1Physiology 472/572 - 2010 - Quantitative modeling of biological systems Lecture 8: Nerve impulse propagation - Part II Cable equations • to analyze action potential, consider the axon as a leaky cable with capacitance • all variables are now functions of x and t - need partial derivatives • define i2 as the current inside the axon associated with the action potential • assume a fixed potential outside the axon (e.g. axon in bath) • then xEr1im22∂∂−= where r2 is the resistance per unit length of the axon • from conservation of charge m2ixi−=∂∂ • combining gives icm2m22iiixEr1+==∂∂ • i.e. )E - (Eg )E - (Eg )E - (Eg tECxEr1LmLKmKNamNam2m22+++∂∂=∂∂ • note that this has form of unsteady diffusion equation with extra terms - a type of reaction-diffusion equation Traveling waves • suppose a function F(x,t) represents a wave traveling with velocity θ • then shape at time t0 is identical to shape at time 0 translated to right a distance θt0 • Define f(x) = F(x,0), then F(x, t) = F(x - θt,0) = f(x - θt)2Traveling wave solution • look for solutions of the form Em = f(x - θt) • then 2m222m2tEθ1xE∂∂=∂∂ • )E - (Eg )E - (Eg )E - (Eg tECtErθ1LmLKmKNamNam2m222+++∂∂=∂∂ • together with the equations for dn/dt, dm/dt and dh/dt this can be treated as a system of ordinary differential equations • set Em slightly different from resting value, n, m and h to resting values • HH integrated this numerically and found that the solution goes to +∞ or -∞ depending on whether θ is too small or too large (i.e. no traveling wave) • correct value of θ found by trial and error Features of predicted action potential • first gNa rises - influx of sodium ions - rapid depolarization (Nernst ENa = +62 mV) • then gK rises - efflux of potassium ions - slower repolarization (Nernst EK = -95 mV) • overshoot and refractory period • flux of ions is small - ion concentrations inside and outside axon do not change significantly during an action
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