1274 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 12, DECEMBER 2008 Synchronization in Networks of Hindmarsh–Rose Neurons Paolo Checco, Member, IEEE, Marco Righero, Student Member, IEEE, Mario Biey, and Ljupco Kocarev, Fellow, IEEE Abstract—Synchronization is deemed to play an important role in information processing in many neuronal systems. In this work, using a well known technique due to Pecora and Carroll, we in- vestigate the existence of a synchronous state and the bifurcation diagram of a network of synaptically coupled neurons described by the Hindmarsh–Rose model. Through the analysis of the bifurca- tion diagram, the different dynamics of the possible synchronous states are evidenced. Furthermore, the influence of the topology on the synchronization properties of the network is shown through an example. Index Terms—Bifurcation, biological systems, networks, non- linear oscillators, synchronization. I. INTRODUCTION D URING the last few years networks of bio-inspired neu- rons have interested an increasing number of researchers in all branches of science. In particular, spiking neurons have attracted the interest because many studies consider this be- havior an essential component in information processing by the brain [1]. In this class of neurons, bursting neurons are of rele- vant interest since they characterize a variety of biological os- cillators. The electrical potential of these neurons, which typ- ically is the state variable that contains the main information, undergoes a succession of alternating active and silent phases in which, respectively, it has a spiking behavior (very fast os- cillations) and it evolves slowly without oscillations. Further- more, the notion of synchronization is related to several central issues of neuroscience [2]; synchronization seems to be a central mechanism for neuronal information processing within a brain area as well as for communication between different brain areas. For example, synchronization between areas of the visual cortex and parietal cortex, and between areas of the parietal and motor cortex was observed during the visual-motor integration task in awake cats [3]. Direct participation of synchrony in a cognitive task was experimentally demonstrated in humans [4]. This mo- tivates the investigation of the conditions for synchronization in networks of bursting neurons [5], [6]. Among many simple bursting models, the Hindmarsh–Rose (HR) neuron [7] is fairly simple and popular. It is described by a system of three coupled first-order differential equations in which the first state variable Manuscript received April 08, 2008; revised June 26, 2008. Current version published December 12, 2008. This work was supported in part by Ministero dell’Università e della Ricerca under PRIN Project 2006093814_003. This paper was recommended by Associate Editor Y. Horio. P. Checco, M. Righero and M. Biey are with the Department of Electronics, Politecnico di Torino, 10129 Torino, Italy (e-mail: marco.righero@polito.it). L. Kocarev is with the Institute of Nonlinear Science, University of California at San Diego, La Jolla, CA 92093 USA. Digital Object Identifier 10.1109/TCSII.2008.2008057 shows the succession of alternating active and silent phases. The synchronization conditions of a network of HR neurons have been studied in several papers (for example see [5], [6], [8]) and more detailed conditions have been recently introduced in [9], [10]. In this paper we focus on a network of synaptically coupled HR neurons and we derive its synchronization conditions, re- sorting to the well established technique proposed by Pecora and Carroll [11]. As a first result of our investigation, using the above-mentioned technique, the approximate results given in [6] are retrieved and their limits are evidenced. Furthermore, it turns out that the synchronous behavior may be different from that of an isolated neuron and it has to be evaluated resorting to a time-domain analysis, using the coupling strength as bifurca- tion parameter. Hence, the second aim of this paper is to give a complete analysis of the possible synchronous states by deter- mining the corresponding bifurcation diagram. Finally, it will be shown that the synchronization properties still depend—even if not strongly—on the topology of the network. II. PRELIMINARIES The HR neuron model [7]—originally proposed to model the synchronization of firing of two snail neurons [12]—can be gen- eralized as follows [13], [14]: (1) where represents the membrane potential, usually consid- ered as the natural output of the cell, and are the re- covery and the adaptation variables taking into account, respec- tively, fast and slow ion currents and dots denote time deriva- tives. The external stimulus is given by constant and input . Furthermore, is the time constant of the slow ion current and the functions , , and are chosen to display the generation of bursts of spikes and are usually third-, second-, and first-order polynomials, respectively. In view of a future comparison, let us use the same parameters as in [6], [8]: , , and , where , , , , , , , , and, for an isolated cell, . It follows that the studied cells are described by the following equations: (2) With the free parameters fixed at the chosen values, the time evolution of the state variables is periodic. The coupling in a 1549-7747/$25.00 © 2008 IEEE