During an action potential current flows across the neuronal membrane. Basically, during depolarization, Na+ goes into the cell, during repolarization K+ goes out. This process for a myelinated axon is shown in Fig. 1.
Action potentials in myelinated neurons jump from node to node, a process called saltatory nerve conduction. These nodes, called nodes of Ranvier, lack myelin and contain Na+ and K+ channels. Outside these nodes there is insulating myelin wrapped around the axon and the axonal membrane lacks Na+ and K+ channels there.
What happens during an action potential at a node is the following: Na+ rushes in at the node. Because of the insulating myelin in the next axonal segment leading up to the next node in line, Na+ can run through the axon to the next node without much signal loss. This passive spread of depolarizing current is much faster than repeatedly generate a new action potential along the length of the axon.
To answer your questions now:
Question: Is it correct to say electric current flows through the extracellular space or cytosol of a nerve during impulse conduction?
Answer: Yes... and no. No, because electric current and notably Na+ and K+ flow perpendicular to the axonal membrane during action potentials. Yes, because the Na+ ions do move through passive current spread through the axon to the next node.
Question: How exactly does the current flow in myelineated nerve fibers?
Answer: Your professor was spot on. During depolarization, ions (notably Na+ in this case) are carried from the extracellular space through Na+ channels in the node of Ranvier to the intracellular space. From there it spreads through passive diffusion to then next node.
Fig. 1. Action potential generation in myelinated axons. source: Purves et al.(2001)
- Purves et al. (ed). Neuroscience. 2nd ed. Sunderland (MA): Sinauer Associates (2001)