The stability analysis of a neural field model with small delay

Abstract

In this study it is elucidated a mathematical framework in which the stability for the neural field model for two neuron populations with small delay is investigated. The primary purpose of this analysis is to provide a unifying mathematical framework for illustrating the effect of small delay considering the cases in Routh-Hurwitz criterion.

Keywords

• Neural Field Model
• Routh- Hurwitz Criterion
• Stability Analysis

References

1. REFERENCES
2. [1] Amari SI. Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern 1977; 27(2): 77-87. https://doi.org/10.1007/BF00337259
3. [2] Atay FM, Hutt A. Stability and bifurcations in neural fields with finite propagation speed and general connectivity. SIAM J. Math. Anal. 2006; 5(4): 670-698. https://doi.org/10.1137/S0036139903430884
4. [3] Cook BJ, Peterson ADH, Woldman W, Terry JR. Neural field models: A mathematical overview and unifying framework. Mathematical Neuroscience and Applications 2022; 2: 1-67. https://doi.org/10.46298/mna.7284
5. [4] Coombes S. Waves, bumps, and patterns in neural field theories. Biol. Cybern 2005; 93(2): 91-108. https://doi.org/10.1007/s00422-005-0574-y
6. [5] Coombes S, Venkov NA, Shiau L, Bojak L, Liley DTJ, Laing CR. Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys. Rev. E. 2007; 76: 051901. https://doi.org/10.1103/PhysRevE.76.051901
7. [6] Faye G, Faugeras O. Some theoretical and numerical results for delayed neural field equations. Phys. D: Nonlinear Phenom 2010; 239(9): 561-578. DOI:10.1016/j.physd.2010.01.010
8. [7] Forde J, Nelson P. Applications of Sturm sequences to bifurcation analysis of delay differential equation models. J. Math. Anal. Appl. 2004; 300: 273-284. DOI:10.1016/j.jmaa.2004.02.063

9. [8] 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 2004; 25(5): 1608-1632. https://doi.org/10.1137/S1064827502409717
10. [9] Insperger T, Stepan G. Semi-discretization for time-delay systems, Stability and engineering applications Springer New York; 2011.
11. [10] Özgür B, Demir A. Some stability charts of a neural field model of two neural populations. Communications in Mathematics and Applications 2016; 7(2): 159-166. https://doi.org/10.26713/cma.v7i2.481
12. [11] Özgür B, Demir A. On the stability of two neuron populations interacting with each other. Rocky Mt. J. Math. 2018; 48(7): 2337-2346. DOI:10.1216/RMJ-2018-48-7-2337
13. [12] Özgür B, Demir A, Erman S. A note on the stability of a neural field model. Hacettepe J. Math. Stat. 2018; 47(6): 1495-1502.
14. [13] Özgür B. Stability switches of a neural field model: An algebraic study on the parameters. Sakarya University Journal of Science 2020;24(1): 178-182. https://doi.org/10.16984/saufenbilder.521545
15. [14] Özgür B. Investigation of stability changes in a neural field model. Kocaeli Journal of Science and Engineering 2021;4(1): 46-50. https://doi.org/10.34088/kojose.852170
16. [15] Stepan, G. Retarded dynamical systems: stability and characteristic functions. Longman Scientific & Technical, England; 1989.
17. [16] Van Gils SA, Janssens SG, Kuznetsov Yu. A, Visser S. On local bifurcations in neural field models with transmission delays. J. Math. Biol 2013;66(4): 837-887. DOI:10.1007/s00285-012-0598-6
18. [17] Veltz R, Faugeras O. Stability of the stationary solutions of neural field equations with propagation delay. J. Math. Neurosci., 2011;1:1. https://doi.org/10.1186/2190-8567-1-1
19. [18] Veltz R. Interplay between synaptic delays and propagation delays in neural field equations. SIAM J. Appl. Dyn. 2013;12(3): 1566-1612. https://doi.org/10.1137/120889253
20. [19] Veltz R, Faugeras O. A center manifold result for delayed neural fields equations. SIAM J. Math. Anal. 2013;45(3): 1527-1562. https://doi.org/10.1137/110856162
21. [20] Wilson H, Cowan J. A Mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybern 1973;13(2): 55-80. https://doi.org/10.1007/BF00288786