Periodic orbits in planar systems modelling neural activity
Authors:
Robert E. Kooij and Fotios Giannakopoulos
Journal:
Quart. Appl. Math. 58 (2000), 437-457
MSC:
Primary 92C20; Secondary 34C25, 34C60
DOI:
https://doi.org/10.1090/qam/1770648
MathSciNet review:
MR1770648
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Abstract: In this paper we will prove certain properties of a planar dynamical system modelling the neural activity of a network consisting of two neurons. At first we show that for a certain region in parameter space (such that there exist three equilibria) the dynamical system has no periodic orbits. To this end we need a new criterion for the nonexistence of limit cycles in a system of Liénard type (Lemma 3.1). Next we derive conditions under which our model system has exactly one periodic orbit, which will be a stable limit cycle. Finally, we cover a part of the parameter space where we can prove that the dynamical system has three equilibria such that around two of the equilibria at most one limit cycle can exist.
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B. Baird, Nonlinear dynamics of pattern formation and pattern recognition in the rabbit olfactory bulb, Physica 22D, 150–175 (1986)
R. M. Borisyuk and A. B. Kirillov, Bifurcation analysis of a neural network model, Biol. Cybern. 66, 319–325 (1992)
J. D. Cowan and G. B. Ermentrout, Some Aspects of the “Eigenbehavior” of Neural Nets, in S. A. Levin: Studies in Mathematical Biology Part I: Cellular Behavior and the Development of Pattern, Studies in Mathematics, vol. 15, Mathematical Association of America, Washington, DC, 1978, pp. 67–117
F. Dumortier, R. Roussarie, and J. Sotomayor, Generic 3-Parameter Families of Planar Fields, Unfoldings of Saddle, Focus and Elliptic Singularities with Nilpotent Linear Parts, in F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek: Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals, Lecture Notes in Mathematics 1480, Springer-Verlag, Berlin, 1991
W. J. Freeman, Tutorial on neurobiology: From single neurons to brain chaos, International Journal of Bifurcations and Chaos 2, 441–482 (1992)
F. Giannakopoulos and O. Oster, Bifurcation properties of a planar system modelling neural activity, Planar Nonlinear Dynamical Systems (Delft, 1995), Differential Equations and Dynamical Systems 5, 229–242 (1997)
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983
M. W. Hirsch, Convergent Activation Dynamics in Continuous Time Networks, Neural Networks 2, 331–349 (1989)
F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, Springer-Verlag, New York, 1997
J. J. Hopfield, Neural Networks and Physical Systems with Emergent Collective Computational Abilities, Proceedings of the National Academy of Sciences, USA 79, 2554–2558 (1982)
N. Levinson and O. K. Smith, A general equation for relaxation oscillation, Duke Math. Journal 9, 382–403 (1942)
A. Liénard, Étude des oscillations entretenues, Rev. Gen. d’Electricité, XXIII, 1928, pp. 901–946 [in French]
R. E. Kooij and Sun Jianhua, A note on “Uniqueness of limit cycles in a Liénard type system", J. Math. Anal. Appl. 208, 260–276 (1997)
J. G. Nicholls, A. R. Martin, and B. G. Wallace, From neuron to brain, Sinauer Associates Inc. Publishers, Sunderland, Massachusetts, USA, 1992
G. Sansone and R. Conti, Nonlinear Differential Equations, Macmillan, New York, 1964
G. M. Shepherd, Neurobiology, Oxford University Press, New York, 1994
H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical J. 12, 1–24 (1972)
Ye Yan-Qian, Theory of limit cycles, Transl. of Math. Monographs, Vol. 66, Amer. Math. Soc., Providence, Rhode Island, 1986
Zhang Zhi-fen, On the uniqueness of limit cycles of certain equations of nonlinear oscillations, Dokl. Akad. Nauk SSSR 119, 659–662 (1958) [in Russian]
Zhang Zhi-fen, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal. 23, 63–76 (1986)
Zhang Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Dong Zhen-xi, Qualitative Theory of Differential Equations, Translations of Math. Monographs, Vol. 101, Amer. Math. Soc., Providence, RI, 1992
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