American Institute of Mathematical Sciences

Advanced
June  2008, 1(2): 263-272. doi: 10.3934/dcdss.2008.1.263

Emergence of travelling waves in smooth nerve fibres

S. Jacquir 1, , S. Binczak 1, , J. P. Gauthier 1, and  J. M. Bilbault 1,

1. 

LE2I UMR CNRS 5158, Aile des Sciences de l’Ingénieur, Université de Bourgogne, BP 47870, 21078 Dijon Cedex, France, France, France, France

Received  October 2006 Revised  May 2007 Published  March 2008

An approximate analytical solution characterizing initial conditions leading to action potential firing in smooth nerve fibres is determined, using the bistable equation. In the first place, we present a non-trivial stationary solution wave. Then, we extract the main features of this solution to obtain a frontier condition between the initiation of the travelling waves and a decay to the resting state. This frontier corresponds to a separatrix in the projected dynamics diagram depending on the width and the amplitude of the stationary wave.
Keywords: nerve fibres, projected dynamics., stationarity.
Mathematics Subject Classification: Primary: 92C20, 35K57; Secondary: 58E3.
Citation: S. Jacquir, S. Binczak, J. P. Gauthier, J. M. Bilbault. Emergence of travelling waves in smooth nerve fibres. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 263-272. doi: 10.3934/dcdss.2008.1.263
[1]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[2]

Gunter M. Ziegler. Projected products of polygons. Electronic Research Announcements, 2004, 10: 122-134.

[3]

Christopher E. Elmer. The stability of stationary fronts for a discrete nerve axon model. Mathematical Biosciences & Engineering, 2007, 4 (1) : 113-129. doi: 10.3934/mbe.2007.4.113
[4]

Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2479-2500. doi: 10.3934/dcdsb.2017127
[5]

David Salas, Lionel Thibault, Emilio Vilches. On smoothness of solutions to projected differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2255-2283. doi: 10.3934/dcds.2019095
[6]

José Antonio Carrillo, Martin Parisot, Zuzanna Szymańska. Mathematical modelling of collagen fibres rearrangement during the tendon healing process. Kinetic and Related Models, 2021, 14 (2) : 283-301. doi: 10.3934/krm.2021005
[7]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[8]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507
[9]

Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial and Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235
[10]

Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042
[11]

Xing Li, Chungen Shen, Lei-Hong Zhang. A projected preconditioned conjugate gradient method for the linear response eigenvalue problem. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 389-412. doi: 10.3934/naco.2018025
[12]

Zhi Guo Feng, Kok Lay Teo, Volker Rehbock. A smoothing approach for semi-infinite programming with projected Newton-type algorithm. Journal of Industrial and Management Optimization, 2009, 5 (1) : 141-151. doi: 10.3934/jimo.2009.5.141
[13]

Rouhollah Tavakoli, Hongchao Zhang. A nonmonotone spectral projected gradient method for large-scale topology optimization problems. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 395-412. doi: 10.3934/naco.2012.2.395
[14]

Brigitte Vallée. Euclidean dynamics. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 281-352. doi: 10.3934/dcds.2006.15.281
[15]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[16]

N. Romero, A. Rovella, F. Vilamajó. Dynamics of vertical delay endomorphisms. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 409-422. doi: 10.3934/dcdsb.2003.3.409
[17]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1
[18]

Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768
[19]

Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1581-1601. doi: 10.3934/cpaa.2015.14.1581
[20]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy