978-1-4673-1572-2/12/$31.00 ©2012 IEEE Abstract — Influence of small time-delays in coupling between noisy excitable systems on the coherence resonance and self-induced stochastic resonance is studied. Parameters of delayed coupled deterministic excitable units are chosen such that the system has only one attractor, namely the stationary state, for any value of the coupling and the time-lag. Addition of white noise induces qualitatively different types of coherent oscillations, and we analyzed the influence of coupling time-delay on the properties of these coherent oscillations. The main conclusion is that time-lag 1, but still smaller than the refractory period, and sufficiently strong coupling drastically change signal-to-noise ratio in the quantitative and qualitative way. An interval of noise values implies quite large signal to noise ratio and different types of noise induced coherence are greatly enhanced. We also observed coincident spiking for small noise intensity and time-lag proportional to the inter-spike interval of the coherent spike trains. On the other hand, time-lags 1 and/or weak coupling induce negligible changes in the properties of the stochastic coherence. — Neurons, Delay, Noise I. INTRODUCTION xcitability is a common property of many physical and biological systems. Although there is no unique definition [1] the intuitive meaning is clear: a small perturbation single stable stationary state can result in a large and long lasting excursion away from the stationary state before the system is returned back asymptotically to equilibrium. Typical example of excitable behavior is provided by the dynamics of neurons. However, realistic models of coupled neurons must include the following two phenomena: (a) influence of different types of noise and (b) different time scales of the creation of impulses on one hand and their transmission between neurons on the other. It is well known that neurons in vivo are function under influences of many sources of noise N. lgrade 68, 11080 Beograd-Zemun, Serbia, e-mail: buric@ipb.ac.rs. I. and N.Vasovi Mathematics, Faculty of Mining and Geology, University of Belgra Box 162, Belgrade, Serbia. K. . is with the Faculty of Transport and Traffic Engineering, Universi 30 . [2]. It is also well known that the noise of an appropriate small intensity can change the systems dynamics by turning the quiescent state of the neuron into the state of periodic firing [3]. There are different types of noise induced coherent oscillations [ , ] that could occur in examples of excitable systems without [6] or with time-delay [7, 8], as will be discussed later. Description of interactions between neurons should include the details of the electrochemical processes in real synapses which occur on much slower time scale then the occurrence of an impulse and its transport along axons [9]. Alternatively, the transport of information between neurons can be phenomenologically described by time-delayed inter- neuronal interaction. It is well known that, depending on the parameters, the time-delay can, but need not, induce drastic qualitative changes on the evolution of coupled deterministic excitable systems (please see for example [10– ]). However, a system of delay-differential equations is infinite dimensional with initial states represented by a vector functions on the interval ( 0). Stability of stochastic delay-differential equations has been studied by mathematicians [16, 17]. Influence of noise on time-delay induced bifurcations and properties of synchronization have been analyzed elsewhere, for example in [18–22]. On the other hand the influence of coupling delay on different types of coherent oscillations that have been induced solely by the noise has been studied much less [23– different attractors of systems with time-delayed feedback has been studied in [26, 27]. Such an analyzes would supply information complementary to the research on the effects of noise on the properties of oscillations and synchrony introduced by sufficient time-lag in the delayed coupling. It is our goal in this paper to study the effects of time-delay in the coupling between two excitable units on different types of noise induced coherent oscillations in each of the units. II. THE MODEL In this paper we shall consider typical type II excitable systems, as modeled by the FitzHugh-Nagumo (FHN) differential equation [1], where the excitable behavior bifurcates into the oscillatory regime via the Hopf bifurcation. white noise that could appear in the model equations in two qualitatively different ways. Thus each neuron is described by the following stochastic differential equations: Coherent Oscillations in Minimal Neural Network of Excitable Systems Induced by Noise and Influenced by Time Delay Nikola Buri E