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 PHY 411-506 Computational Physics 2 1 Monday, March 3
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. PHY 411-506 Computational Physics 2 2 Monday, March 3
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. PHY 411-506 Computational Physics 2 3 Monday, March 3
The I V characterisitics of a small patch of axon membrane are determined by the following equations: I = C M dv dt + ḡ K n4 ( V V K ) + ḡ Na m 3 h ( V V Na ) + ḡl (V V l ) PHY 411-506 Computational Physics 2 4 Monday, March 3
dn dt = α n(1 n) β n n dm dt = α m(1 m) β m m dh dt = α h(1 h) β h h 0.01(V + 10) α n = exp [ ] V +10 10 1 0.1(V + 25) α m = exp [ ] V +25 10 1 ( ) V α h = 0.07 exp 20 PHY 411-506 Computational Physics 2 5 Monday, March 3
( ) V β n = 0.125 exp 80 ( ) V β m = 4 exp 18 1 β h = exp [ ] V +30 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: PHY 411-506 Computational Physics 2 6 Monday, March 3
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 I = C M dv dt + ḡ K n4 ( V V K ) + ḡ Na m 3 h ( V V Na ) + ḡl (V V l ) is solved with I = 0 and the initial conditions that V = V 0 and m, n and h have their steady state resting values, at t = 0. 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: PHY 411-506 Computational Physics 2 7 Monday, March 3
The continuum limit of an infinite number of circuit elements results in a partial differential equation a 2 V 2R 2 x = C V 2 M t + ḡ ( ) K n4 V V K + ḡ Na m 3 h ( ) V V Na + ḡl (V V l ) see Eq. (29) in Hodgkin-Huxley, where x measures longitudinal distance along the axon, and R 2 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 2 V x = 1 2 V 2 θ 2 t 2 PHY 411-506 Computational Physics 2 8 Monday, March 3
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 d 2 V 2R 2 θ 2 dt 2 = C M dv dt + ḡ K n4 ( V V K ) + ḡ Na m 3 h ( V V Na ) + ḡl (V V l ) 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 import math import sys from tools.odeint import RK4_adaptive_step hodgkin-huxley.py PHY 411-506 Computational Physics 2 9 Monday, March 3
# 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): PHY 411-506 Computational Physics 2 10 Monday, March 3
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] PHY 411-506 Computational Physics 2 11 Monday, March 3
h = Vnmht[3] t = Vnmht[4] flow = [0] * 5 flow[0] = ( I(t) - g_k * n**4 * (V - V_K) - g_na * m**3 * h * (V - V_Na) - g_l * (V - V_l) ) / C_M flow[1] = alpha_n(v) * (1 - n) - beta_n(v) * n flow[2] = alpha_m(v) * (1 - m) - beta_m(v) * m flow[3] = alpha_h(v) * (1 - h) - beta_h(v) * h flow[4] = 1 return flow 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 PHY 411-506 Computational Physics 2 12 Monday, March 3
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) PHY 411-506 Computational Physics 2 13 Monday, March 3
print(" Data in file hodgkin-huxley.data") PHY 411-506 Computational Physics 2 14 Monday, March 3