Investigation Of Stability Changes In A Neural Field Model

Abstract

In this paper, the stability analysis of the neural field model is studied. The special case for three neuron populations is considered. The work is conducted by finding the characteristic equation of the system first and then investigating the characteristic roots of the third-order equation by using the Routh-Hurwitz criterion and Sturm sequence. The main analysis is given in two parts considering the nonexistence and existence of the delay term. Some basic stability criteria in terms of coefficients of the system are given in the theorems.

Keywords

• Characteristic Equation
• Neural Field Model
• Routh-Hurwitz Criterion
• Stability Analysis
• Sturm Sequence

References

1. [1] Wilson H., Cowan J., 1973. A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue. Biological Cybernetics, 13(2), pp. 55-80.
2. [2] Amari S.I., 1977. Dynamics of Pattern Formation in Lateral-inhibition Type Neural Fields. Biological Cybernetics, 27(2), pp. 77-87.
3. [3] Coombes S., 2005. Waves, Bumps, and Patterns in Neural Field Theories. Biological Cybernetics, 93(2), pp. 91-108.
4. [4] Atay F.M., Hutt A., 2006. Stability and Bifurcations in Neural Fields with Finite Propagation Speed and General Connectivity. Siam Journal on Mathematical Analysis, 5(4), pp. 670-698.
5. [5] Coombes S., Venkov N.A., Shiau L., Bojak L., Liley D.T.J., Laing C.R., 2007. Modeling Electrocortical Activity Through Improved Local Approximations of Integral Neural Field Equations. Physical Review E , 76, 051901.
6. [6] Faye G., Faugeras O., 2010. Some Theoretical and Numerical Results for Delayed Neural Field Equations. Physica D: Nonlinear Phenomena, 239(9), pp. 561-578.
7. [7] Veltz R., Faugeras O., 2011. Stability of the Stationary Solutions of Neural Field Equations with Propagation Delay. Journal of Mathematical Neuroscience, 1, 1, pp. 1-28.
8. [8] Van Gils S.A., Janssens S.G., Kuznetsov Yu. A., Visser S., 2013. On Local Bifurcations in Neural Field Models with Transmission Delays. Journal of Mathematical Biology, 66(4), pp. 837-887.

9. [9] Veltz R., 2013. Interplay Between Synaptic Delays and Propagation Delays in Neural Field Equations. Siam Journal of Applied Dynamical Systems, 12(3), pp. 1566-1612.
10. [10] Veltz R. Faugeras O., 2013. A Center Manifold Result for Delayed Neural Fields Equations. Siam Journal on Mathematical Analysis, 45(3), pp. 1527-1562.
11. [11] Özgür B., Demir A., 2016. Some Stability Charts of a Neural Field Model of Two Neural Populations. Communications in Mathematics and Applications, 7(2), pp. 159-166.
12. [12] 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), pp. 1495-1502.
13. [13] Özgür B., Demir A., 2018. On the Stability of Two Neuron Populations Interacting with Each Other. Rocky Mountain Journal of Mathematics, 48(7), pp. 2337-2346.
14. [14] Özgür B., 2019. Some stability notes of a neural field model. IDES 2019 International Design and Engineering Symposium, İzmir, Turkey, 10-12 October, pp. 123-126.
15. [15] Özgür B., 2020. Stability Switches in a Neural Field Model: An Algebraic Study on the Parameters. Sakarya University Journal of Science, 24(1), pp. 178-182.
16. [16] Forde J., Nelson P., 2004. Applications of Sturm Sequences to Bifurcation Analysis of Delay Differential Equation Models. Journal of Mathematical Analysis and Applications, 300, pp. 273-284.