Hodgin & Huxley • The problem: Explain action potentials • The preparation:

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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: – V o – V i – I i (injected current, measured with I-mon) • Advantages – Space clamp – axial wires used – – Can effectively eliminate I c – V is fixed – Used to isolate time dependent changes in I

Voltage clamp currents in Original presentation: - Vm relative to rest -referenced to inside of cell -amplitude & polarity appropriate for necessary charging of membrane

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Modern convention:

Isolation of the “outward current”

• Sigmoid onset • Noninactivating • Exponential offset g K(t)

Model of g K

g

K 

g k n

4

n c

 

n n o

d

n

 

n

( 1 

n

)  

n n

d

t

With

dn

/

dt

 0 , i.e.

at steady state,

n

 

n

 

n



n

If

n n

 

n



n o

when

t

 (

n

 

n o

 0 : ) 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 and 

n



n

 

n

 ( 1  /

n

  )

n

, / 

n

Equilibrium n(V), n oo • Similar to a Boltzmann distribution

Rate constants for gate n • Derived from onset or offset of g K upon D V 

n



n

 

n

 ( 1  / 

n

,

n

 ) / 

n

gK fitted to HH equation • Reasonable fit to onset, offset & steady state

Isolate

i Na

by algebraic • Appears Ohmic subtraction • Sigmoidal onset • Increase in g Na is reversible • g(V) is independent of i sign

Current flow through p Na Ohmic • Open channel I/V curve • Instantaneous conductance is

g Na kinetics • Both activation and inactivation speed up with depolarization

Model of g Na

h c m c

     

h h m



m h o m o g

Na  at rest

g Na m

3

h m o

 0 and with strong depolariza

g

Na  tion

h

  1

g

'

Na

[ 1  exp( 

t

/ 

m

)] 3 exp( 

t

/ 

h

) where

g

'

Na



g Na m

3 

h o m



m

  (

m

 

m o

) exp( 

t

/ 

m

)

h



h

  (

h

 

h o

) exp( 

t

/ 

h

) obtain  h and  m to give best fit to I Na  obtain

h

 (

V

) from prepulse data obtain m  from 3 '

g Na

Then 

h



h

 / 

h

, and 

h

Then 

m

and 

m

  ( 1  

m



h

 /  ) /

m

 , ( 1 

m

 ) /

h



m

h oo • Determined with prepulse experiments

Rate constants for gate h • Derived from onset or offset of g Na D V upon

Rate constants for gate m • Derived from onset or offset of g Na D V upon

Summary of equilibrium states and time constants for HH gates

HH model equations

I



C m

d

V

d

t



g K n

4 (

V



V K

) 

g Na m

3

h

(

V



V Na

) 

g l

(

V



V l

) d

h

 

h

( 1 

h

)  

h h

d

t

d

m

 

m

( 1 

m

)  

m m

d

t

d

n

 

n

( 1 

n

)  

n n

d

t

- 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 V 1 =V(t+ D t)

HH fit to expermentally determined g Na

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