Branching Law for Axons

What determines the caliber of axonal branches? We pursue the hypothesis that the axonal caliber has evolved to minimize signal propagation delays, while keeping arbor volume to a minimum. We show that for a general cost function the optimal diameters of mother ($d_0$) and daughter ($d_1$, $d_2$) branches at a bifurcation obey a branching law: $d_{0}^{\nu+2}=d_{1}^{\nu+2} + d_{2}^{\nu+2}$. The derivation relies on the fact that the conduction speed scales with the axon diameter to the power $\nu$ ($\nu=1$ for myelinated axons and $\nu=0.5$ for non-myelinated axons). We test the branching law on the available experimental data and find a reasonable agreement.