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

Asymptotic effects of boundary perturbations in excitable systems

Monica De Angelis 1, and  Pasquale Renno 1,

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.
Keywords: fundamental solution, Neumann problem., Laplace transform, Reaction-diffusion system, parabolic equation.
Mathematics Subject Classification: 44A10, 35K57, 35A0.
Citation: Monica De Angelis, Pasquale Renno. Asymptotic effects of boundary perturbations in excitable systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2039-2045. doi: 10.3934/dcdsb.2014.19.2039
References:
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L. Berg, Introduction to the Operational Calculus, North Holland Publ. Comp., 1967.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50.  Google Scholar

[7]

M. De Angelis, A priori estimates for excitable models, Meccanica, 48 (2013), 2491-2496. doi: 10.1007/s11012-013-9763-2.  Google Scholar

[8]

M. De Angelis, On exponentially shaped Josephson junctions, Acta. Appl. Math., 122 (2012), 179-189. doi: 10.1007/s10440-012-9736-9.  Google Scholar

[9]

M. De Angelis, Asymptotic estimates related to an integro differential equation, Nonlinear Dynamics and Systems Theory, 13 (2013), 217-228.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

J. P. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, N.Y, 1998.  Google Scholar

[17]

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT press. England, 2007.  Google Scholar

[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.  Google Scholar

[19]

J. D. Murray, Mathematical Biology, I, II, Springer-Verlag, N.Y. 2002.  Google Scholar

[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.  Google Scholar

[21]

S. Rionero, On the stability of nonautonomous binary dynamical systems of partial differential equations, Att. Acc. Pelor. Per. (AAPP), 91 (2013).  Google Scholar

[22]

Alwyn C. Scott, The Nonlinear Universe: Chaos, Emergence, Life, Springer-Verlag New York, 2007, Google Scholar

[23]

Alwyn C. Scott, Neuroscience A Mathematical Primer, Springer-Verlag New York, 2002.  Google Scholar

[24]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[25]

Torcicollo I., On the Dynamics of the nonlinear duopoly game, International Journal of Non-Linear Mechanics, 57 (2013), 31-38. Google Scholar

show all references

References:
[1]

L. Berg, Introduction to the Operational Calculus, North Holland Publ. Comp., 1967.  Google Scholar

[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.  Google Scholar

[3]

J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley Publishing Company, 1984. doi: 10.1017/CBO9781139086967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50.  Google Scholar

[7]

M. De Angelis, A priori estimates for excitable models, Meccanica, 48 (2013), 2491-2496. doi: 10.1007/s11012-013-9763-2.  Google Scholar

[8]

M. De Angelis, On exponentially shaped Josephson junctions, Acta. Appl. Math., 122 (2012), 179-189. doi: 10.1007/s10440-012-9736-9.  Google Scholar

[9]

M. De Angelis, Asymptotic estimates related to an integro differential equation, Nonlinear Dynamics and Systems Theory, 13 (2013), 217-228.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

J. P. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, N.Y, 1998.  Google Scholar

[17]

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, The MIT press. England, 2007.  Google Scholar

[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.  Google Scholar

[19]

J. D. Murray, Mathematical Biology, I, II, Springer-Verlag, N.Y. 2002.  Google Scholar

[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.  Google Scholar

[21]

S. Rionero, On the stability of nonautonomous binary dynamical systems of partial differential equations, Att. Acc. Pelor. Per. (AAPP), 91 (2013).  Google Scholar

[22]

Alwyn C. Scott, The Nonlinear Universe: Chaos, Emergence, Life, Springer-Verlag New York, 2007, Google Scholar

[23]

Alwyn C. Scott, Neuroscience A Mathematical Primer, Springer-Verlag New York, 2002.  Google Scholar

[24]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[25]

Torcicollo I., On the Dynamics of the nonlinear duopoly game, International Journal of Non-Linear Mechanics, 57 (2013), 31-38. Google Scholar

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