HH Neuron

:opt no-lint

First Cut At HH Neuron

See below for plot and table data.

Doesn't look quite right but I'm not totally sure. One or more of the myriad constants could be wrong. I thought it should go back to about -65mv.

The code, graph and table output are shown below.

• The code itself is dead simple .. It's all in the constnts and the difference method of solving the 'nested' DEs.

• Code Breakdown

  • The setup
    • There are the constants .
    • The channel conductances ..
    • The membrane currents ..
    • The derivatives
  • The Main Functions
    • step .. simply increments everything one time step
    • simulate .. recursive routing that does the work. Nothing magic.
    • showResults .. Generates a table of times and action potentials.
    • graphResults .. Displays the results in graph form.
• Note that each function has (not required I put it there for clarity) a type definition
  in the form "foo :: Float -> Float" .. This means that foo is a function that takes a float and returns a float (the last entry is the return value).

--module HodgkinHuxley where

import Numeric.LinearAlgebra
import Numeric.LinearAlgebra.Data
import qualified Data.ByteString as B

import Text.Printf
:l GPlot2D
import GPlot2D

-- Constants
c_m = 1.0

g_Na = 120.0

g_K = 36.0

g_L = 0.3

e_Na = 115.0

e_K = -12.0

e_L = 10.566

-- Channel conductances

alpha_m, beta_m, alpha_n, beta_n :: Float -> Float

alpha_m t = 0.1 * (25.0 - t) / (exp (2.5 - 0.1 * t) - 1.0)

beta_m t = 4.0 * exp (- t / 18.0)

alpha_h t = 0.07 * exp (- t / 20.0)

beta_h t = 1.0 / (exp (3.0 - 0.1 * t) + 1.0)

alpha_n t = 0.01 * (10.0 - t) / (exp (1.0 - 0.1 * t) - 1.0)

beta_n t = 0.125 * exp (- t / 80.0)

-- Membrane current

i_Na :: Float -> Float -> Float -> Float
i_Na v m h = g_Na * (m ** 3) * h * (v - e_Na)

i_K :: Float -> Float -> Float
i_K v n = g_K * (n ** 4) * (v - e_K)

i_L :: Float -> Float
i_L v = g_L * (v - e_L)

i_ion :: Float -> Float -> Float -> Float -> Float
i_ion v m h n = i_Na v m h + i_K v n + i_L v

-- Derivatives
dvdt :: Float -> Float -> Float -> Float -> Float
dvdt v m h n = (-1.0 / c_m) * i_ion v m h n

dmdt :: Float -> Float -> Float
dmdt v m = alpha_m v * (1.0 - m) - beta_m v * m

dhdt :: Float -> Float -> Float
dhdt v h = alpha_h v * (1.0 - h) - beta_h v * h

dndt :: Float -> Float -> Float
dndt v n = alpha_n v * (1.0 - n) - beta_n v * n

-- Time stepping
step ::
  Float -> -- time
  Float -> -- voltage
  Float -> -- m
  Float -> -- h
  Float -> -- n
  Float -> -- delta t
  (Float, Float, Float, Float, Float)
step t v m h n dt = (t + dt, v + dvdt v m h n * dt, 
                     m + dmdt v m * dt, h + dhdt v h * dt,
                         n + dndt v n * dt)

-- Simulation
simulate ::
  Float -> -- time
  Float -> -- voltage
  Float -> -- m
  Float -> -- h
  Float -> -- n
  Float -> -- end time
  Float -> -- delta t
  ([Float], [Float], [Float], [Float], [Float])
simulate t_0 v_0 m_0 h_0 n_0 t_final dt =
  let (t, v, m, h, n) = step t_0 v_0 m_0 h_0 n_0 dt
  in   if t >= t_final
          then ([], [], [], [], []) --ts, vs, ms, hs, ns)
          else
             let (t', v', m', h', n') = simulate t v m h n t_final dt
             in (t : t', v : v', m : m', h : h', n : n')

-- Show the output
showResults :: [Float] -> [Float] -> IO ()
showResults ts vs = do
  putStrLn "Action potential simulation using the Hodgkin-Huxley model:"
  putStrLn "\n    Time(ms)    Membrane Potential (mV)"
  mapM_ (\(t, v) -> (printf "%10.2f       %10.2f\n" t v)) $ zip ts vs

graphResults :: [Float] -> [Float] -> IO (B.ByteString)
graphResults ts vs = do
    writeFile "hh_solution01.dat" 
      (concatMap (\(a,b) -> show a ++ "  " ++ show b ++ "\n") $ zip ts vs)
    runGNUPlot "hh_solution01.dat" "Hodgkins-Huxley Neuronal Model"
                   "Time (ms)" "Activation Voltage" "hh_solution01.png"

-- Main function
--main = do
t_0 = 0.0

v_0 = -65.0

m_0 = 0.05

--
h_0 = 0.6

n_0 = 0.32

t_final = 20.0

dt = 0.01

(ts, vs, ms, hs, ns) = simulate t_0 v_0 m_0 h_0 n_0 t_final dt

----------------------
-- Display or Graph --
----------------------

-- Graph the results
graphResults ts vs

-- Display the results 
showResults ts vs

Action potential simulation using the Hodgkin-Huxley model:

    Time(ms)    Membrane Potential (mV)
      0.01           -64.56
      0.02           -64.14
      0.03           -63.72
      0.04           -63.31
      0.05           -62.90
      0.06           -62.50
      0.07           -62.10
      0.08           -61.71
      0.09           -61.32
      0.10           -60.94
      0.11           -60.56
      0.12           -60.18
      0.13           -59.81
      0.14           -59.44
      0.15           -59.08
      0.16           -58.72
      0.17           -58.36
      0.18           -58.01
      0.19           -57.66
      0.20           -57.31
      0.21           -56.97
      0.22           -56.63
      0.23           -56.30
      0.24           -55.97
      0.25           -55.64
      0.26           -55.31
      0.27           -54.99
      0.28           -54.67
      0.29           -54.36
      0.30           -54.05
      0.31           -53.74
      0.32           -53.43
      0.33           -53.13
      0.34           -52.82
      0.35           -52.53
      0.36           -52.23
      0.37           -51.94
      0.38           -51.65
      0.39           -51.36
      0.40           -51.07
      0.41           -50.79
      0.42           -50.51
      0.43           -50.23
      0.44           -49.96
      0.45           -49.68
      0.46           -49.41
      0.47           -49.14
      0.48           -48.88
      0.49           -48.61
      0.50           -48.35
      0.51           -48.09
      0.52           -47.83
      0.53           -47.58
      0.54           -47.32
      0.55           -47.07
      0.56           -46.82
      0.57           -46.57
      0.58           -46.33
      0.59           -46.09
      0.60           -45.84
      0.61           -45.60
      0.62           -45.37
      0.63           -45.13
      0.64           -44.89
      0.65           -44.66
      0.66           -44.43
      0.67           -44.20
      0.68           -43.97
      0.69           -43.75
      0.70           -43.52
      0.71           -43.30
      0.72           -43.08
      0.73           -42.86
      0.74           -42.64
      0.75           -42.42
      0.76           -42.21
      0.77           -41.99
      0.78           -41.78
      0.79           -41.57
      0.80           -41.36
      0.81           -41.16
      0.82           -40.95
      0.83           -40.74
      0.84           -40.54
      0.85           -40.34
      0.86           -40.14
      0.87           -39.94
      0.88           -39.74
      0.89           -39.54
      0.90           -39.35
      0.91           -39.15
      0.92           -38.96
      0.93           -38.77
      0.94           -38.58
      0.95           -38.39
      0.96           -38.20
      0.97           -38.01
      0.98           -37.82
      0.99           -37.64
      1.00           -37.46
      1.01           -37.27
      1.02           -37.09
      1.03           -36.91
      1.04           -36.73
      1.05           -36.55
      1.06           -36.38
      1.07           -36.20
      1.08           -36.03
      1.09           -35.85
      1.10           -35.68
      1.11           -35.51
      1.12           -35.34
      1.13           -35.17
      1.14           -35.00
      1.15           -34.83
      1.16           -34.66
      1.17           -34.49
      1.18           -34.33
      1.19           -34.16
      1.20           -34.00
      1.21           -33.84
      1.22           -33.68
      1.23           -33.52
      1.24           -33.36
      1.25           -33.20
      1.26           -33.04
      1.27           -32.88
      1.28           -32.73
      1.29           -32.57
      1.30           -32.42
      1.31           -32.26
      1.32           -32.11
      1.33           -31.96
      1.34           -31.80
      1.35           -31.65
      1.36           -31.50
      1.37           -31.35
      1.38           -31.21
      1.39           -31.06
      1.40           -30.91
      1.41           -30.77
      1.42           -30.62
      1.43           -30.48
      1.44           -30.33
      1.45           -30.19
      1.46           -30.05
      1.47           -29.90
      1.48           -29.76
      1.49           -29.62
      1.50           -29.48
      1.51           -29.34
      1.52           -29.21
      1.53           -29.07
      1.54           -28.93
      1.55           -28.80
      1.56           -28.66
      1.57           -28.53
      1.58           -28.39
      1.59           -28.26
      1.60           -28.12
      1.61           -27.99
      1.62           -27.86
      1.63           -27.73
      1.64           -27.60
      1.65           -27.47
      1.66           -27.34
      1.67           -27.21
      1.68           -27.08
      1.69           -26.96
      1.70           -26.83
      1.71           -26.70
      1.72           -26.58
      1.73           -26.45
      1.74           -26.33
      1.75           -26.20
      1.76           -26.08
      1.77           -25.96
      1.78           -25.83
      1.79           -25.71
      1.80           -25.59
      1.81           -25.47
      1.82           -25.35
      1.83           -25.23
      1.84           -25.11
      1.85           -24.99
      1.86           -24.87
      1.87           -24.76
      1.88           -24.64
      1.89           -24.52
      1.90           -24.41
      1.91           -24.29
      1.92           -24.18
      1.93           -24.06
      1.94           -23.95
      1.95           -23.83
      1.96           -23.72
      1.97           -23.61
      1.98           -23.50
      1.99           -23.38
      2.00           -23.27
      2.01           -23.16
      2.02           -23.05
      2.03           -22.94
      2.04           -22.83
      2.05           -22.72
      2.06           -22.62
      2.07           -22.51
      2.08           -22.40
      2.09           -22.29
      2.10           -22.19
      2.11           -22.08
      2.12           -21.98
      2.13           -21.87
      2.14           -21.77
      2.15           -21.66
      2.16           -21.56
      2.17           -21.45
      2.18           -21.35
      2.19           -21.25
      2.20           -21.15
      2.21           -21.04
      2.22           -20.94
      2.23           -20.84
      2.24           -20.74
      2.25           -20.64
      2.26           -20.54
      2.27           -20.44
      2.28           -20.34
      2.29           -20.24
      2.30           -20.14
      2.31           -20.05
      2.32           -19.95
      2.33           -19.85
      2.34           -19.75
      2.35           -19.66
      2.36           -19.56
      2.37           -19.47
      2.38           -19.37
      2.39           -19.28
      2.40           -19.18
      2.41           -19.09
      2.42           -18.99
      2.43           -18.90
      2.44           -18.81
      2.45           -18.72
      2.46           -18.62
      2.47           -18.53
      2.48           -18.44
      2.49           -18.35
      2.50           -18.26
      2.51           -18.17
      2.52           -18.08
      2.53           -17.99
      2.54           -17.90
      2.55           -17.81
      2.56           -17.72
      2.57           -17.63
      2.58           -17.54
      2.59           -17.45
      2.60           -17.37
      2.61           -17.28
      2.62           -17.19
      2.63           -17.11
      2.64           -17.02
      2.65           -16.93
      2.66           -16.85
      2.67           -16.76
      2.68           -16.68
      2.69           -16.59
      2.70           -16.51
      2.71           -16.42
      2.72           -16.34
      2.73           -16.26
      2.74           -16.17
      2.75           -16.09
      2.76           -16.01
      2.77           -15.93
      2.78           -15.84
      2.79           -15.76
      2.80           -15.68
      2.81           -15.60
      2.82           -15.52
      2.83           -15.44
      2.84           -15.36
      2.85           -15.28
      2.86           -15.20
      2.87           -15.12
      2.88           -15.04
      2.89           -14.96
      2.90           -14.89
      2.91           -14.81
      2.92           -14.73
      2.93           -14.65
      2.94           -14.57
      2.95           -14.50
      2.96           -14.42
      2.97           -14.34
      2.98           -14.27
      2.99           -14.19
      3.00           -14.12
      3.01           -14.04
      3.02           -13.97
      3.03           -13.89
      3.04           -13.82
      3.05           -13.74
      3.06           -13.67
      3.07           -13.60
      3.08           -13.52
      3.09           -13.45
      3.10           -13.38
      3.11           -13.30
      3.12           -13.23
      3.13           -13.16
      3.14           -13.09
      3.15           -13.01
      3.16           -12.94
      3.17           -12.87
      3.18           -12.80
      3.19           -12.73
      3.20           -12.66
      3.21           -12.59
      3.22           -12.52
      3.23           -12.45
      3.24           -12.38
      3.25           -12.31
      3.26           -12.24
      3.27           -12.17
      3.28           -12.11
      3.29           -12.04
      3.30           -11.97
      3.31           -11.90
      3.32           -11.83
      3.33           -11.77
      3.34           -11.70
      3.35           -11.63
      3.36           -11.57
      3.37           -11.50
      3.38           -11.43
      3.39           -11.37
      3.40           -11.30
      3.41           -11.24
      3.42           -11.17
      3.43           -11.11
      3.44           -11.04
      3.45           -10.98
      3.46           -10.91
      3.47           -10.85
      3.48           -10.78
      3.49           -10.72
      3.50           -10.66
      3.51           -10.59
      3.52           -10.53
      3.53           -10.47
      3.54           -10.41
      3.55           -10.34
      3.56           -10.28
      3.57           -10.22
      3.58           -10.16
      3.59           -10.10
      3.60           -10.04
      3.61            -9.97
      3.62            -9.91
      3.63            -9.85
      3.64            -9.79
      3.65            -9.73
      3.66            -9.67
      3.67            -9.61
      3.68            -9.55
      3.69            -9.49
      3.70            -9.43
      3.71            -9.37
      3.72            -9.32
      3.73            -9.26
      3.74            -9.20
      3.75            -9.14
      3.76            -9.08
      3.77            -9.02
      3.78            -8.97
      3.79            -8.91
      3.80            -8.85
      3.81            -8.79
      3.82            -8.74
      3.83            -8.68
      3.84            -8.62
      3.85            -8.57
      3.86            -8.51
      3.87            -8.45
      3.88            -8.40
      3.89            -8.34
      3.90            -8.29
      3.91            -8.23
      3.92            -8.18
      3.93            -8.12
      3.94            -8.07
      3.95            -8.01
      3.96            -7.96
      3.97            -7.90
      3.98            -7.85
      3.99            -7.80
      4.00            -7.74
      4.01            -7.69
      4.02            -7.64
      4.03            -7.58
      4.04            -7.53
      4.05            -7.48
      4.06            -7.42
      4.07            -7.37
      4.08            -7.32
      4.09            -7.27
      4.10            -7.22
      4.11            -7.16
      4.12            -7.11
      4.13            -7.06
      4.14            -7.01
      4.15            -6.96
      4.16            -6.91
      4.17            -6.86
      4.18            -6.81
      4.19            -6.75
      4.20            -6.70
      4.21            -6.65
      4.22            -6.60
      4.23            -6.55
      4.24            -6.50
      4.25            -6.45
      4.26            -6.41
      4.27            -6.36
      4.28            -6.31
      4.29            -6.26
      4.30            -6.21
      4.31            -6.16
      4.32            -6.11
      4.33            -6.06
      4.34            -6.02
      4.35            -5.97
      4.36            -5.92
      4.37            -5.87
      4.38            -5.82
      4.39            -5.78
      4.40            -5.73
      4.41            -5.68
      4.42            -5.63
      4.43            -5.59
      4.44            -5.54
      4.45            -5.49
      4.46            -5.45
      4.47            -5.40
      4.48            -5.36
      4.49            -5.31
      4.50            -5.26
      4.51            -5.22
      4.52            -5.17
      4.53            -5.13
      4.54            -5.08
      4.55            -5.04
      4.56            -4.99
      4.57            -4.94
      4.58            -4.90
      4.59            -4.86
      4.60            -4.81
      4.61            -4.77
      4.62            -4.72
      4.63            -4.68
      4.64            -4.63
      4.65            -4.59
      4.66            -4.55
      4.67            -4.50
      4.68            -4.46
      4.69            -4.41
      4.70            -4.37
      4.71            -4.33
      4.72            -4.28
      4.73            -4.24
      4.74            -4.20
      4.75            -4.16
      4.76            -4.11
      4.77            -4.07
      4.78            -4.03
      4.79            -3.98
      4.80            -3.94
      4.81            -3.90
      4.82            -3.86
      4.83            -3.82
      4.84            -3.77
      4.85            -3.73
      4.86            -3.69
      4.87            -3.65
      4.88            -3.61
      4.89            -3.57
      4.90            -3.53
      4.91            -3.48
      4.92            -3.44
      4.93            -3.40
      4.94            -3.36
      4.95            -3.32
      4.96            -3.28
      4.97            -3.24
      4.98            -3.20
      4.99            -3.16
      5.00            -3.12
      5.01            -3.08
      5.02            -3.04
      5.03            -3.00
      5.04            -2.96
      5.05            -2.92
      5.06            -2.88
      5.07            -2.84
      5.08            -2.80
      5.09            -2.76
      5.10            -2.72
      5.11            -2.68
      5.12            -2.64
      5.13            -2.60
      5.14            -2.57
      5.15            -2.53
      5.16            -2.49
      5.17            -2.45
      5.18            -2.41
      5.19            -2.37
      5.20            -2.33
      5.21            -2.29
      5.22            -2.26
      5.23            -2.22
      5.24            -2.18
      5.25            -2.14
      5.26            -2.10
      5.27            -2.07
      5.28            -2.03
      5.29            -1.99
      5.30            -1.95
      5.31            -1.91
      5.32            -1.88
      5.33            -1.84
      5.34            -1.80
      5.35            -1.76
      5.36            -1.73
      5.37            -1.69
      5.38            -1.65
      5.39            -1.61
      5.40            -1.58
      5.41            -1.54
      5.42            -1.50
      5.43            -1.47
      5.44            -1.43
      5.45            -1.39
      5.46            -1.36
      5.47            -1.32
      5.48            -1.28
      5.49            -1.25
      5.50            -1.21
      5.51            -1.17
      5.52            -1.14
      5.53            -1.10
      5.54            -1.06
      5.55            -1.03
      5.56            -0.99
      5.57            -0.95
      5.58            -0.92
      5.59            -0.88
      5.60            -0.85
      5.61            -0.81
      5.62            -0.77
      5.63            -0.74
      5.64            -0.70
      5.65            -0.67
      5.66            -0.63
      5.67            -0.59
      5.68            -0.56
      5.69            -0.52
      5.70            -0.49
      5.71            -0.45
      5.72            -0.41
      5.73            -0.38
      5.74            -0.34
      5.75            -0.31
      5.76            -0.27
      5.77            -0.24
      5.78            -0.20
      5.79            -0.16
      5.80            -0.13
      5.81            -0.09
      5.82            -0.06
      5.83            -0.02
      5.84             0.01
      5.85             0.05
      5.86             0.08
      5.87             0.12
      5.88             0.16
      5.89             0.19
      5.90             0.23
      5.91             0.26
      5.92             0.30
      5.93             0.33
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import Data.List -- Simulation parameters v_0 = -64.0 t_0 = 0.0 m_0 = 0.05 h_0 = 0.6 n_0 = 0.32 t_final = 100.0 dt = 0.001 -- Time constant functions alpha_m t = 0.1 * (v t + 40.0) / (1.0 - exp (-0.1 * (v t + 40.0))) beta_m t = 4.0 * exp (-0.0556 * (v t + 65.0)) alpha_h t = 0.07 * exp (-0.05 * (v t + 65.0)) beta_h t = 1.0 / (1.0 + exp (-0.1 * (v t + 35.0))) alpha_n t = 0.01 * (v t + 55.0) / (1.0 - exp (-0.1 * (v t + 55.0))) beta_n t = 0.125 * exp (-0.0125 * (v t + 65.0)) -- Steady-state functions m_inf t = alpha_m t / (alpha_m t + beta_m t) h_inf t = alpha_h t / (alpha_h t + beta_h t) n_inf t = alpha_n t / (alpha_n t + beta_n t) -- Voltage-dependent functions i_Na t v m h = 120.0 * m^3 * h * (v - 115.0) i_K t v n = 36.0 * n^4 * (v + 12.0) i_L t v = 0.3 * (v + 65.0) -- ODE functions v' t v m h n = (1.0 / cm) * (i_Stim t - i_Na t v m h - i_K t v n - i_L t v) m' t v m = alpha_m t * (1.0 - m) - beta_m t * m h' t v h = alpha_h t * (1.0 - h) - beta_h t * h n' t v n = alpha_n t * (1.0 - n) - beta_n t * n -- Stimulus current i_Stim t | t >= 10.0 && t <= 10.1 = 40.0 | otherwise = 0.0 -- Specific membrane capacitance cm = 1.0 -- Voltage-gated ion channels --v t = v t --m t = m t --h t = h t --n t = n t -- Time step for the simulation step t v m h n dt = (t + dt, v + dt * v', m + dt * m', h + dt * h', n + dt * n') -- Simulation function simulate :: Double -> Double -> Double -> Double -> Double -> Double -> Double -> ([Double], [Double], [Double], [Double], [Double]) simulate t v m h n t_final dt | t > t_final = ([], [], [], [], []) | otherwise = (t:ts, v:vs, m:ms, h:hs, n:ns) where t_next = t + dt v_next = v + dt * v_inf(t) * (v - v_rest) / tau_v(t) m_next = m + dt * alpha_m(v) * (1 - m) - dt * beta_m(v) * m h_next = h + dt * alpha_h(v) * (1 - h) - dt * beta_h(v) * h n_next = n + dt * alpha_n(v) * (1 - n) - dt * beta_n(v) * n (ts, vs, ms, hs, ns) = simulate t_next v_next m_next h_next n_next t_final dt showResult :: (Double, Double) -> String showResult (t, v) = (show t) ++ "\t\t" ++ (show v) -- Main function --main = do (time, v, m, h, n) = simulate t_0 v_0 m_0 h_0 n_0 t_final dt mapM_ (putStrLn . showResult) (zip time v)