Journal article
Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network
Physica. D, Vol.239(9), pp.504-514
2010
DOI: 10.1016/j.physd.2009.12.010
Abstract
Recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators [R. Curtu, A. Shpiro, N. Rubin, J. Rinzel, Mechanisms for frequency control in neuronal competition models, SIAM J. Appl. Dyn. Syst. 7 (2) (2008) 609–649] showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics. Several types of behavior were identified such as: fusion, equivalent to a steady state of identical activity levels for both neural units; oscillations due to either an
escape or a
release mechanism; and a winner-take-all state of bistability. The model consists of two neural populations interacting through reciprocal inhibition, each endowed with a slow negative-feedback process in the form of spike frequency adaptation. In this paper we report the occurrence of another complex oscillatory pattern, the mixed-mode oscillations (MMOs). They exist in the model at the transition between the relaxation oscillator dynamical regime and the winner-take-all regime. The system distinguishes itself from other neuronal models where MMOs were found by the following interesting feature: there is no autocatalysis involved (as in the examples of voltage-gated persistent inward currents and/or intrapopulation recurrent excitation) and therefore the two cells in the network are
not intrinsic oscillators; the oscillations are instead a combined result of the mutual inhibition and the adaptation. We prove that the MMOs are due to a
singular Hopf bifurcation point situated in close distance to the transition point to the winner-take-all case. We also show that in the vicinity of the singular Hopf other types of bifurcations exist and we construct numerically the corresponding diagrams.
Details
- Title: Subtitle
- Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network
- Creators
- Rodica Curtu - Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242, United States
- Resource Type
- Journal article
- Publication Details
- Physica. D, Vol.239(9), pp.504-514
- Publisher
- Elsevier B.V
- DOI
- 10.1016/j.physd.2009.12.010
- ISSN
- 0167-2789
- eISSN
- 1872-8022
- Language
- English
- Date published
- 2010
- Academic Unit
- Mathematics; Iowa Neuroscience Institute
- Record Identifier
- 9984070744902771
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