# American Institute of Mathematical Sciences

December  2009, 2(4): 873-895. doi: 10.3934/dcdss.2009.2.873

## Survival of subthreshold oscillations: The interplay of noise, bifurcation structure, and return mechanism

 1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada, Canada

Received  September 2008 Revised  April 2009 Published  September 2009

Mixed mode oscillations (MMO's) composed of subthreshold oscillations (STO's) and spikes appear via a variety of mechanisms in models of neural dynamics. Two key elements that can influence the prominence of the STO's are multiple time scales and time varying parameters near critical points. These features can lead to dynamics associated with bifurcation delay, and we consider three systems with this behavior. While it is well known that bifurcation delay related to a slow time scale is sensitive to noise, we compare other aspects of the noise-sensitivity in the context of MMO's, where not only bifurcation delay, but also coherence resonance and dynamics in the interspike interval play a role. Noise can play a role in amplifying the STO's but it can also drive the system into repetitive spiking without STO's. In particular we compare integrate and fire models with models that capture both spike and STO dynamics. The interplay of the underlying bifurcation structure and the modeling of the return mechanism following the spike are major factors in the robustness and noise sensitivity of the STO's in the context of multiple time scales.
Citation: Rachel Kuske, Peter Borowski. Survival of subthreshold oscillations: The interplay of noise, bifurcation structure, and return mechanism. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 873-895. doi: 10.3934/dcdss.2009.2.873
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