Journal
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
Volume 14, Issue 4, Pages 1808-1844Publisher
SIAM PUBLICATIONS
DOI: 10.1137/140981770
Keywords
geometric singular perturbation theory; multiple time scales; averaging; folded singularities; torus canards; bursting; neuronal dynamics
Categories
Funding
- ARC [FT120100309]
- NSF DMS awards [1021701, 1312508]
- University of Pittsburgh Center for International Studies
- Australian Research Council [FT120100309] Funding Source: Australian Research Council
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1312508] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1021701] Funding Source: National Science Foundation
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In this work, we identify generic bifurcation scenarios corresponding to transitions between bursting and tonic spiking solutions in a model for a coupled pair of burst-capable neurons, and we elucidate the central role of folded singularities in these scenarios. The folded singularities in our work arise in the context of fast-slow averaging, and hence our results link with the study of torus canards, a recently identified class of ordinary differential equation (ODE) solutions featuring oscillatory excursions along repelling structures in phase space [J. Burke et al., J. Math. Neurosci., 2 (2012), pp. 1-30]; in particular, our work extends this study to systems featuring two slow variables and symmetry and goes significantly beyond the analysis of activity transitions presented by Best et al. [SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 1107-1139].
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