Commun Nonlinear Sci Numer Simulat 56 (2018) 161–176 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Research paper Spiking and bursting patterns of fractional-order Izhikevich model Wondimu W. Teka a,∗ , Ranjit Kumar Upadhyay b , Argha Mondal b a Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202, USA b Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India a r t i c l e i n f o Article history: Received 11 January 2017 Revised 23 July 2017 Accepted 27 July 2017 Available online 7 August 2017 Keywords: Fractional-order Memory Plasticity Model a b s t r a c t Bursting and spiking oscillations play major roles in processing and transmitting informa- tion in the brain through cortical neurons that respond differently to the same signal. These oscillations display complex dynamics that might be produced by using neuronal models and varying many model parameters. Recent studies have shown that models with fractional order can produce several types of history-dependent neuronal activities without the adjustment of several parameters. We studied the fractional-order Izhikevich model and analyzed different kinds of oscillations that emerge from the fractional dynamics. The model produces a wide range of neuronal spike responses, including regular spiking, fast spiking, intrinsic bursting, mixed mode oscillations, regular bursting and chattering, by ad- justing only the fractional order. Both the active and silent phase of the burst increase when the fractional-order model further deviates from the classical model. For smaller fractional order, the model produces memory dependent spiking activity after the pulse signal turned off. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. On the network level, the response of the neuronal network shifts from random to scale-free spiking. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Spiking and bursting oscillations are the most common dynamic states of excitable cells such as neurons and many endocrine cells [1–3]. Different excitable cells, spontaneously or in response to external stimulation, generate different os- cillation patterns, for example, regular spiking, fast spiking, intrinsic bursting, chattering and regular bursting [4,5]. Bursting oscillation is characterized by periods of fast spiking activity separated by periods of silent phase, and it is controlled by the interplay of fast ionic currents responsible for fast spiking activity and slower currents that modulate the silent phase [6,7]. Different types of mathematical models developed based on ordinary differential equations (ODEs) are used to describe and study the electrical activity of excitable cells [8–10]. These models behave as a Markov process that can capture only the immediate past. On the other side, models with fractional-order derivatives are capable of capturing the long-term history dependent activity caused by the long-range interaction of ionic conductances. ∗ Corresponding author. E-mail addresses: wondimuwub@gmail.com, wwt08@my.fsu.edu (W.W. Teka). http://dx.doi.org/10.1016/j.cnsns.2017.07.026 1007-5704/© 2017 Elsevier B.V. All rights reserved.