A mathematical model for conduction of action potentials along bifurcating axons.

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Abstract
1. A mathematical model based on the Hodgkin-Huxley equations is derived to describe quantitatively the propagation of action potentials in a branching axon. 2. The model treats the case of a bifurcating axon with branches of different diameters. The solution takes into account the changes in space constant in the different regions. 3. The model allows for investigating parameters leading to preferential conduction of action potentials in one daughter branch as seen experimentally. 4. Assuming that the only difference between the various daughter branches is in their diameters, conduction blocks should occur simultaneously rather than differentially into all daughter branches when the geometrical ratio is greater than 10. 5. In order to obtain differential conduction into the two branches changes in ionic concentrations due to the repetitive action potentials had to be introduced into the equations. 6. We find that conditions which allow differential buildup of K concentration around the two branches, produce differential conduction block. These conditions may be different periaxonal spaces around the branches or different time constant for recovery processes that eliminate K from the periaxonal space. 7. The effects of an inexcitable branch on conduction of action potentials in the second branch are described. 8. We find that the membrane current which is associated with the action potential is much more sensitive than the action potential itself and shows more distinct changes near regions of inhomogeneity such as a branch point, a step increase in diameter or an inexcitable branch.