  The threshold voltage for action potential generation is a key regulator of
neuronal signal transduction, 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 neuron 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.
