Download The Hodgkin-Huxley Neuron Model

Topic 3
Polymers and Neurons
Lecture 4
The Hodgkin-Huxley Neuron Model
Neuron Structure and Function
Neurons are nerve cells with a cell body or soma which contains the nucleus and protein manufacturing
apparatus, an axon which propagates action potentials and distributes them to other neurons, and a system
of dendrites which collect signals from other neurons.
The axon is essentially a coaxial conducting cable with an ionic conducting medium inside, surrounded by
the cell membrane which is a leaky insulator, and immersed in the extrcellular ionic conducting fluid. It has
similar electrical properties to an undersea communications cable.
In a living neuron, energy from ATP molecules is continually used to power ion pumps which maintain a
resting potential difference of approximately −70 mV across the cell membrane.
When the axon is subjected to a small depolarizing excitation, it responds in a linear fashion to dissipate
the signal and quickly revert to its resting state.
When the axon is subjected to a large depolarizing excitation, it responds in a nonlinear fashion by generating
a large voltage spike or action potential which propagates down the axon at constant speed without changing
its shape.
Neural computation in the brain is based largely on spike trains propagating through a neural network
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consisting of billions of neurons each connected to tens of thousands of other neurons.
Ion Channels
Living cells maintain a potential gradient across their membranes using Ion-channel pumps powered by
energy stored in ATP molecules. The Na-K-Pump maintains a resting potential difference of approximately
−70 mV across the membrane of a nerve axon.
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Lecture 4
The figure shows the Crystal structure of the sodium-potassium pump which contains three polymer subunits.
The trans-membrane potential is determined by the concentration gradient of ions, according to the
Goldman-Hodgkin-Katz voltage equation.
Hodgkin-Huxley Equations
In the final paper of a series of 5 articles, A.L Hodgkin and A.F. Huxley, ”A quantitative description of
membrane current and its application to conduction and excitation in nerve”, J. Physiol. 117(4), 500-544
(1952) proposed a model to describe the generation and propagation of an action potential in a neuron.
In this series of experimental, theoretical and computational studies, they measured the membrane properties
of the giant axon of the common squid Loligo not to be confused with the Giant squid, deduced a set of
theoretical equations, and showed numerically how they explained the propagation of action potentials.
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The I − V characterisitics of a small patch of axon membrane are determined by the following equations:
dV
4
3
I = CM
+ ḡ Kn V − V K + ḡ Nam h V − V Na + ḡl (V − Vl )
dt
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dn
= αn(1 − n) − βnn
dt
dm
= αm(1 − m) − βmm
dt
dh
= αh(1 − h) − βhh
dt
0.01(V + 10)
+10 −1
exp V 10
0.1(V + 25)
+25 αm =
exp V 10
−1
V
αh = 0.07 exp
20
αn =
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V
βn = 0.125 exp
80
V
βm = 4 exp
18
1
V +30 βh =
exp 10 + 1
The values of the physical parameters in these equations were determined by their experiments, and are
summarized in Table 3 in their article:
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The Membrane Action Potential and Propagated Action Potential
Hodgkin and Huxley solve these equations numerically under two different experimental conditions.
1. Constant Uniform Membrane Potential
The potential is held constant and uniform over the whole length of the axon. This is done by inserting a
wire axially through the length of the axon and holding it at a constant potential. There is no current along
the cylinder axis. The net membrane current must therefore always be zero, except during a stimulus. The
stimulus is taken to be a short shock at t = 0. The equation
dV
+ ḡ Kn4 V − V K + ḡ Nam3h V − V Na + ḡl (V − Vl )
dt
is solved with I = 0 and the initial conditions that V = V0 and m, n and h have their steady state resting
values, at t = 0.
I = CM
2. Propagated Action Potential
An axon at rest in a living organism is excited at its junction with the cell body. The excitation generates a
spike, which propagates down the length of the axon. To model this propagated action potential, the axon
is represented by segments of Hodgkin-Huxley circuit elements connected in series by longitudinal resistors:
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The continuum limit of an infinite number of circuit elements results in a partial differential equation
a ∂ 2V
∂V
4
3
= CM
+ ḡ Kn V − V K + ḡ Nam h V − V Na + ḡl (V − Vl )
2R2 ∂x2
∂t
see Eq. (29) in Hodgkin-Huxley, where x measures longitudinal distance along the axon, and R2 is the
specific resistance of the axoplasm (cytosol). This form of partial differential equation is called the Telegrapher’s equation. For a derivation and further information, see Wikipedia Cable theory and Scholarpedia
Neuronal cable theory.
Hodgkin and Huxley suggest solving this partial differential equation in the steady state approximation.
