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 , 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=109 W 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.