Hodgkin-Huxley Model

This formula is used to calculate the membrane potential assuming some initial state. The calculation is based on Sodium ion flow, Potassium ion flow and leakage ion flow (which is a nice way of saying all the insignificant ions which cross the membrane.) If there is a mother of computational neuroscience this is it. It resulted in an Noble Prize for the authors. (Hodgkin, A. L. and Huxley, A. F. (1952) "A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve" Journal of Physiology 117: 500-544)

The Main formula

               3                   4
          I = m  h G  (E - E  ) + n  G (E - E ) + G (E - E )
                    Na      Na        K      K     L      L

The Variable Definitions

The parameter names in bold are fixed variables.

I : the total ionic current across the membrane
m : the probability that 1 of the 3 required activation particles has contributed to the activation of the Na gate (m^3 : the probability that all 3 activation particles have produced an open channel)
h : the probability that the 1 inactivation particle has not caused the Na gate to close
G_Na : Maximum possible Sodium Conductance (about 120 mOhms^-1/cm2)
E : total membrane potential (about -60 mV)
E_Na : Na membrane potential (about 55 mV)
n : the probability that 1 of 4 activation particles has influenced the state of the K gate.
G_K : Maximum possible Potassium Conductance (about 36 mOhms^-1/cm2)
E_K : K membrane potential (about -72 mV)
G_L : Maximum possible Leakage Conductance (about .3 mOhms^-1/cm2)
E_L : Leakage membrane potential (about -50 mV)

Parameters and Values
Parameter	Value
G_Na	120 mOhms^-1/cm2
G_K	36 mOhms^-1/cm2
G_L	.3 mOhms^-1/cm2
E	-60 mV
E_Na	55 mV
E_K	-72 mV
E_L	-50 mV

The Activation Particle probability(m) for Sodium

                                  m    - m
         dm                        inf
        ---- = a (1 - m) - b m = ----------
         dt     m           m       T
                                     m

                                         a
                 1                        m
        T  = ---------         m    = ---------
         m    a  + b            inf    a  + b
               m    m                   m    m

The Variable Definitions

m : the probability that 1 of the 3 required activation particles has contributed to the activation of the Na gate (m^3 : the probability that all 3 activation particles have produced an open channel)
t : time in msec
a_m : rate constant for particle not activating a gate
b_m : rate constant for particle activating a gate
m_inf : the steady state value of m
T_m : the time constant of m

The Empirical Formula for the workings of the Sodium gate activation (m)

(from Hodgkin and Huxley, 1952)

                       .1 (25 + E)
          a (E) = ----------------------
           m            -(25 + E)
                   exp(-----------) - 1
                            10


                              E
          b (E) = 4 exp(-----------)
           m                 18

The Variable Definitions

E : total membrane voltage

a_m : rate constant for particle not activating a gate

b_m : rate constant for particle activating a gate

The InActivation Particle probability(h) for Sodium

                                  h    - h
         dh                        inf
        ---- = a (1 - h) - b h = ----------
         dt     h           h       T
                                     h

                                         a
                 1                        h
        T  = ---------         h    = ---------
         h    a  + b            inf    a  + b
               h    h                   h    h

The Variable Definitions

h : the probability that the 1 inactivation particle has not caused the Na gate to close
t : time in msec
a_h : rate constant for particle inactivating a gate
b_h : rate constant for particle not inactivating a gate
h_inf : the steady state value of h
T_h : the time constant of h

The Empirical Formula for the workings of the Sodium gate inactivation (h)

(from Hodgkin and Huxley, 1952)

                                E
          a (E) = .07 exp( ----------- )
           h                   20



                              1
          b (E) = (------------------------)
           h               (30 + E)
                    exp( ----------- ) + 1
                              10

The Variable Definitions

E : total membrane voltage

a_h : rate constant for particle inactivating a gate

b_h : rate constant for particle not inactivating a gate

The Activation Particle probability(n) for Potassium

                                  n    - n
         dn                        inf
        ---- = a (1 - n) - b n = ----------
         dt     n           n       T
                                     n

                                         a
                 1                        n
        T  = ---------         n    = ---------
         n    a  + b            inf    a  + b
               n    n                   n    n

The Variable Definitions

n : the probability that 1 of 4 activation particles has influenced the state of the K gate.
t : time in msec
a_n : rate constant for particle not activating a gate
b_n : rate constant for particle activating a gate
n_inf : the steady state value of n
T_n : the time constant of n

The Empirical Formula for the workings of the Potassium gate activation (n)

(from Hodgkin and Huxley, 1952)

                       .01 (10 + E)
          a (E) = ----------------------
           n            -(10 + E)
                   exp(-----------) - 1
                            10


                                E
          b (E) = .125 exp(-----------)
           n                    80

The Variable Definitions

E : total membrane voltage

a_n : rate constant for particle not activating a gate

b_n : rate constant for particle activating a gate

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