Abstract
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.
Highlights
The dynamic threshold plays an important role in neuronal information processing, its mechanism has not been well characterized
In investigating of the quantitative laws of AP generation, Hodgkin and Huxley found that the threshold could be increased by Na+ channel inactivation and K+ channel activation and suggested that the threshold might be a function of the membrane potential[20,21]
Several experimental works on the threshold variation adopted similar methods or obtained results that were derived from these concepts; threshold was defined as the threshold voltage value at the time point that a stimulus is switched off[9,24,28], and simple dynamic threshold equations similar to equation (1) were adopted[6,9,18]
Summary
The dynamic threshold plays an important role in neuronal information processing, its mechanism has not been well characterized. The biophysical mechanism has not been well studied by neuroscientists, mathematical classifications and mechanisms describing the neuronal threshold, especially the separatrix concept and quasi-threshold phenomena, were proposed by FitzHugh in 1950s26. These mechanisms have not captured the attention of scientists in a long time, until recently, when similar concepts have been hypothesized or reintroduced by researchers[19,27,28]. 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. According to the form of a separatrix, the general threshold evolving equation versus time is deduced and transformed into equation (1) for the simplest case
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
