  We introduce a method for computing probabilities for spontaneous activity
and propagation fail- ure of the action potential in spatially extended,
conductance-based neuronal models subject to channel noise, based on
statistical properties of the membrane potential. We compare different
estimators with respect to the quality of detection, computational costs and
robustness and propose the integral of the membrane potential along the axon as
an appropriate estimator to detect both spontaneous activity and propagation
failure. Performing a model reduction we achieve a simplified analytical
expression based on the linearization at the resting potential (resp. the
traveling action potential). This allows to approximate the probabilities for
spontaneous activity and propagation failure in terms of (classical) hitting
probabilities of one-dimensional linear stochastic differential equations. The
quality of the approximation with respect to the noise amplitude is discussed
and illustrated with numerical results for the spatially extended
Hodgkin-Huxley.
