American Institute of Mathematical Sciences

Advanced
October  2004, 10(4): 925-940. doi: 10.3934/dcds.2004.10.925

A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium

Jonathan E. Rubin 1,

1. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States

Received  September 2002 Revised  October 2003 Published  March 2004

Past work on stability analysis of traveling waves in neuronal media has mostly focused on linearization around perturbations of spike times and has been done in the context of a restricted class of models. In theory, stability of such solutions could be affected by more general forms of perturbations. In the main result of this paper, linearization about more general perturbations is used to derive an eigenvalue problem for the stability of a traveling wave solution in the biophysically derived theta model, for which stability of waves has not previously been considered. The resulting eigenvalue problem is a nonlocal equation. This can be integrated to yield a reduced integral equation relating eigenvalues and wave speed, which is itself related to the Evans function for the nonlocal eigenvalue problem. I show that one solution to the nonlocal equation is the derivative of the wave, corresponding to translation invariance. Further, I establish that there is no unstable essential spectrum for this problem, that the magnitude of eigenvalues is bounded, and that for a special but commonly assumed form of coupling, any possible eigenfunctions for real, positive eigenvalues are nonmonotone on $(-\infty,0)$.
Keywords: eigenvalues, traveling wave, Neuronal dynamics, stability, nonlocal equations., integrodifferential equations.
Mathematics Subject Classification: 34L05, 37B55, 45J05, 92C2.
Citation: Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925
[1]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[2]

Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160
[3]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
[4]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
[5]

Min He. A class of integrodifferential equations and applications. Conference Publications, 2005, 2005 (Special) : 386-396. doi: 10.3934/proc.2005.2005.386
[6]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[7]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[8]

Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048
[9]

Xiaoxiao Zheng, Hui Wu. Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components. Mathematical Foundations of Computing, 2020, 3 (1) : 11-24. doi: 10.3934/mfc.2020002
[10]

Guangying Lv, Mingxin Wang. Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 415-433. doi: 10.3934/dcdsb.2010.13.415
[11]

Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681
[12]

Stephan Didas, Joachim Weickert. Integrodifferential equations for continuous multiscale wavelet shrinkage. Inverse Problems & Imaging, 2007, 1 (1) : 47-62. doi: 10.3934/ipi.2007.1.47
[13]

Paola Loreti, Daniela Sforza. Inverse observability inequalities for integrodifferential equations in square domains. Evolution Equations & Control Theory, 2018, 7 (1) : 61-77. doi: 10.3934/eect.2018004
[14]

Mathias Nikolai Arnesen. Existence of solitary-wave solutions to nonlocal equations. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3483-3510. doi: 10.3934/dcds.2016.36.3483
[15]

Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
[16]

Albert Erkip, Abba I. Ramadan. Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2125-2132. doi: 10.3934/cpaa.2017105
[17]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
[18]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
[19]

Felicia Maria G. Magpantay, Xingfu Zou. Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections. Mathematical Biosciences & Engineering, 2010, 7 (2) : 421-442. doi: 10.3934/mbe.2010.7.421
[20]

Lijun Zhang, Yixia Shi, Maoan Han. Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2917-2926. doi: 10.3934/dcdss.2020217

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences