An unmyelinated invertebrate axon has radius

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch The nerve impulse

Problem

a = 90\,\mathrm{\mu m}

, specific membrane resistance

R_m = 2000\,\Omega\,\mathrm{cm^2}

, specific membrane capacitance

C_m = 1\,\mathrm{\mu F\,cm^{-2}}

, axial resistivity

R_i = 50\,\Omega\,\mathrm{cm}

, and a sodium activation time

\tau_{\rm act} = 0.3\,\mathrm{ms}

set by channel kinetics rather than geometry. ((a)) Starting from Kirchhoff's current law applied to a ladder of axial resistors and membrane resistor-capacitor pairs, derive the cable equation

\lambda^2V''=\tau V_t+V

and give

\lambda=\sqrt{aR_m/2R_i}

and

\tau=R_mC_m

in terms of the specific quantities

R_i

R_m

C_m

and

a

. ((b)) Ahead of a wave of fixed shape moving at speed

v

V(x-vt)

, the resting membrane must be charged towards threshold within the sodium activation time

\tau_{\rm act}

, much shorter than

\tau

. Treating the cable over that short time as a diffusion equation with

D=\lambda^2/\tau

, show that

v\sim\sqrt{D/\tau_{\rm act}}=\lambda/\sqrt{\tau\,\tau_{\rm act}}\propto\sqrt a

at fixed

R_i

C_m

\tau_{\rm act}

, and explain why

R_m

drops out. ((c)) Compute

\lambda

\tau

and the estimated conduction velocity of this axon. ((d)) A thicker axon of the same species, with the same

R_m

R_i

C_m

and

\tau_{\rm act}

, conducts at exactly twice this speed. Use the scaling law of part (b) to find its radius.

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