# American Institute of Mathematical Sciences

September  2014, 19(7): 2039-2045. doi: 10.3934/dcdsb.2014.19.2039

## Asymptotic effects of boundary perturbations in excitable systems

 1 University of Naples Federico II, Via Claudio, 21, Naples, 80121, Italy, Italy

Received  April 2013 Revised  March 2014 Published  August 2014

A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation in a non linear integral equation permits to deduce a priori estimates for the solution. A complete asymptotic analysis shows that for large $t$ the effects of the initial data vanish while the effects of boundary disturbances $\varphi_1 (t),$ $\varphi_2(t)$ depend on the properties of the data. When $\varphi_1,\,\, \varphi_2$ are convergent for large $t$, the solution is everywhere bounded and depends on the asymptotic values of $\varphi_1 ,$ $\varphi_2$. More, when $\varphi_i \in L^1 (0,\infty) (i=1,2)$ too, the effects are vanishing.
Citation: Monica De Angelis, Pasquale Renno. Asymptotic effects of boundary perturbations in excitable systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2039-2045. doi: 10.3934/dcdsb.2014.19.2039
##### References:
 [1] L. Berg, Introduction to the Operational Calculus, North Holland Publ. Comp., 1967. [2] B. Buonomo A. d Onofrio and D.Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl., 404 (2013), 385-398. doi: 10.1016/j.jmaa.2013.02.063. [3] J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley Publishing Company, 1984. doi: 10.1017/CBO9781139086967. [4] F. Capone, V. De Cataldis and R. De Luca, On the nonlinear stability of an epidemic SEIR reaction-diffusion model, Ricerche di Matematica, 62 (2013), 161-181. doi: 10.1007/s11587-013-0151-y. [5] A. D'Anna, M. De Angelis and G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta. Appl. Math., 122 (2012), 255-267. doi: 10.1007/s10440-012-9741-z. [6] M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50. [7] M. De Angelis, A priori estimates for excitable models, Meccanica, 48 (2013), 2491-2496. doi: 10.1007/s11012-013-9763-2. [8] M. De Angelis, On exponentially shaped Josephson junctions, Acta. Appl. Math., 122 (2012), 179-189. doi: 10.1007/s10440-012-9736-9. [9] M. De Angelis, Asymptotic estimates related to an integro differential equation, Nonlinear Dynamics and Systems Theory, 13 (2013), 217-228. [10] M. D. Angelis and G. Fiore, Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect, J. Math. Anal. Appl., 404 (2013), 477-490. doi: 10.1016/j.jmaa.2013.03.029. [11] M. De Angelis and G. Fiore, Diffusion effects in a superconductive model, Communications on Pure and Applied Analysis, 13 (2014), 217-223. doi: 10.3934/cpaa.2014.13.217. [12] M. De Angelis, A. Maio and E. Mazziotti, Existence and uniqueness results for a class of non linear models, in Mathematical Physics Models and Engineering Sciences (eds. Liguori, Italy), 2008, 191-202. [13] M. De Angelis and P. Renno, Existence, uniqueness and a priori estimates for a non linear integro-differential equation, Ric. Mat., 57 (2008), 95-109. doi: 10.1007/s11587-008-0028-7. [14] M. De Angelis and P. Renno, On the FitzHugh-Nagumo model, in WASCOM 2007 4th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2008, 193-198, doi: 10.1142/9789812772350_0029. [15] M. De Angelis and P. Renno, On asymptotic effects of boundary perturbations in exponentially shaped Josephson junction, Acta Appl. Math.,DOI 10.1007/s10440-014-9898-8, 2014. [16] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, N.Y, 1998. [17] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT press. England, 2007. [18] B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geiere, Effects of noise in excitable systems, Physics Reports, 392 (2004), 321-424. doi: 10.1016/j.physrep.2003.10.015. [19] J. D. Murray, Mathematical Biology, I, II, Springer-Verlag, N.Y. 2002. [20] O. Nekhamkina and M. Sheintuch, Boundary-induced spatiotemporal complex patterns in excitable systems, Phys. Rev., E73 (2006), 66224-66228. doi: 10.1103/PhysRevE.73.066224. [21] S. Rionero, On the stability of nonautonomous binary dynamical systems of partial differential equations, Att. Acc. Pelor. Per. (AAPP), 91 (2013). [22] Alwyn C. Scott, The Nonlinear Universe: Chaos, Emergence, Life, Springer-Verlag New York, 2007, [23] Alwyn C. Scott, Neuroscience A Mathematical Primer, Springer-Verlag New York, 2002. [24] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [25] Torcicollo I., On the Dynamics of the nonlinear duopoly game, International Journal of Non-Linear Mechanics, 57 (2013), 31-38.

