Asymptotic construction of pulses in the Hodgkin Huxley model for myelinated nerves

Understanding wave propagation in discrete excitable media is challenging because of poorly understood phenomena associated with spatial discreteness [1, 2, 3, 4, 5]. The study of the transmission of nerve impulses along myelinated axons is a paradigmatic example. Myelinated nerve fibers, such as the motor axons of vertebrates, are covered almost everywhere by a thick insulating coat of myelin. Only a fraction of the active membrane is exposed, at small active nodes called Ranvier nodes. The myelinated axons of motor nerves can be very long, and contain hundreds or thousands of nodes [6]. The wave activity jumps from one node to the next one giving rise to “saltatory” propagation of impulses [7]. Saltatory conduction on myelinated nerve models has two important features. One is the possibility of increasing the speed of the nerve impulse while decreasing the diameter of the nerve fiber [8]. The other is propagation failure when the myelin coat is damaged [9], which causes diseases such as multiple sclerosis.