Front. Phys. DOI 10.1007/s11467-013-0365-1 RESEARCH ARTICLE Firing rates of coupled noisy excitable elements Shuai Liu 1,2 , Zhi-Wei He 1,2 , Meng Zhan 1,† 1 Wuhan Center for Magnetic Resonance, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 2 University of the Chinese Academy of Sciences, Beijing 100049, China E-mail: † zhanmeng@wipm.ac.cn Received May 10, 2013; accepted June 24, 2013 The dynamics of coupled excitable FitzHugh–Nagumo systems under external noisy driving is stud- ied. Different from most of previous work focusing on the noise-induced regularity in the framework of coherence resonance, here the average frequency (or firing rate) of coupled excitable elements is of much more concern. We find that (i) their frequencies first increase and then decrease with the increase of the coupling, and there is a clear crossover from a rush increase to a smooth increase with the increase of noise strength, and (ii) for nonidentical cases, all elements transit to an identical frequency simultaneously only after a certain coupling strength is achieved. These first-increase-then- decrease non-monotonic frequency behavior and isochronous frequency synchronization are believed to be two basic behaviors in coupled noisy excitable systems. Keywords coupled excitable elements, firing rate, noise, coherence resonance PACS numbers 05.45.Xt, 05.40.Ca, 87.19.lm 1 Introduction The collective behaviors of coupled nonlinear elements [1–3] are of great significance for our understanding of many dynamical processes in biological systems, such as the circadian rhythms showing that many living organ- isms synchronize to the day-night cycle [4, 5] and the coherent beats synchronously created by the intestinal muscles in heart [6]. A clear fact is that all these col- lective behaviors are created by a network of more or less similar coupled systems, not a single one. Thus, it is clear that the theoretical study of coupled nonlinear oscillators in the nonlinear dynamics field can provide a powerful and helpful means for these problems. On the other hand, for many realistic systems, the effects of various fluctuations, such as the influences of intrinsic (intracellular), extrinsic (intercellular) and external (en- vironmental) noises in multicellular systems, are always unavoidable. Quite opposite to the idea that noise can only be a nuisance, with the development of the theory of stochastic resonance [7–12], researchers have already found that noise may even induce coherent motion and have many positive roles in biological functions, for ex- ample, regulation in genetic networks, maintenance of the quantitative individuality of cells, noise-driven diver- gence of cell fates, noise-induced amplification of signals and so on. Indeed, many other regularity effects induced by noise or other types of disorder have been reported and intensively studied, including stochastic resonance without an external periodic force [13], coherence reso- nance in excitable systems [14], array-enhanced stochas- tic resonance [15], array-enhanced coherence resonance [16, 17], vibrational resonance [18–21], diversity-induced resonance [22–24], ghost resonance [25, 26], etc. There- fore, it is only natural to see that the study of noise effect on coupled nonlinear systems has continued being a hot topic for several decades. In a very recent paper [27], the collective dynamics of coupled excitable FitzHugh–Nagumo (FHN) elements in the presence of noise was studied. In particular, the authors focused on how the average frequency (or firing rate) changes with the variation of the coupling strength, and they found an unexpected peak in the frequency vari- ation before reaching synchronization. This model study has well reproduced the unexpected peak in the varia- tion of the beating rates observed in cultured cardiac cells experiments. Following their work, in the present c  Higher Education Press and Springer-Verlag Berlin Heidelberg 2013