Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 147282, 6 pages
http://dx.doi.org/10.1155/2015/147282

Research Article

Nonautonomous Discrete Neuron Model with Multiple Periodic and Eventually Periodic Solutions

Alexander N. Pisarchik,1,2 Michael A. Radin,³ and Ryan Vogt³

¹Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 León, GTO, Mexico
²Center for Biomedical Technology, Technical University of Madrid, Campus de Montegancedo, 28223 Madrid, Spain
³College 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 , and

when 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.

Figure 1: Bifurcation diagram with as a control parameter for .

Figure 2: Bifurcation diagram with as a control parameter for .

Next, by letting be constant, we get the parabola shown in Figure 2, which bounds different stability regions.

From the linear stability analysis of the autonomous systems and from the two bifurcation diagrams shown in Figures 1 and 2, we obtain the following stability conditions:

(i)stability conditions:

(ii)stable periodic conditions:

(iii)instability and bifurcations

5. Time Series

To illustrate the map dynamics, we will present some graphical examples for various parameters in different regions of the stability diagrams shown in Figures 1 and 2.

Example 1. In this example, we assume that condition (i) mentioned above is satisfied, where , , , and . Then, we obtain an eventually steady periodic cycle in Figure 3.
We see that in this case the solution becomes eventually periodic.

Figure 3: Eventually steady periodic cycle satisfies condition (i).

Example 2. In this example, we assume that (ii) is satisfied, where , , , and . This gives us the periodic orbit shown in Figure 4.
Now, we observe that even though we are in the stability region, the solution is eventually periodic instead of being eventually constant as we have complex eigenvalues when .

Figure 4: Periodic orbit satisfies condition (i).

Example 3. In this example, we assume that (ii) is satisfied, where , , , and , which gives us the graph in Figure 5.
Now, we observe that the periodic character of the solutions changes as we have complex eigenvalues when .

Figure 5: Periodic orbit satisfies condition (i).

Example 4. In this example, we assume that (ii) fails by letting , , , and which gives us the graph in Figure 6.
Notice that this is eventually periodic orbit with a different period compared to Example 4 due to the fact that the stability conditions fail.

Figure 6: Chaotic orbit satisfies condition (ii).

Example 5. In this example, we will assume that (ii) fails by letting , , , and which gives us the graph in Figure 7.
Notice that the periodicity is quite substantially different compared to the previous examples due to the fact that the stability conditions fail.

Figure 7: Unbounded solutions satisfy condition (ii).

Example 6. In this example, we will assume that (ii) fails by letting , , , and which gives us the graph in Figure 8.
Notice that the periodicity is quite substantially different compared to the previous examples due to the fact that the stability conditions fail. Furthermore, we observe that when , the periodic character of the solutions changes, unstable periodic orbits appear, and chaotic behavior appears as well.

Figure 8: Eventually periodic orbit satisfies condition (ii).

6. Conclusions and Future Works

On the base of an autonomous Rulkov map, we designed a nonautonomous discrete neuron model. We demonstrated steady state, periodic, and chaotic character of solutions and performed a linear stability analysis of the equilibrium points of both the autonomous and nonautonomous Rulkov maps.

Our future goal is to generalize the results when is periodic with period . In particular, it would be interesting to compare similarities and differences that will arise with the results of this paper when is periodic with period 2. Furthermore, it would be of paramount interest to introduce the periodic parameter to increase the interaction between the neurons of the model and make the model much more accurate and, moreover, to show how the interaction between the terms of and determines the stability of solutions, periods of solutions, and boundedness of solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This study was supported by the BBVA-UPM Isaac Peral BioTech Program.

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