Assuming the spike propagates like a soliton without changing its shape, V (x, t) as a function of x at any
fixed time has the same functional form as V (x, t) as a function of t at any fixed position x. Thus
∂ 2V
1 ∂ 2V
= 2 2
∂x2
θ ∂t
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where θ is the speed of the spike. Assuming the circuit constants and conductances do not depend on x
results in an ordinary differential equation
a d2V
dV
4
3
= CM
+ ḡ Kn V − V K + ḡ Nam h V − V Na + ḡl (V − Vl )
2R2θ2 dt2
dt
Because θ is not known in advance, they guess a value of θ and solve this equation starting from the resting
state at the foot of the action potential. They then find that V tends to either +∞ or −∞ if the guess is
either too small or too large. The correct value of θ results in V tending to zero when the action potential
is over. This value can be found using a root-finding algorithm.
The partial differential equations can be solved without the assumption of soliton-like behavior, see Wikipedia
Hodgkin-Huxley model: Mathematical properties and the existence of the stable propagating solutions can
be proven rigorously.
Solving the Hodgkin-Huxley Model Equations
hodgkin-huxley.py
import math
import sys
from tools.odeint import RK4_adaptive_step
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# Membrane constants from Table 3
C_M = 1.0
V_Na = -115
V_K = +12
V_l = -10.613
g_Na = 120
g_K = 36
g_l = 0.3
#
#
#
#
#
#
#
membrane capacitance per unit area
sodium Nernst potential
potassium Nernst potential
leakage potential
sodium conductance
potassium conductance
leakage conductance
# Voltage-dependent rate constants (constant in time)
def alpha_n(V):
return 0.01 * (V + 10) / (math.exp((V + 10) / 10) - 1)
def beta_n(V):
return 0.125 * math.exp(V / 80)
def alpha_m(V):
return 0.1 * (V + 25)
/ (math.exp((V + 25) / 10) - 1)
def beta_m(V):
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return 4 * math.exp(V / 18)
def alpha_h(V):
return 0.07 * math.exp(V / 20)
def beta_h(V):
return 1 / (math.exp((V + 30) / 10) + 1)
# Membrane current as function of time
def I(t):
# In a voltage clamp experiment I = 0, see page 522 of H-H article
return 0
# For propagated action potential see Eqs. (30,31) in the article
# Hodgkin-Huxley equations
def HH_equations(Vnmht):
# returns flow vector given extended solution vector [V, n, m, h, t]
V = Vnmht[0]
n = Vnmht[1]
m = Vnmht[2]
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h = Vnmht[3]
t = Vnmht[4]
flow = [0] * 5
flow[0] = ( I(t) - g_K
g_l * (V flow[1] = alpha_n(V) *
flow[2] = alpha_m(V) *
flow[3] = alpha_h(V) *
flow[4] = 1
return flow
Polymers and Neurons
Lecture 4
* n**4 * (V - V_K) - g_Na * m**3 * h * (V - V_Na) V_l) ) / C_M
(1 - n) - beta_n(V) * n
(1 - m) - beta_m(V) * m
(1 - h) - beta_h(V) * h
print(" Hodgkin-Huxley Fig. 12")
# resting state is defined by V = 0, dn/dt = dm/dt = dh/dt = 0
# calculate the resting conductances
n_0 = alpha_n(0) / (alpha_n(0) + beta_n(0))
m_0 = alpha_m(0) / (alpha_m(0) + beta_m(0))
h_0 = alpha_h(0) / (alpha_h(0) + beta_h(0))
V_0 = -90
print(" Initial depolarization V(0) =", V_0, "mV")
t = 0
Vnmht = [ V_0, n_0, m_0, h_0, t ]
t_max = 6
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dt = 0.01
dt_min, dt_max = [dt, dt]
print(" Integrating using RK4 with adaptive step size dt =", dt)
print(" t
V(t)
n(t)
m(t)
h(t)")
print(" ----------------------------------------------------------------")
skip_steps = 10
step = 0
file = open("hodgkin-huxley.data", "w")
while t < t_max + dt:
if step % skip_steps == 0:
print(" ", t, Vnmht[0], Vnmht[1], Vnmht[2], Vnmht[3])
data = str(t)
for i in range(4):
data += ’\t’ + str(Vnmht[i])
file.write(data + ’\n’)
dt = RK4_adaptive_step(Vnmht, dt, HH_equations)
dt_min = min(dt_min, dt)
dt_max = max(dt_max, dt)
t = Vnmht[4]
step += 1
file.close()
print(" min, max adaptive dt =", dt_min, dt_max)
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print(" Data in file hodgkin-huxley.data")
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