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Action Potential - Review

Action Potential - Review. V m = V Na G Na + V K G K + V Cl G Cl G Na + G K + G Cl. Current Paths. Response to an injected step current charge Capacitor (I Rm = 0) Transmembrane Ionic Flux (I Rm ) Along Axoplasm ( D V). Current Flow - Initial.

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Action Potential - Review

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  1. Action Potential - Review • Vm = VNa GNa + VK GK + VCl GCl • GNa + GK + GCl

  2. Current Paths • Response to an injected step current charge • Capacitor (IRm = 0) • Transmembrane Ionic Flux (IRm) • Along Axoplasm (DV)

  3. Current Flow - Initial • All current flows thru the capacitor • (low resistant path to injected step current) • Redistributed charges change Vm • Current begins to flow thru Rm and spreads laterally, • affecting adjacent membrane capacitance. • At injection point, dv/dt 0, Ic = 0 • Transmembrane current carried only by Rm, • remainder of current spread laterally along axon. • (Rm relatively high resistance, axon relatively low)

  4. Current Flow - Sequential • The process is repeated at adjacent membrane • due to influence of the lateral current along axon. • As the capacitor initially accumulates charge, • Vm changes, current flows thru Rm, dv/dt 0, Ic = 0. • Transmembrane current carried only by Rm, • remainder of current spreads laterally along axon. • etc, etc, etc, etc.

  5. Passive Membrane - Analytical • Note: T = RmC and VIN = RmIIN • Response to step current for • C DV/dt + V/R = IIN (0 < t < Dt) • Vm(t) = Vr + VIN(1 - e-t/T) • Response to removal of step current for • C DV/dt + V/R = 0 (t > 0) • Vm(t’) = Vr + VIN(1 - e-Dt / T) e-t’/ T

  6. Cable Equation Passive Membrane • Propagating Voltage V’ = Vm - VResting • Current Im = -dIin/dX = dIout/dX • General Cable Equation d2V’/dX2 = (Rout + Rin) Im • Passive Membrane Im = V’/Rm + C(dV’/dt) • V’ = Vq e-X/l where l = [ Rm / (Rout + Rin) ]1/2

  7. Cable Equation (Passive) - continued • General Cable Equation d2V’/dX2 = (Rout + Rin) Im • Passive Membrane Im = V’/Rm + C(dV’/dt) • Action Potential Equation (by substituting from above) • (Rout + Rin)-1 d2V’/dX2 = V’/Rm + C(dV’/dt)

  8. Cable Equation Active Membrane • d2V’/dX2 = (Rout + Rin) Im • since Rout >> Rin d2V’/dX2 = Rout Im • Assumption • Action potential travels at constant velocity q • so X = q t • d2V’/dX2 = d2V’/d(q t)2 = (1/d2)d2V’/dt2

  9. Cable Equation (Active) - continued • From • (Rout + Rin)-1 d2V’/dX2 = V’/Rm + C(dV’/dt) • d2V’/dX2 = Rout Im • d2V’/dX2 = d2V’/d(q t)2 = (1/d2)d2V’/dt2 • Substituting and rearranging • (Rinq2)-1(d2V’/dt2) - C(dV’/dt) - V’/Rm = 0 • Im - IC - IRm = 0 • Note: Differential Potential V’ = Vm - VResting • is the propagating potential.

  10. Cable Equation (Active) - continued • (Rinq2)-1(d2V’)/dt2 - C(dV’/dt) - V’/Rm = 0 • d2V’/dt2 - (Rinq2) C(dV’/dt) - (Rinq2)/Rm V’ = 0 • Solving the differential equation and using typical values for C=10-13 F, Rin=109W and Rm = 1010 W • and q = 100 m/s (1 m/s < q < 100 m/s) • and boundary conditions (t=¥,V’=0) and (t=0, V’=Va) • V’ = Vae-.916t

  11. Propagating Action Potential

  12. Action Potential • If a stimulus exceeds threshold voltage, then • a characteristic non-linear response occurs. • An voltage waveform the so called electrogenic • “Action Potential” is generated due to a change in the membrane permeability to sodium and potassium ions. • The action potential is propagated undiminished and with constant velocity along the nerve axon.

  13. Hodgkin-Huxley Equation • Unit Membrane Model • Longitudinal resistance of axoplasm per unit length • Resistance = Resistivity / Cross Sectional Area • Membrane Current Density (Flux) • Currents (Capacitive, Sodium, Potassium, Others) • Uses Conductances rather than Resistances • Variable Permeabilities as a function of Vm’(t) • Sodium GNa = GNa M3H • Potassium GK = GK N4

  14. H & H - continued • Conductances Gna and GK are variable and are defined by their respective permeabilities. • Sodium Gna = GNa M3H • Potassium GK = GK N4 • M is the hypothetical process that activates GNa • H is the hypothetical process that deactivates GNa • N is is the hypothetical process that activates GK • M, H, N are membrane potential and time dependent • G = G Max

  15. H & H - Concluding Remark • The Hodgkin-Huxley Model was first developed in the 1940’s and published in the 1950’s. • It does not explain how or why the membrane permeabilities change, but it does model the shape and speed of the action potential quite faithfully. • Empirical values were developed for the GNa, GK, GL • as well as the hypothetical permeability relationships for M, H, N using the giant squid axon.

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