Hodgkin-Huxley Neuron Model

Although integrate-and-fire neuron models show key features of neuronal spiking patterns, they are not conductance-based models. A conductance-based model would allow us to look at or predict changes in membrane voltage in response to the different types of conductances (there are many different types of ion channels present in different densities in the neuronal membrane). In 1952, Alan Lloyd Hodgkin and Andrew Huxley developed a model that was conductance based. This general type of model is widely used even today. They described the neuronal membrane by the circuit below.

The lipid bilayer is an excellent capacitor and has a capacitance (c_m) of about 10 fF/μm^2. In the original Hodgkin-Huxley model, there were only three conductances: voltage-dependent sodium current, voltage-dependent potassium current, and leak current. These are each represented in the circuit diagram as I_{Na}, I_K, and I_L, respectively. The sodium and potassium conductances vary with the voltage, but the leak conductance does not. They are all described in the following equations, which also include an added input current (I_{app}):

 

c_m\frac{dV}{dt}=I_{Na}+I_K+I_L+I_{app}

or

c_m\frac{dV}{dt}=\bar{G}_{Na}m^3h(E_{Na}-V)+\bar{G}_Kn^4(E_K-V)+G_L(E_L-V)+I_{app}

\frac{dm}{dt}=\frac{m_\infty-m}{\tau_m}

         

\frac{dh}{dt}=\frac{h_\infty-h}{\tau_h}

         

\frac{dn}{dt}=\frac{n_\infty-n}{\tau_n}

These equations make up the basis of the Hodgkin-Huxley model. The top two equations are equivalent, both representing each of the currents in the same order. E_{Na}, E_K, and E_L are the reversal potentials for the sodium, potassium, and leak currents, respectively, and are found using the Nernst Equation. \bar{G}_{Na} and \bar{G}_K are the maximal conductances of their respective ions, and G_L is the leak conductance, which remains constant the entire time. Since the actual conductance at any given point in time varies with the voltage, we add in m, h, and n as gating variables. These each range between 0 and 1. Too see how these gating variables change depending on voltage, we must add a few more equations:

x_{\infty}=\alpha_x/(\alpha_x+\beta_x)

         and          

\tau_x=1/(\alpha_x+\beta_x)

,

where x=m, h, or n.

\alpha_m=9.6401\exp(0.0578V)          \beta_m=0.1081\exp(-0.0556V)

\alpha_h=0.0027\exp(-0.0500V)          \beta_h=7.2634\exp(0.0768V)

\alpha_n=1.1709\exp(0.0461V)          \beta_n=0.0555\exp(-0.0125V)

Thus, \alpha_x and \beta_x vary with voltage. Using these equations, we have a flexible, biologically relevant model of neuronal membrane voltage changes over time. Additional conductances, or ion channels, can be included as long as their maximal conductances, gating variables, and reversal potentials have been characterized. Synaptic conductances and noise can also be included. I used a MATLAB's ODE solver with the above equations and applied step currents (all starting at t=0) to produce the voltage traces shown below. If0 is the current step required to produce repeated spiking. Happy modeling!

P.S. Do you have a favorite software package for neuronal modeling?

Posted March 11th, 2011 in Neuroscience, Science.