A, The steady-state threshold curve (red curve) is well approximated by a piecewise linear curve determined by Na channel properties (top dashed black curve), where V

_{i} is the half-inactivation voltage and V

_{T} is the non-inactivated threshold. The slope of the linear asymptote is k

_{a}/k

_{i} (resp. activation and inactivation slope parameters). Na channel properties in this figure were taken from Kuba et al. (2009). The spike threshold is variable only when

, and very variable when (additionally)

. B, The non-inactivated threshold V

_{T} is determined by the maximum Na conductance g

_{Na}, relative to the leak conductance g

_{L}. As the ratio

increases, the steady-state threshold curve

shifts downward (red curves; r = 0.4; 2; 10) and threshold variability is reduced. C, Trajectory of the model in the

phase plane (blue), superimposed on the steady-state threshold curve (red). Spikes are initiated when

(dashed line:

), but the empirical measurement overestimates the threshold. The spike threshold is highly variable in this example (−50 to −10 mV). D, Trajectory of the model in the

phase plane (blue), superimposed on the Na inactivation curve (black). The threshold is very variable when most Na channels are inactivated. E, Voltage trace (black curve) and spike threshold

(red curve;

) in the inactivating exponential model driven by a fluctuating input (see ), where black dots represent empirical measurement of spike onsets (first derivative method, k

_{th} = 5 mV/ms). Note that the membrane potential can exceed threshold without triggering a spike because the threshold is soft (unlike in integrate-and-fire models).

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