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We introduce a method for computing probabilities for spontaneous activityand propagation fail- ure of the action potential in spatially extended,conductance-based neuronal models subject to channel noise, based onstatistical properties of the membrane potential. We compare differentestimators with respect to the quality of detection, computational costs androbustness and propose the integral of the membrane potential along the axon asan appropriate estimator to detect both spontaneous activity and propagationfailure. Performing a model reduction we achieve a simplified analyticalexpression based on the linearization at the resting potential (resp. thetraveling action potential). This allows to approximate the probabilities forspontaneous activity and propagation failure in terms of (classical) hittingprobabilities of one-dimensional linear stochastic differential equations. Thequality of the approximation with respect to the noise amplitude is discussedand illustrated with numerical results for the spatially extendedHodgkin-Huxley.

Reliability of signal transmission in stochastic nerve axon equations