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Hodgin & Huxley
• The problem: Explain action potentials
• The preparation: loligo giant axons
• What was known:
– Time dependent conductance: Curtis & Cole
– Multiple batteries in play
• Likely players Na+, K+ : Hodgkin & Katz
• A new method: Voltage Clamp
Action Potentials “Overshoot”
Hodgkin & Huxley, 1939 Nature 144:473-96
Loligo forbesi
Parallel conductance model
How to study the process of action
potential generation
Voltage Clamp
• 3 electrodes used:
– Vo
– Vi
– Ii (injected current, measured with I-mon)
• Advantages
– Space clamp – axial wires used –
– Can effectively eliminate Ic – V is fixed
– Used to isolate time dependent changes in I
Voltage clamp currents in
loligoModern convention:
Original presentation:
- Vm relative to rest
-referenced to inside of cell
-amplitude & polarity appropriate
for necessary charging of membrane
Isolation of the “outward current”
gK(t)
• Sigmoid onset
• Noninactivating
• Exponential offset
Model of gK
gK  gk n4
n
nc 


no
n
dn
  n (1  n)   n n
dt
With dn / dt  0,
i.e. at steady state,
n
n
n  n
If n  no when t  0 :
n  n  (n  no ) exp( t /  n )
where  n  1 /( n   n )
obtain  n to give best fit to I K 
obtain n from g K /g K max
Then  n  n /  n ,
and  n  (1  n ) /  n
Equilibrium n(V), noo
• Similar to a Boltzmann distribution
Rate constants for gate n
• Derived from onset or offset of gK upon DV
 n  n /  n ,
 n  (1  n ) /  n
gK fitted to HH equation
• Reasonable fit to onset, offset & steady
state
Isolate iNa by algebraic
subtraction
• Appears Ohmic
• Sigmoidal onset
• Increase in gNa is
reversible
• g(V) is independent of i
sign
Current flow through pNa is
Ohmic
• Open channel I/V curve
• Instantaneous conductance
gNa kinetics
• Both activation
and inactivation
speed up with
depolarization
Model of gNa
h
m  m  (m  mo ) exp( t /  m )
h
m
h  h  (h  ho ) exp( t /  h )
hc 

 h o
mc 
 mo

m
obtain  h and  m to give best fit to I Na 
g Na  g Na m h
3
obtain h (V ) from prepulse data
at rest mo  0
and with strong depolariza tion h  1
g Na  g [1  exp( t /  m )] exp( t /  h )
'
Na
where g
'
Na
3
 g Na m h
3
 o
obtain m  from
3
'
g Na
Then  h  h /  h ,
and  h  (1  h ) /  h
Then  m  m /  m ,
and  m  (1  m ) /  m
hoo
• Determined with prepulse experiments
Rate constants for gate h
• Derived from onset or offset of gNa upon
DV
Rate constants for gate m
• Derived from onset or offset of gNa upon
DV
Summary of equilibrium states
and time constants for HH gates
HH model equations
I  Cm dV  g K n4 (V  VK )  g Na m3h(V  VNa )  gl (V  Vl )
dt
dh
  h (1  h)   h h
dt
dm
  m (1  m)   m m
dt
dn
  n (1  n)   n n
dt
- All s and s are dependent on voltage but not time
- Calculate I from sum of leak, Na, K
- Can calculate dV/dt, and approximate V1 =V(t+Dt)
HH fit to expermentally
determined gNa
Voltage clamp currents are
reproduced by simulations
…as are action potentials
Evolution of channel gates
during action potential
Modern view of voltage gated
ion channels
Markov model of states &
transitions
• Allosteric model of Taddese & Bean
– Only 2 voltage dependent rates
Allosteric model results
• Reproduces transient & sustained current
Generality of model
• Many ion channels described in different
neuronal systems
• Each has unique
–
–
–
–
Equilibrium V activation range
Equilibrium V inactivation range
Kinetics of activation and inactivation
Reversal potential
• These contribute to modification of spike firing in
different V and f domains