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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DOI:
https://doi.org/10.1090/qam/1770648

Article copyright:
© Copyright 2000
American Mathematical Society