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##### References:
 [1] L. Berg, Introduction to the Operational Calculus, North Holland Publ. Comp., 1967. [2] B. Buonomo A. d Onofrio and D.Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl., 404 (2013), 385-398. doi: 10.1016/j.jmaa.2013.02.063. [3] J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley Publishing Company, 1984. doi: 10.1017/CBO9781139086967. [4] F. Capone, V. De Cataldis and R. De Luca, On the nonlinear stability of an epidemic SEIR reaction-diffusion model, Ricerche di Matematica, 62 (2013), 161-181. doi: 10.1007/s11587-013-0151-y. [5] A. D'Anna, M. De Angelis and G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta. Appl. Math., 122 (2012), 255-267. doi: 10.1007/s10440-012-9741-z. [6] M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50. [7] M. De Angelis, A priori estimates for excitable models, Meccanica, 48 (2013), 2491-2496. doi: 10.1007/s11012-013-9763-2. [8] M. De Angelis, On exponentially shaped Josephson junctions, Acta. Appl. Math., 122 (2012), 179-189. doi: 10.1007/s10440-012-9736-9. [9] M. De Angelis, Asymptotic estimates related to an integro differential equation, Nonlinear Dynamics and Systems Theory, 13 (2013), 217-228. [10] M. D. Angelis and G. Fiore, Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect, J. Math. Anal. Appl., 404 (2013), 477-490. doi: 10.1016/j.jmaa.2013.03.029. [11] M. De Angelis and G. Fiore, Diffusion effects in a superconductive model, Communications on Pure and Applied Analysis, 13 (2014), 217-223. doi: 10.3934/cpaa.2014.13.217. [12] M. De Angelis, A. Maio and E. Mazziotti, Existence and uniqueness results for a class of non linear models, in Mathematical Physics Models and Engineering Sciences (eds. Liguori, Italy), 2008, 191-202. [13] M. De Angelis and P. Renno, Existence, uniqueness and a priori estimates for a non linear integro-differential equation, Ric. Mat., 57 (2008), 95-109. doi: 10.1007/s11587-008-0028-7. [14] M. De Angelis and P. Renno, On the FitzHugh-Nagumo model, in WASCOM 2007 4th Conference on Waves and Stability in Continuous Media, World Sci. Publ., Hackensack, NJ, 2008, 193-198, doi: 10.1142/9789812772350_0029. [15] M. De Angelis and P. Renno, On asymptotic effects of boundary perturbations in exponentially shaped Josephson junction, Acta Appl. Math.,DOI 10.1007/s10440-014-9898-8, 2014. [16] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, N.Y, 1998. [17] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT press. England, 2007. [18] B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geiere, Effects of noise in excitable systems, Physics Reports, 392 (2004), 321-424. doi: 10.1016/j.physrep.2003.10.015. [19] J. D. Murray, Mathematical Biology, I, II, Springer-Verlag, N.Y. 2002. [20] O. Nekhamkina and M. Sheintuch, Boundary-induced spatiotemporal complex patterns in excitable systems, Phys. Rev., E73 (2006), 66224-66228. doi: 10.1103/PhysRevE.73.066224. [21] S. Rionero, On the stability of nonautonomous binary dynamical systems of partial differential equations, Att. Acc. Pelor. Per. (AAPP), 91 (2013). [22] Alwyn C. Scott, The Nonlinear Universe: Chaos, Emergence, Life, Springer-Verlag New York, 2007, [23] Alwyn C. Scott, Neuroscience A Mathematical Primer, Springer-Verlag New York, 2002. [24] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [25] Torcicollo I., On the Dynamics of the nonlinear duopoly game, International Journal of Non-Linear Mechanics, 57 (2013), 31-38.
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