Dynamic analysis of the discrete fractional-order Rulkov neuron map

Gayathri Vivekanandhan; Hamid Reza Abdolmohammadi; Hayder Natiq; Karthikeyan Rajagopal; Sajad Jafari; Hamidreza Namazi; Gayathri Vivekanandhan; Hamid Reza Abdolmohammadi; Hayder Natiq; Karthikeyan Rajagopal; Sajad Jafari; Hamidreza Namazi

doi:10.3934/mbe.2023220

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 4760-4781. doi: 10.3934/mbe.2023220

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Dynamic analysis of the discrete fractional-order Rulkov neuron map

1.
Department of Computerscience Engineering, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
2.
Electrical and Computer Engineering Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan, 87717-67498, Iran
3.
Department of Computer Technology Engineering, College of Information Technology, Imam Ja'afar Al-Sadiq University, Baghdad, Iraq
4.
Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
5.
Department of Biomedical Engineering, Amirkabir University of Technology (Tehran polytechnic), Iran
6.
Health Technology Research Institute, Amirkabir University of Technology (Tehran polytechnic), Iran
7.
College of Engineering and Science, Victoria University, Melbourne, Australia
8.
School of Engineering, Monash University, Selangor, Malaysia

Received: 25 November 2022 Revised: 20 December 2022 Accepted: 23 December 2022 Published: 30 December 2022

Human evolution is carried out by two genetic systems based on DNA and another based on the transmission of information through the functions of the nervous system. In computational neuroscience, mathematical neural models are used to describe the biological function of the brain. Discrete-time neural models have received particular attention due to their simple analysis and low computational costs. From the concept of neuroscience, discrete fractional order neuron models incorporate the memory in a dynamic model. This paper introduces the fractional order discrete Rulkov neuron map. The presented model is analyzed dynamically and also in terms of synchronization ability. First, the Rulkov neuron map is examined in terms of phase plane, bifurcation diagram, and Lyapunov exponent. The biological behaviors of the Rulkov neuron map, such as silence, bursting, and chaotic firing, also exist in its discrete fractional-order version. The bifurcation diagrams of the proposed model are investigated under the effect of the neuron model's parameters and the fractional order. The stability regions of the system are theoretically and numerically obtained, and it is shown that increasing the order of the fractional order decreases the stable areas. Finally, the synchronization behavior of two fractional-order models is investigated. The results represent that the fractional-order systems cannot reach complete synchronization.
- discrete fractional-order,
- Rulkov neuron map,
- stability analysis,
- bifurcation,
- synchronization
Citation: Gayathri Vivekanandhan, Hamid Reza Abdolmohammadi, Hayder Natiq, Karthikeyan Rajagopal, Sajad Jafari, Hamidreza Namazi. Dynamic analysis of the discrete fractional-order Rulkov neuron map[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4760-4781. doi: 10.3934/mbe.2023220

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Abstract

Human evolution is carried out by two genetic systems based on DNA and another based on the transmission of information through the functions of the nervous system. In computational neuroscience, mathematical neural models are used to describe the biological function of the brain. Discrete-time neural models have received particular attention due to their simple analysis and low computational costs. From the concept of neuroscience, discrete fractional order neuron models incorporate the memory in a dynamic model. This paper introduces the fractional order discrete Rulkov neuron map. The presented model is analyzed dynamically and also in terms of synchronization ability. First, the Rulkov neuron map is examined in terms of phase plane, bifurcation diagram, and Lyapunov exponent. The biological behaviors of the Rulkov neuron map, such as silence, bursting, and chaotic firing, also exist in its discrete fractional-order version. The bifurcation diagrams of the proposed model are investigated under the effect of the neuron model's parameters and the fractional order. The stability regions of the system are theoretically and numerically obtained, and it is shown that increasing the order of the fractional order decreases the stable areas. Finally, the synchronization behavior of two fractional-order models is investigated. The results represent that the fractional-order systems cannot reach complete synchronization.

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