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A cylindrical dielectric model is used to compute transmembrane potential changes and evaluate the axial electric field magnitudes produced within a nerve by a high-intensity relatively short electrical pulse. For concreteness, the pulse was taken to have a duration of about 700 ns and large current magnitudes in keeping with ongoing experimental studies within our group. Interest in this quantitative analysis arises from probing the possibility of triggering bioeffects at intracellular organelles in tissues (or even whole animals) through such electric stimulation. Almost all other studies have focused on simple spherical cells. This paper provides a theoretical framework for computing electric fields (especially the axial components) within such cylindrical geometries (e.g., nerve cells). It is shown that fields can become sufficiently high within microseconds and initiate electroporation, modulate electrochemical processes (e.g., calcium release), or trigger secondary biochemical effects depending on the electrical pulsing parameters.