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.
Parameters and Values
Parameter Value
G_Na120 mOhms^-1/cm2
G_K36 mOhms^-1/cm2
G_L.3 mOhms^-1/cm2
E-60 mV
E_Na55 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

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

    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

    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 Back to Models and Math

    Back to general retina list

    123 visit(s) to this page.

    © 1995 Lance Hahn (lance@retina.anatomy.upenn.edu)