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.

show all references

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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