Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 147282, 6 pages

http://dx.doi.org/10.1155/2015/147282

## Nonautonomous Discrete Neuron Model with Multiple Periodic and Eventually Periodic Solutions

^{1}Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 León, GTO, Mexico^{2}Center for Biomedical Technology, Technical University of Madrid, Campus de Montegancedo, 28223 Madrid, Spain^{3}College of Science, Rochester Institute of Technology, 1 Lomb Memorial Drive, Rochester, NY 14623, USA

Received 20 April 2015; Accepted 25 May 2015

Academic Editor: Lu Zhen

Copyright © 2015 Alexander N. Pisarchik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a nonautonomous discrete neuron model based on the Rulkov map and investigate its dynamics. Using both the linear stability and bifurcation analyses of the system of piecewise difference equations, we determine dynamical bifurcations and parameter regions of steady-state and periodic solutions.

#### 1. Introduction

Piecewise difference equations exhibit very rich dynamics because the lack of differentiability makes their solutions either eventually constant, eventually periodic of various periods, or eventually chaotic [1]. The conjecture proposed by Lothar Collatz in 1937 and the tent map were the first considered piecewise linear difference equations [2, 3]. Later, piecewise difference equations have been used as mathematical models for various applications, including neurons (for extensive review, see [4] and references therein).

In this paper, we focus on the Rulkov map as one of the simplest systems to model neuron dynamics [5]. The original Rulkov map is the autonomous system which reproduces the spiking behavior similar to biological neurons. Although this model does not have real parameters, it is computationally less costly than neurophysiological models, such as, the Hodgkin-Huxley model [6], and hence it can be easily used for simulation of a complex network of synaptically coupled neurons. Being a part of the complex neural network, a neuron is not separated; its dynamics is affected by oscillations of the neighboring neurons through synapses. Therefore, the neuron can be considered as a nonautonomous system.

#### 2. Model

The autonomous Rulkov map is the system of piecewise difference equations that consists of three components [5, 7]:when ,when and , andwhen or .

To convert the autonomous Rulkov map equations (1)–(3) into a nonautonomous system, we introduce two periodic parameters and as follows: It is our goal to investigate monotonic, periodic, and chaotic characters of solutions. We will start with a linear stability analysis of equilibrium points of the autonomous system equations (1)–(5).

#### 3. Linear Stability Analysis of the Autonomous System

We will analyze the local stability of the equilibrium points of the autonomous map linearizing each of the three components individually. The first component is the following system:when . By setting we get the following equilibrium point: Now, let Then, So, we get the following Jacobian matrix : Thus, the eigenvalues of are the roots of the following characteristic polynomial: Hence, we see that and of if and only if and if and only if (i) (ii) (iii) Next, we will analyze the local stability of the equilibrium points of the second component of the autonomous system:when and . By setting we get the following equilibrium point: Now, let Then, So, we get the following Jacobian matrix : Thus, the eigenvalues of are the roots of the following characteristic polynomial: Hence, we see that and of if and only if and if and only if Finally, we will analyze the local stability of the equilibrium points of the third component of the autonomous system:when or . By setting we get the following equilibrium point: Now, let Then, So, we get the following Jacobian matrix : Thus, the eigenvalues of are and .

#### 4. Stability of the Nonautonomous System

From the linearized stability analysis, we decompose the nonautonomous system into six components and apply the linearized stability analysis on each one as previously done from which we obtain the following stability conditions:(1)stability conditions: (2)stable periodic conditions: These conditions result in two bifurcation diagrams shown in Figures 1 and 2. By letting be constant, we get the straight line which bounds different stability regions.