American Institute of Mathematical Sciences

Advanced
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:
[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.  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.  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.  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.   Google Scholar

[7]

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

[8]

M. De Angelis, On exponentially shaped Josephson junctions,, Acta. Appl. Math., 122 (2012), 179.  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.   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.  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.  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, (2008), 191.   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.  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, (2007), 193.  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., (2014), 10440.   Google Scholar

[16]

J. P. Keener and J. Sneyd, Mathematical Physiology,, Springer-Verlag, (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.  doi: 10.1016/j.physrep.2003.10.015.  Google Scholar

[19]

J. D. Murray, Mathematical Biology,, I, (2002).   Google Scholar

[20]

O. Nekhamkina and M. Sheintuch, Boundary-induced spatiotemporal complex patterns in excitable systems,, Phys. Rev., E73 (2006), 66224.  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, (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.   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.  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.  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.  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.   Google Scholar

[7]

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

[8]

M. De Angelis, On exponentially shaped Josephson junctions,, Acta. Appl. Math., 122 (2012), 179.  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.   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.  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.  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, (2008), 191.   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.  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, (2007), 193.  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., (2014), 10440.   Google Scholar

[16]

J. P. Keener and J. Sneyd, Mathematical Physiology,, Springer-Verlag, (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.  doi: 10.1016/j.physrep.2003.10.015.  Google Scholar

[19]

J. D. Murray, Mathematical Biology,, I, (2002).   Google Scholar

[20]

O. Nekhamkina and M. Sheintuch, Boundary-induced spatiotemporal complex patterns in excitable systems,, Phys. Rev., E73 (2006), 66224.  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, (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.   Google Scholar

[1]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[2]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083
[3]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[4]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369
[5]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033
[6]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[7]

Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516
[8]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[9]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[10]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
[11]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587
[12]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191
[13]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[14]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[15]

Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105
[16]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[17]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[18]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[19]

Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227
[20]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences