Hodgkin–Huxley model

Voltage-gated ion channels

Using a series of voltage clamp experiments and by varying extracellular sodium and potassium concentrations, Hodgkin and Huxley developed a model in which the properties of an excitable cell are described by a set of four ordinary differential equations.^[1] Together with the equation for the total current mentioned above, these are:

${\displaystyle I=C_{m}{\frac {{\mathrm {d} }V_{m}}{{\mathrm {d} }t}}+{\bar {g}}_{\text{K}}n^{4}(V_{m}-V_{K})+{\bar {g}}_{\text{Na}}m^{3}h(V_{m}-V_{Na})+{\bar {g}}_{l}(V_{m}-V_{l}),}$

${\displaystyle {\frac {dn}{dt}}=\alpha _{n}(V_{m})(1-n)-\beta _{n}(V_{m})n}$

${\displaystyle {\frac {dm}{dt}}=\alpha _{m}(V_{m})(1-m)-\beta _{m}(V_{m})m}$

${\displaystyle {\frac {dh}{dt}}=\alpha _{h}(V_{m})(1-h)-\beta _{h}(V_{m})h}$

where I is the current per unit area and ${\displaystyle \alpha _{i}}$ and ${\displaystyle \beta _{i}}$ are rate constants for the i-th ion channel, which depend on voltage but not time. ${\displaystyle {\bar {g}}_{n}}$ is the maximal value of the conductance. n, m, and h are dimensionless probabilities between 0 and 1 that are associated with potassium channel subunit activation, sodium channel subunit activation, and sodium channel subunit inactivation, respectively. For instance, given that potassium channels in squid giant axon are made up of four subunits which all need to be in the open state for the channel to allow the passage of potassium ions, the n needs to be raised to the fourth power. For ${\displaystyle p=(n,m,h)}$, ${\displaystyle \alpha _{p}}$ and ${\displaystyle \beta _{p}}$ take the form

${\displaystyle \alpha _{p}(V_{m})=p_{\infty }(V_{m})/\tau _{p}}$

${\displaystyle \beta _{p}(V_{m})=(1-p_{\infty }(V_{m}))/\tau _{p}.}$

${\displaystyle p_{\infty }}$ and ${\displaystyle (1-p_{\infty })}$ are the steady state values for activation and inactivation, respectively, and are usually represented by Boltzmann equations as functions of ${\displaystyle V_{m}}$. In the original paper by Hodgkin and Huxley,^[1] the functions ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are given by

${\displaystyle {\begin{array}{lll}\alpha _{n}(V_{m})={\frac {0.01(10-V)}{\exp {\big (}{\frac {10-V}{10}}{\big )}-1}}&\alpha _{m}(V_{m})={\frac {0.1(25-V)}{\exp {\big (}{\frac {25-V}{10}}{\big )}-1}}&\alpha _{h}(V_{m})=0.07\exp {\bigg (}-{\frac {V}{20}}{\bigg )}\\\beta _{n}(V_{m})=0.125\exp {\bigg (}-{\frac {V}{80}}{\bigg )}&\beta _{m}(V_{m})=4\exp {\bigg (}-{\frac {V}{18}}{\bigg )}&\beta _{h}(V_{m})={\frac {1}{\exp {\big (}{\frac {30-V}{10}}{\big )}+1}}\end{array}}}$

where ${\displaystyle V=V_{rest}-V_{m}}$ denotes the negative depolarization in mV.

In many current software programs ^[2] Hodgkin–Huxley type models generalize ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ to

${\displaystyle {\frac {A_{p}(V_{m}-B_{p})}{\exp {\big (}{\frac {V_{m}-B_{p}}{C_{p}}}{\big )}-D_{p}}}}$

In order to characterize voltage-gated channels, the equations can be fitted to voltage clamp data. For a derivation of the Hodgkin–Huxley equations under voltage-clamp, see.^[3] Briefly, when the membrane potential is held at a constant value (i.e., with a voltage clamp), for each value of the membrane potential the nonlinear gating equations reduce to equations of the form:

${\displaystyle m(t)=m_{0}-[(m_{0}-m_{\infty })(1-e^{-t/\tau _{m}})]\,}$

${\displaystyle h(t)=h_{0}-[(h_{0}-h_{\infty })(1-e^{-t/\tau _{h}})]\,}$

${\displaystyle n(t)=n_{0}-[(n_{0}-n_{\infty })(1-e^{-t/\tau _{n}})]\,}$

Thus, for every value of membrane potential ${\displaystyle V_{m}}$ the sodium and potassium currents can be described by

${\displaystyle I_{\mathrm {Na} }(t)={\bar {g}}_{\mathrm {Na} }m(V_{m})^{3}h(V_{m})(V_{m}-E_{\mathrm {Na} }),}$

${\displaystyle I_{\mathrm {K} }(t)={\bar {g}}_{\mathrm {K} }n(V_{m})^{4}(V_{m}-E_{\mathrm {K} }).}$

In order to arrive at the complete solution for a propagated action potential, one must write the current term I on the left-hand side of the first differential equation in terms of V, so that the equation becomes an equation for voltage alone. The relation between I and V can be derived from cable theory and is given by

${\displaystyle I={\frac {a}{2R}}{\frac {\partial ^{2}V}{\partial x^{2}}},}$

where a is the radius of the axon, R is the specific resistance of the axoplasm, and x is the position along the nerve fiber. Substitution of this expression for I transforms the original set of equations into a set of partial differential equations, because the voltage becomes a function of both x and t.

The Levenberg–Marquardt algorithm is often used to fit these equations to voltage-clamp data.^[4]

While the original experiments involved only sodium and potassium channels, the Hodgkin–Huxley model can also be extended to account for other species of ion channels.

Leak channels

Leak channels account for the natural permeability of the membrane to ions and take the form of the equation for voltage-gated channels, where the conductance ${\displaystyle g_{leak}}$ is a constant. Thus, the leak current due to passive leak ion channels in the Hodgkin-Huxley formalism is ${\displaystyle I_{l}=g_{leak}(V-V_{leak})}$.

Pumps and exchangers

The membrane potential depends upon the maintenance of ionic concentration gradients across it. The maintenance of these concentration gradients requires active transport of ionic species. The sodium-potassium and sodium-calcium exchangers are the best known of these. Some of the basic properties of the Na/Ca exchanger have already been well-established: the stoichiometry of exchange is 3 Na⁺: 1 Ca²⁺ and the exchanger is electrogenic and voltage-sensitive. The Na/K exchanger has also been described in detail, with a 3 Na⁺: 2 K⁺ stoichiometry.^[5][6]

Center manifold

Because there are four state variables, visualizing the path in phase space can be difficult. Usually two variables are chosen, voltage ${\displaystyle V_{m}(t)}$ and the potassium gating variable ${\displaystyle n(t)}$, allowing one to visualize the limit cycle. However, one must be careful because this is an ad-hoc method of visualizing the 4-dimensional system. This does not prove the existence of the limit cycle.

A better projection can be constructed from a careful analysis of the Jacobian of the system, evaluated at the equilibrium point. Specifically, the eigenvalues of the Jacobian are indicative of the center manifold's existence. Likewise, the eigenvectors of the Jacobian reveal the center manifold's orientation. The Hodgkin–Huxley model has two negative eigenvalues and two complex eigenvalues with slightly positive real parts. The eigenvectors associated with the two negative eigenvalues will reduce to zero as time t increases. The remaining two complex eigenvectors define the center manifold. In other words, the 4-dimensional system collapses onto a 2-dimensional plane. Any solution starting off the center manifold will decay towards the center manifold. Furthermore, the limit cycle is contained on the center manifold.

Bifurcations

If the injected current ${\displaystyle I}$ were used as a bifurcation parameter, then the Hodgkin–Huxley model undergoes a Hopf bifurcation. As with most neuronal models, increasing the injected current will increase the firing rate of the neuron. One consequence of the Hopf bifurcation is that there is a minimum firing rate. This means that either the neuron is not firing at all (corresponding to zero frequency), or firing at the minimum firing rate. Because of the all-or-none principle, there is no smooth increase in action potential amplitude, but rather there is a sudden "jump" in amplitude. The resulting transition is known as a canard.