A note on the stability of a neural field model

Berrak Özgür; Ali Demir; Sertaç Erman

Research Article

A note on the stability of a neural field model

Year 2018, Volume: 47 Issue: 6, 1495 - 1502, 12.12.2018

Abstract

In this paper we consider the neural field model for two neural populations. We investigate the properties of D-curves and we give some conditions for asymptotic stability. The asymptotic stability region is determined by using the Stépan's formula. Taking various delay terms into account, the effect of delay on the stability is investigated. Moreover we study on the stability cases by considering the real roots of the characteristic equation.

Keywords

Neural field model, asymptotic stability

References

Amari, SI. Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern. 27 (2), 77-87, 1977.
Erman, S., Demir, A. An analysis on the stability of a state dependent delay differential equation, Open Math., 14 (1), 425-435 ,2016.
Faye G. Symmetry breaking and pattern formation in some neural field equations, PhD thesis, University of Nice- Sophia Antipolis, 2012.
Faye, G., Faugeras, O. Some theoretical and numerical results for delayed neural field equations, Physica D, 239 (9), 561-578, 2010.
Huang, C., Vandewalle, S. An analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays, SIAM J. Sci. Comput., 25 (5), 1608-1632, 2004.
Insperger, T., St\'{e}p\'{a}n, G. Semi-discretization for time-delay systems, Stability and engineering applications, New York, Springer, 2011.
McCormick, DA., Connors, BW., Lighthall, JW., Prince, DA. Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex, J Neurophysiol, 54 (4), 782-806, 1985.
Stépan, G. Retarded dynamical systems: stability and characteristic functions, England, Longman Scientific \& Technical, 1989.
Veltz, R. Interplay between synaptic delays and propagation delays in neural field equations. Siam J Appl Dyn Syst, 12 (3), 1566-1612, 2013.
Veltz, R., Faugeras, O. Stability of the stationary solutions of neural field equations with propagation delay. Journal of Mathematical Neuroscience 1:1, 2011.
Veltz, R., Faugeras, O. A center manifold result for delayed neural fields equations, Siam J Math Anal 45 (3), 1527-1562, 2013.
Wilson, H., Cowan, J. A Mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybern. 13 (2), 55-80, 1973.

Year 2018, Volume: 47 Issue: 6, 1495 - 1502, 12.12.2018

Berrak Özgür Ali Demir , Sertaç Erman

Abstract

References

Amari, SI. Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern. 27 (2), 77-87, 1977.
Erman, S., Demir, A. An analysis on the stability of a state dependent delay differential equation, Open Math., 14 (1), 425-435 ,2016.
Faye G. Symmetry breaking and pattern formation in some neural field equations, PhD thesis, University of Nice- Sophia Antipolis, 2012.
Faye, G., Faugeras, O. Some theoretical and numerical results for delayed neural field equations, Physica D, 239 (9), 561-578, 2010.
Huang, C., Vandewalle, S. An analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays, SIAM J. Sci. Comput., 25 (5), 1608-1632, 2004.
Insperger, T., St\'{e}p\'{a}n, G. Semi-discretization for time-delay systems, Stability and engineering applications, New York, Springer, 2011.
McCormick, DA., Connors, BW., Lighthall, JW., Prince, DA. Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex, J Neurophysiol, 54 (4), 782-806, 1985.
Stépan, G. Retarded dynamical systems: stability and characteristic functions, England, Longman Scientific \& Technical, 1989.
Veltz, R. Interplay between synaptic delays and propagation delays in neural field equations. Siam J Appl Dyn Syst, 12 (3), 1566-1612, 2013.
Veltz, R., Faugeras, O. Stability of the stationary solutions of neural field equations with propagation delay. Journal of Mathematical Neuroscience 1:1, 2011.
Veltz, R., Faugeras, O. A center manifold result for delayed neural fields equations, Siam J Math Anal 45 (3), 1527-1562, 2013.
Wilson, H., Cowan, J. A Mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybern. 13 (2), 55-80, 1973.

There are 12 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Berrak Özgür This is me Ali Demir Sertaç Erman This is me
Publication Date	December 12, 2018
Published in Issue	Year 2018 Volume: 47 Issue: 6

Cite

APA	Özgür, B., Demir, A., & Erman, S. (2018). A note on the stability of a neural field model. Hacettepe Journal of Mathematics and Statistics, 47(6), 1495-1502.
AMA	Özgür B, Demir A, Erman S. A note on the stability of a neural field model. Hacettepe Journal of Mathematics and Statistics. December 2018;47(6):1495-1502.
Chicago	Özgür, Berrak, Ali Demir, and Sertaç Erman. “A Note on the Stability of a Neural Field Model”. Hacettepe Journal of Mathematics and Statistics 47, no. 6 (December 2018): 1495-1502.
EndNote	Özgür B, Demir A, Erman S (December 1, 2018) A note on the stability of a neural field model. Hacettepe Journal of Mathematics and Statistics 47 6 1495–1502.
IEEE	B. Özgür, A. Demir, and S. Erman, “A note on the stability of a neural field model”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, pp. 1495–1502, 2018.
ISNAD	Özgür, Berrak et al. “A Note on the Stability of a Neural Field Model”. Hacettepe Journal of Mathematics and Statistics 47/6 (December 2018), 1495-1502.
JAMA	Özgür B, Demir A, Erman S. A note on the stability of a neural field model. Hacettepe Journal of Mathematics and Statistics. 2018;47:1495–1502.
MLA	Özgür, Berrak et al. “A Note on the Stability of a Neural Field Model”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, 2018, pp. 1495-02.
Vancouver	Özgür B, Demir A, Erman S. A note on the stability of a neural field model. Hacettepe Journal of Mathematics and Statistics. 2018;47(6):1495-502.