuronal network, is a hot issue due to its importance to the processing and transmission of information [4]. Many insightful findings have been reported, these findings indicate that the neuronal network displays a cluster organization, synchronized state always be more easily reached in one cluster [5]. The neuronal network is an active network, and it can generate activity by itself in the absence of external signals [2]. The synchronized dynamics of neuronal network is dependent on some critical bifurcation parameters in the network, such as topology structure, noise, delay, coupling intensity, etc. [6–10]. For example, it is found that the synchronization and coherence can be enhanced by increasing the randomness of the network topology. Many spatiotemporal patterns, such as spiral wave, are maintained by irregular connections between the neurons [11]. In different coupled neuron systems, the constructive effect of noise has been identified in optimizing synchronization as well as coherence [12,13]. In reality, the dynamics of a neuronal network often involves time delay due to the finite signal propagation time in biological networks [14]. Recently, neuronal networks with time delay have received considerable attention. Several delay-induced phenomena in neuronal networks are identified. Delay-induced coherent oscillation [15] is found in neuronal network as well as other coupled systems. Delay-enhanced synchronization [7,16] may be relevant for neuronal networks for establishing a concept of collective information processing in the presence of delayed information transmission. Currently, Wang et al. find that information transmission delay in a * tjuns1979@yahoo.com.cn(J.Tang) scale-free neuronal network induces synchronization transi- tions [7]. Although considerable improvements in the study of delay have been achieved, more complex topology structure or other characters of the neuronal networks need to be concerned. Diversity is omnipresent in nature. It means not all units are identical in coupled systems and can crucially influence the dynamics of the systems [17–19]. It is well known neurons are very diverse in terms of morphology and function [2], and di- versity in neuronal network attracts more and more theoretical concerns [20–22]. It has been reported that coherent pattern can be sustained taking advantage of diversity in a regular network [22], that is similar to diversity-induced resonance [18,20]. The influence of diversity on the synchronization of coupled oscillators has been investigated by Winfree [23] and Kuramoto [24]. The results can be the basis for the studies on synchronization in neuronal networks. The neuron connectivity in the cortex and other brain regions is mainly local, with relatively sparse long distance projections, which suggests a small-world (SW) topological structure rather than regular one [25]. Otherwise, to our knowledge, the studies concerning diversity in neuronal networks mainly focus on the influence of diversity on coherence of the networks, how diversity influences synchronization is not very clear. In this paper, to extend the above-mentioned investigations, our aim is to find out how delay and diversity influence the synchronized behaviors in a noisy SW neuronal network by calculating a synchronization measure and plotting the firing patterns. The remainder of this paper is organized as follows. In Sec. II, a noisy SW neuronal network and corresponding equations are introduced. In Sec. III, ignoring the diversity, the synchronized behaviors of the homogeneous neuronal network are studied by calculating the synchronization measure for different time delay. In Sec. V, the influence of diversity on synchronization of the neuronal network with time delay is studied. The paper ends with conclusions in Sec. V. II. MODEL AND METHODS The dynamical properties of neurons in the network are described by two-variable FitzHugh-Nagumo (FHN) [26] -type equations and N neurons are connected. A SW network topology is implemented as Ref. [27]. In a regular network, 046207-1 1539-3755/2011/83(4)/046207(6) ©2011 American Physical Society