Someone argues: "The passive cable equation gives a potential that

PHYSICS · Subjective · 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

Someone argues: "The passive cable equation gives a potential

V(x,t)

that, held on long enough, approaches the steady profile

V_0e^{-x/\lambda}

. So if I inject the current for a full second instead of one millisecond, far more charge has flowed in, and the potential

5\,\mathrm{cm}

from the electrode should be much larger, perhaps even large enough to trigger a spike, since I have given it

1000

times longer to build up." For an axon with

\lambda = 7.2\,\mathrm{mm}

\tau = 1.5\,\mathrm{ms}

and an injected steady depolarisation

V_0 = 40\,\mathrm{mV}

at the electrode, find the error in this argument and compute the actual potential at

x=5\,\mathrm{cm}

once the response has settled.

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