Neurobiophysics

1
Neurobiophysics

• Cable Equation (Passive Membrane)Action
  Potential Propagation Hodgkin Huxley Action
  Potential Model

2
Action Potential - Review

• Vm VNa GNa VK GK VCl GCl
• GNa GK GCl

3
Current Paths

• Response to an injected step current charge
• Capacitor (IRm 0)
• Transmembrane Ionic Flux (IRm)
• Along Axoplasm (DV)

4
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)

5
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.

6
Passive Membrane - Analytical

• Note T RmC and VIN RmIIN
• Response to step current for
• C DV/dt V/R IIN (0 lt t lt Dt)
• Vm(t) Vr VIN(1 - e-t/T)
• Response to removal of step current for
• C DV/dt V/R 0 (t gt 0)
• Vm(t) Vr VIN(1 - e-Dt / T) e-t/ T

7
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

8
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)

9
Cable Equation Active Membrane

• d2V/dX2 (Rout Rin) Im
• since Rout ltlt Rin d2V/dX2 Rin Im
• Assumption
• Action potential travels at constant velocity q
• so X q t
• d2V/dX2 d2V/d(q t)2 (1/d2) d2V/dt2

10
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.

11
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 C10-13 F, Rin109 W and Rm
  1010 W
• and q 100 m/s (1 m/s lt q lt 100 m/s)
• and boundary conditions (t, V0) and (t0,
  VVa)
• V Vae-.916t

12
Propagating Action Potential
13
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.

14
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

15
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

16
H H - Concluding Remarks

• The Hodgkin-Huxley Model was first developed in
  the 1940s and published in the 1950s.
• 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.