Singularities and Canards

Kuehn, Christian

doi:10.1007/978-3-319-12316-5_8

Cited by 1 publication

(2 citation statements)

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“…These phenomena were referred to as QTP by FitzHugh, and can be called canard threshold phenomenon (CTP) on the recent advances 29 33 40 . The Fenichel’s Theorem ensures the general existence of quasi-separatrix 29 30 33 41 . The typical type I and type II neurons belong to the type with more than one equilibrium and the type with only one equilibrium locally, respectively.…”

Section: Resultsmentioning

confidence: 99%

“…However, the general mechanism of dynamic threshold for AP generation in a specified neuron has not been well described, especially when the neuron is subjected to differing stimuli, such as a rectangular pulse or a ramp current 9 24 28 . To connect the theory and experiment results, and considering the recent advances on the threshold phenomena associated with large difference between the time scales of fast and slow variables 29 30 that explain the quasi-threshold well 26 , we introduce the general notion of separatrix , which is a boundary separating two different modes of dynamic behavior in state space. The general separatrix is equivalent to the concept of ‘threshold manifold’ in mathematics 31 32 33 and the concept of ‘switching manifold’ in control sciences 19 .…”

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Spike-Threshold Variability Originated from Separatrix-Crossing in Neuronal Dynamics

Wang

et al. 2016

Sci Rep

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The threshold voltage for action potential generation is a key regulator of neuronal signal processing, yet the mechanism of its dynamic variation is still not well described. In this paper, we propose that threshold phenomena can be classified as parameter thresholds and state thresholds. Voltage thresholds which belong to the state threshold are determined by the ‘general separatrix’ in state space. We demonstrate that the separatrix generally exists in the state space of neuron models. The general form of separatrix was assumed as the function of both states and stimuli and the previously assumed threshold evolving equation versus time is naturally deduced from the separatrix. In terms of neuronal dynamics, the threshold voltage variation, which is affected by different stimuli, is determined by crossing the separatrix at different points in state space. We suggest that the separatrix-crossing mechanism in state space is the intrinsic dynamic mechanism for threshold voltages and post-stimulus threshold phenomena. These proposals are also systematically verified in example models, three of which have analytic separatrices and one is the classic Hodgkin-Huxley model. The separatrix-crossing framework provides an overview of the neuronal threshold and will facilitate understanding of the nature of threshold variability.

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