Discontinuity, Nonlinearity, and Complexity
Bifurcation Analysis of WilsonCowan Model with Singular Impulses
Discontinuity, Nonlinearity, and Complexity 10(1) (2021) 161172  DOI:10.5890/DNC.2021.03.012
Marat Akhmet , Sabahattincag
Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey
Download Full Text PDF
Abstract
The paper concerns with WilsonCowan neural model. The main novelty of the study is that besides the traditional singularity of the model, we consider singular impulses. A new technique of analysis of the phenomenon is suggested. This allows to consider the existence of solutions of the model and bifurcation in ultimate neural behavior observed through numerical simulations. The bifurcations are reasoned by impulses and singularity in the model and they concern the structure of attractors, which consist of newly introduced sets in the phase space such that medusas and rings.
References

[1] 
Wilson, H.R. and Cowan, J.D. (1972), Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, 12, 124.


[2] 
Kilpatrick, Z.P. (2013), Wilsoncowan model, In: Jaeger, D. and Jung, R. (Eds.), Encyclopedia of Computational Neuroscience, pp. 15, Springer: New York.


[3] 
Wang, Y., Goodfellow, M., Taylor, P.N. and Baier, G. (2014), Dynamic mechanisms of neocortical focal seizure onset, PLoS Comput Biol, 10(8), 118.


[4] 
Aka, H., Alassar, R., Covachev, V., Covacheva, Z. and AlZahrani, E. (2004), Continuoustime additive hopeldtype neural networks with impulses, Journal of Mathematical Analysis and Applications, 290(2), 436451.


[5] 
Segel, L.A. and Slemrod, M. (1989), The quasisteady state assumption: a case study in perturbation, SIAM Review, 31, 446477.


[6] 
Hek, G. (2010), Geometric singular perturbation theory in biological practice, J. Math. Biol., 60, 347386.


[7] 
Owen, M.R. and Lewis, M.A. (2001), How predation can slow, stop, or reverse a prey invasion, Bulletin of Mathematical Biology, 63, 655684.


[8] 
Terman, D. (2002), Dynamics of singularly perturbed neural networks, In: An introduction to mathematical modeling in physiology, cell biology, and immunology (New Orleans, LA, 2001), Vol. 59 of Proc. Sympos. Appl. Math., pp. 132, Amer. Math. Soc., Providence: RI.


[9] 
Damiano, E.R. and Rabbitt, R.D. (1996), A singular perturbation model of fluid dynamics in the vestibular semicircular canal and ampulla, Journal of Fluid Mechanics, 307, 333372.


[10] 
Kokotovic, P.V. (1984), Applications of singular perturbation techniques to control problems, SIAM Review 26, 501550.


[11] 
Gondal, I. (1988), On the application of singular perturbation techniques to nuclear engineering control problems, IEEE Transactions on Nuclear Science, 35, 10801085.


[12] 
Kuang, Y. (1993), Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, Elsevier Science.


[13] 
Hoppensteadt, F. C. and Izhikevich, E. M. (1997), Weakly Connected Neural Networks, Applied Mathematical Sciences, Springer: New York.


[14] 
Fasoli, D., Cattani, A. and Panzeri, S. (2006), The complexity of dynamics in small neural circuits, PLoS Computational Biology, 12(8), e1004992.


[15] 
Dias, A.P.S. and Lamb, J.S. (2006), Local bifurcation in symmetric coupled cell networks: Linear theory, Physica D: Nonlinear Phenomena, 223(1), 93108.


[16] 
Krupa, M. and Szmolyan, P. (2001), Relaxation oscillation and canard explosion, Journal of Differential Equations, 174(2), 312368.


[17] 
Szmolyan, P. and Wechselberger, M. (2001), Canards in r3, Journal of Differential Equations, 177(2), 419453.


[18] 
De Maesschalck, P., Popovi\{c}, N. and Kaper, T.J. (2009), Canards and bifurcation delays of spatially homogeneous and inhomogeneous types in reactiondiffusion equations, Advances in Differential Equations, 14(910), 943962.


[19] 
Benoit, E., Callot, J. L., Diener, F., Diener, M. et al. (1981), Chasse au canard (premi\{e}re partie), Collectanea Mathematica, 32(1), 3776.


[20] 
Campbell, S.A., Stone, E. Erneux, T. (2009), Delay induced canards in a model of high speed machining, Dynamical Systems, 24(3), 373392.


[21]  Akhmet, M., Cag, S. (2018), Tikhonov theorem for differential equations with singular impulses Discontinuity, Nonlinearity, and Complexity, 7, 291303.


[22] 
Srinivasan, R., Thorpe, S. and Nunez, P. (2013), Topdown in
uences on local networks: Basic theory with experimental implications, Frontiers in Computational Neuroscience, 7, 29.
