Document Type : Full Research Paper


1 Ph.D. Student, Bioelectric Department, Biomedical Engineering Faculty, Amirkabir University of Technology, Tehran, Iran

2 Assistant Professor, Bioelectric Department, Biomedical Engineering Faculty, Amirkabir University of Technology, Tehran, Iran


Spiral wave is a particular spatiotemporal pattern, observed in a wide range of complex systems such as neuronal network. Appearance of these waves is related to the network structure as well as the dynamics of its blocks. In this paper, we propose a new modified Hindmarsh-Rose neuron model. The proposed model uses a hyperbolic memductance function as the monotonically differentiable magnetic flux. An external electromagnetic excitation is also considered in the model. Firstly, we study the dynamics of the proposed neuron model through bifurcation diagram and Lyapunov spectrum, in two cases of no excitation and periodic excitation. The bifurcation diagram shows the property of antimonotonicity, which has not been observed in the previous models. Then a square network is constructed and we investigate the spatiotemporal pattrens. By varying the parameters values, spiral waves are observed in specific ranges. The formation of these waves depends on the interaction of all parameters simultaneously. 


Main Subjects

[1]   R. Albert R, A-L. Barabási. “Statistical mechanics of complex networks”, Rev. Mod. Phys., vol. 74, pp. 47, 2002.
[2]   S. Boccaletti, et. al., “Complex networks: Structure and dynamics”, Phys. Rep., vol. 424, pp. 175-308, 2006.
[3]   ME Newman, “The structure and function of complex networks”, SIAM Rev., vol. 45, pp. 167-256, 2003.
[4]   V. Berec, “Complexity and dynamics of topological and community structure in complex networks”, Eur. Phys. J. Spec. Top., vol. 226, pp. 2205-18, 2017.
[5]   M. Jalili, M. Perc, “Information cascades in complex networks”, J. Complex Networks, vol. 5, pp. 665-93, 2017.
[6]   RV Solé, S. Valverde, “Information theory of complex networks: on evolution and architectural constraints”, Complex networks, pp. 189-207, 2004.
[7]   Y. Ji, et. al., “Dynamical analysis of periodic bursting in piece-wise linear planar neuron model”, Cogn. Neurodynamics, vol. 9, pp. 573-9, 2017.
[8]   E. Yilmaz, et. al., “Enhancement of pacemaker induced stochastic resonance by an autapse in a scale-free neuronal network”, Sci. China Technol. Sc., vol. 59, pp. 364-70, 2016.
[9]   M. Jalili. “Spike phase synchronization in multiplex cortical neural networks”. Physica A, vol. 446, pp. 325-33, 2017.
[10]J. Ma, et. al., “Synchronization behaviors of coupled neurons under electromagnetic radiation”, Int. J. Mod. Phys. B, vol. 31, pp. 1650251, 2017.
[11]Y. Xu, et. al., “Synchronization between neurons coupled by memristor”. Chaos. Soliton. Fract. vol. 104, pp. 435-42, 2017.
[12]S. Majhi, et. al., “Chimera states in a multilayer network of coupled and uncoupled neurons”. Chaos, vol. 27, pp. 073109, 2017.
[13]Z. Wei, et. al., “Nonstationary chimeras in a neuronal network”, Europhys. Lett., vol. 123, pp. 48003, 2018.
[14]Z. Rostami, S. Jafari, “Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction”, Cogn. Neurodynamics, vol. 12, pp. 235-54, 2018.
[15]Z. Rostami, et. al., “Elimination of spiral waves in excitable media by magnetic induction”, Nonlinear Dyn., pp. 1-14, 2018.
[16]Y. Yao, et. al., “Impact of bounded noise on the formation and instability of spiral wave in a 2D Lattice of neurons”, Sci. Rep., vol. 7, pp. 43151, 2017.
[17]H. Ashikaga, RG. James, “Hidden structures of information transport underlying spiral wave dynamics”, Chaos, vol. 27, pp. 013106, 2017.
[18]T. Epanchintsev,, “Spiral Wave Drift Induced by High-Frequency Forcing. Parallel Simulation in the Luo–Rudy Anisotropic Model of Cardiac Tissue”, Proc.  International Conference on Computational Science: Springer, pp. 378-91, 2018.
[19]Y. Wang, et. al., “Effect of network structural perturbations on spiral wave patterns”, Nonlinear Dyn., vol. 93, pp. 1671–1680, 2018.
[20]MO. Gani, T. Ogawa, “Spiral breakup in a RD system of cardiac excitation due to front–back interaction”, Wave Motion, vol. 79, pp. 73-83, 2018.
[21]G. Yuan,, “Feedback-controlled dynamics of spiral waves in the complex Ginzburg–Landau equation”, Nonlinear Dyn., vol. 90, pp. 2745-53, 2017.
[22]G. Zhang,, “Selection of spatial pattern on resonant network of coupled memristor and Josephson junction”, Commun. Non-linear Sci. Numer. Simul., vol. 65, pp. 79-90, 2018.
[23]R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane”, Biophys. J., vol. 1, pp. 445-66, 1961. 
[24]JL. Hindmarsh JL, R. Rose, “A model of neuronal bursting using three coupled first order differential equations” Proc. R Soc. Lond. B., vol. 221, pp. 87-102, 1984. 
[25]AL. Hodgkin, AF. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve”, J. Physiol., vol. 117, pp. 500-44, 1952. 
[26]EM. Izhikevich, “Simple model of spiking neurons” IEEE Trans. on neural networks, vol. 14, pp. 1569-72, 2003.
[27]F. Zhan, S. Liu, “Response of Electrical Activity in an Improved Neuron Model under Electromagnetic Radiation and Noise”, Front. Comput. Neurosci., vol. 11, pp. 107, 2017.
[28]M. Shi, Z. Wang, “Abundant bursting patterns of a fractional-order Morris–Lecar neuron model”, Commun. Nonlinear Sci. Numer. Simul., vol. 19, pp. 1956-69, 2014.
[29]B. Bao, et. al., “AC-induced coexisting asymmetric bursters in the improved Hindmarsh–Rose model”, Nonlinear Dyn. vol. 92, pp. 1695–1706, 2018.
[30]J. Hindmarsh, P. Cornelius, “The development of the Hindmarsh-Rose model for bursting”, Bursting: the genesis of rhythm in the nervous system: World Scientific, pp. 3-18, 2005.
[31]M. Lv, J. Ma, “Multiple modes of electrical activities in a new neuron model under electromagnetic radiation”, Neurocomputing, vol. 205, pp. 375-81, 2016.
[32]A. Wolf A,, “Determining Lyapunov exponents from a time series”, Physica D, vol. 16, pp. 285-317, 1985.
[33]M. Nabaei, N. Fatouraee, “Microstructural modelling of cerebral aneurysm evolution through mural-cell-mediated destructive remodeling,” J. Theor. Biol., vol. 354, pp. 60-71, Aug. 2014.