Nonlinear Dynamics

, Volume 95, Issue 1, pp 43–56 | Cite as

Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction

  • Han BaoEmail author
  • Wenbo Liu
  • Aihuang Hu
Original Paper


The membrane potential difference between two adjacent neurons can induce an electromagnetic induction current that behaves like a memristor synapse effect using to couple these two neurons. Based on two-dimensional (2D) Hindmarsh–Rose (HR) neuron and non-ideal threshold memristor, this paper presents a five-dimensional (5D) neuron model of two adjacent neurons coupled by memristive electromagnetic induction. With the 5D memristor-coupled neuron model, the equilibrium states and their stabilities are investigated by qualitative analyses, and the coexisting phenomena of multiple firing patterns and initials-depending bifurcation routes along with extreme events are then uncovered by numerical simulations. Due to complex stabilities of the seven equilibrium states, the attraction basin of one neuron in the 5D memristor-coupled neuron model is not only related to the coupling strength of memristor synapse but also associated with another neuron initials, leading to that the coexisting multiple firing patterns are emerged from different regions in the attraction basin and the bifurcation routes are closely dependent to the initials. Furthermore, circuit syntheses using the off-the-shelf components and breadboard experiments for the proposed 5D memristor-coupled neuron model are executed so that the coexisting multiple firing patterns and extreme events are then conformed perfectly, which have not yet been previously reported in the coupled HR neuron model.


Neuron model Memristor synapse Coexisting multiple firing patterns Initials Stability 



This work was supported by the grants from the National Natural Science Foundations of China under 51777016, 61471191, and 61601062, and the Aeronautical Science Foundation of China under 20152052026.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest. These authors contribute equally to this work.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of Automation EngineeringNanjing University of Aeronautics and AstronauticsNanjingChina
  2. 2.School of Information Science and EngineeringChangzhou UniversityChangzhouChina

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