  We use the qualitative insight of a planar neuronal phase portrait to detect
an excitability switch in arbitrary conductance-based models from a simple
mathematical condition. The condition expresses a balance between ion channels
that provide a negative feedback at resting potential (restorative channels)
and those that provide a positive feedback at resting potential (regenerative
channels). Geometrically, the condition imposes a transcritical bifurcation
that rules the switch of excitability through the variation of a single
physiological parameter. Our analysis of six different published conductance
based models always finds the transcritical bifurcation and the associated
switch in excitability, which suggests that the mathematical predictions have a
physiological relevance and that a same regulatory mechanism is potentially
involved in the excitability and signaling of many neurons.
