Eventually periodic solutions of single neuron model. (English) Zbl 1451.39010
Summary: In this paper, we consider a nonautonomous piecewise linear difference equation that describes a discrete version of a single neuron model with a periodic (period two and period three) internal decay rate. We investigated the periodic behavior of solutions relative to the periodic internal decay rate in our previous papers. Our goal is to prove that this model contains a large quantity of initial conditions that generate eventually periodic solutions. We will show that only periodic solutions and eventually periodic solutions exist in several cases.
MSC:
| 39A23 | Periodic solutions of difference equations |
| 39A30 | Stability theory for difference equations |
| 39A60 | Applications of difference equations |
| 92C20 | Neural biology |
References:
| [1] | A.M. Amleh, J. Hoag, G. Ladas, A difference equation with eventually periodic solutions, Comput. Math. Appl.,36:401-404, 1998. · Zbl 0933.39030 |
| [2] | A. Anisimova, M. Avotina, I. Bula, Periodic orbits of single neuron models with internal decay rate0< β61,Math. Model. Anal.,18:325-345, 2013. · Zbl 1277.39027 |
| [3] | A. Anisimova, M. Avotina, I. Bula, Periodic and chaotic orbits of a neuron model,Math. Model. Anal.,20:30-52, 2015. · Zbl 1277.39027 |
| [4] | A. Ashyralyev, O. Yildirim, On the numerical solution of hyperbolic IBVP with high-order stable finite difference schemes,Boundary Value Prob.,2013:1-36, 2013. · Zbl 1283.65068 |
| [5] | H. Bin, L. Huang, G. Zhang, Convergence and periodicity of solutions for a class of difference systems,Adv. Difference Equ.,2006:070461, 2006. · Zbl 1139.39010 |
| [6] | I. Bula, M.A. Radin, Periodic orbits of a neuron model with periodic internal decay rate,Appl. Math. Comput.,266:293-303, 2015. · Zbl 1410.39023 |
| [7] | I. Bula, M.A. Radin, N. Wilkins,Neuron model with a period three internal decay rate, Electron. J. Qual. Theory Differ. Equ.,2017:46, 2017. · Zbl 1413.39032 |
| [8] | Y. Chen, All solutions of a class of difference equations are truncated periodic,Appl. Math. Lett.,15:975-979, 2002. · Zbl 1029.39003 |
| [9] | D.W. Cranston, C.M. Kent, On the boundedness of positive solutions of the reciprocal MAXtype difference equationxn= max{x;xn−1;...,Atn−1}with periodic parameters,Appl. |
| [10] | S.N. Elaydi,An Introduction to Difference Equations,2nd. ed., Springer, New York, 1999. · Zbl 0930.39001 |
| [11] | S.N. Elaydi,Discrete Chaos: With Applications in Science and Engineering,2nd. ed., ChapmanHall/CRC, Boca Raton, FL, 2008. · Zbl 1153.39002 |
| [12] | C. Hou, S.S. Cheng,Eventually periodic solutions for difference equations with periodic coefficients and nonlinear control functions,Discrete Dyn. Nat. Soc.,2008:179589, 2008. · Zbl 1160.39305 |
| [13] | B. Ibaz, J.M. Casado, M.A. Sanjuan, Map-based models in neuronal dynamics,Phys. Rep., 501:1-74, 2011. |
| [14] | A.N. Pisarchik, M.A. Radin, R. Vogt, Nonautonomous discrete neuron model with multiple periodic and eventually periodic solutions,Discrete Dyn. Nat. Soc.,2015:147282, 2015. · Zbl 1418.92027 |
| [15] | X. Qian, S. Qi-hong, Eventually periodic solutions of a max-type equation,Math. Comput. Model.,57:992-996, 2013. · Zbl 1305.39007 |
| [16] | S. Steviˇc, On some periodic systems of max-type difference equations,Appl. Math. Comput., 218:11483-11487, 2012. · Zbl 1280.39012 |
| [17] | Z. Wei, L. Huang, Y. Meng, Unboundedness and periodicity of solutions for a discrete-time network model of three neurons,Appl. Math. Model.,32:1463-1474, 2008. · Zbl 1176.39012 |
| [18] | J. Wu,Introduction to Neural Dynamics and Signal Transmission Delay, De Gruyter, Berlin, 2001. · Zbl 0977.34069 |
| [19] | O. Yildirim, M. Uzun, On fourth-order stable difference scheme for hyperbolic multipoint NBVP,Numer. Funct. Anal. Optim.,38(10):1305-1324, 2017. · Zbl 1383.65098 |
| [20] | Z. Yuan, L. Huang, All solutions of a class of discrete-time systems are eventually periodic, Appl. Math. Comput.,158:537-546, 2004. · Zbl 1059.93080 |
| [21] | Z. Yuan, L. Huang, Y. Chen, Convergence and periodicity of solutions for a discrete-time network model of two neurons,Math. Comput. Model.,35:941-950, 2002. · Zbl 1045.92002 |
| [22] | Z. Zhou, Periodic orbits on discrete dynamical systems,Comput. Math. Appl.,45:1155-1161, 2003. · Zbl 1052.39016 |
| [23] | Z. Zhou, J. Wu, Stable periodic orbits in nonlinear discrete-time neural networks with delayd feedback,Comput. Math. Appl.,45:935-942, 2003. · Zbl 1051.39016 |
| [24] | Z. Zhou, J. Yu, L. Huang, Asimptotic behavior of delay difference systems,Comput. Math. Appl.,42:283-290, 2001. · Zbl 0998.39004 |
| [25] | H. Zhu, L. Huang, Dynamics of a class of nonlinear discrete- · Zbl 1072.39018 |
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