A steady current injected into an unmyelinated axon of radius (length constant )…

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

Forces, Machines and Signals, Ch The nerve impulse

Problem

A steady current injected into an unmyelinated axon of radius

a_1 = 0.25\,\mathrm{mm}

(length constant

\lambda_1 = 7.2\,\mathrm{mm}

) produces a resting depolarisation of

V_0 = 40\,\mathrm{mV}

at the injection site, decaying passively as

V(x) = V_0 e^{-x/\lambda}

. Suppose the local threshold for triggering an active spike, once active channels are present, is

V_{\rm th} = 15\,\mathrm{mV}

above rest. ((a)) Find the distance

x_{\rm th}

at which the passive potential has decayed to exactly

V_{\rm th}

. ((b)) The axon radius is now quadrupled,

a_2 = 4a_1

, with

R_m

and

R_i

unchanged. Find the new length constant

\lambda_2

and the new

x_{\rm th}

. ((c)) Even with the larger axon, explain in one or two sentences why passive spread of this kind still cannot carry a usable signal a metre, referring to how

x_{\rm th}

compares with

1\,\mathrm{m}

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