Hodgkin-Huxley model

The Hodgkin-Huxley model is the most famous model for the simulation of neurons . It was developed in 1952 by Alan Lloyd Hodgkin and Andrew Fielding Huxley , originally to describe the formation of action potentials in the giant axon of the squid. Both scientists and Sir John Carew Eccles received the Nobel Prize in Physiology or Medicine in 1963 "for their discoveries about the ion mechanism that takes place in the peripheral and central areas of the nerve cell membrane during excitation and inhibition".

The model simulates neurons close to the biological conditions, e.g. B. individual action potentials (spikes) are modeled. This model is also often referred to as the basic model , electrical model or cable model. The Hodgkin-Huxley model is, however, poorly suited for modeling large networks because it is complex. Instead, a more imprecise, simpler model is chosen, for example the FitzHugh-Nagumo model .

Modeling by a circuit

Electrical schematic for the Hodgkin-Huxley model

In the Hodgkin-Huxley model, the voltage-dependent membrane resistances for the ion currents within a membrane are modeled with memristors . The electrical membrane capacitance C is simulated by a capacitor . The total membrane current is the sum of the potassium , sodium and leakage currents I _K , I _Na and I _L and the external current I (e.g. a current applied by an external electrode) that charges the membrane. The membrane potential U then obeys the differential equation

${\displaystyle C\cdot {\frac {\mathrm {d} U}{\mathrm {d} t}}=I-I_{K}-I_{Na}-I_{L}}$.

The currents I _K , I _Na and I _L result from the difference between the membrane voltage and the respective equilibrium potential U _x and the associated conductivity g _x :

${\displaystyle I_{K}=g_{K}\cdot (U-U_{K})}$,

${\displaystyle I_{Na}=g_{Na}\cdot (U-U_{Na})}$ and

${\displaystyle I_{L}=g_{L}\cdot (U-U_{L})}$.

The values U _K = -77mV, U _Na = 50mV and U _L = -54mV were found as absolute values ​​for the equilibrium potentials for squid neurons.

By definition, an external current (e.g. through an electrode) is defined as a positive inward current , while a membrane current is considered a positive outward current . This explains the choice of signs .

The conductivities g _K and g _Na are time-dependent and therefore responsible for the creation of an action potential, while the leakage conductivity g _L remains constant, since the resistance of the membrane can be assumed to be unchangeable.

Gating variables

Hodgkin and Huxley postulated gating variables (English: gate: gate), which simulate the (probabilistic) dynamics of the ion channels. These variables indicate the proportion of channels that are currently open. Experimental findings led the researchers to infer three gating variables, which they named n , m, and h . Together with the maximum conductivity for the type of ion per unit area of ​​the membrane, the gating variables describe the current flowing through the unit area of ​​the membrane: ${\displaystyle {\hat {g}}_{x}}$ ${\displaystyle x}$

${\displaystyle I_{K}={\hat {g}}_{K}\cdot n^{4}\cdot (U-U_{K})}$ and

${\displaystyle I_{Na}={\hat {g}}_{Na}\cdot m^{3}\cdot h\cdot (U-U_{Na})}$

The three gating variables are subject to the following dynamics:

${\displaystyle {\frac {\mathrm {d} n}{\mathrm {d} t}}=\alpha _{n}(U)\cdot (1-n)-\beta _{n}(U)\cdot n}$ ,

${\displaystyle {\frac {\mathrm {d} m}{\mathrm {d} t}}=\alpha _{m}(U)\cdot (1-m)-\beta _{m}(U)\cdot m}$ ,

${\displaystyle {\frac {\mathrm {d} h}{\mathrm {d} t}}=\alpha _{h}(U)\cdot (1-h)-\beta _{h}(U)\cdot h}$ .

The functions are each exponential functions, the parameters of which were in turn adjusted through experiments. The resulting coupled, nonlinear differential equation system cannot generally be solved analytically, but only through numerical approximation methods. ${\displaystyle \alpha _{n}(U),\alpha _{m}(U),\alpha _{h}(U),\beta _{n}(U),\beta _{m}(U),\beta _{h}(U)}$

The postulated gating variables later turned out to be a real structural property of the voltage-dependent ion channels. The sodium channel has three different states, only one of which lets sodium ions through. It is subject to an activation, which is described by the gating variable m , and a time-shifted inactivation, which is described by h . The channel can be in three states: closed (inactive), closed (active), open (active).

literature

• AL Hodgkin, AF Huxley: A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve . In: The Journal of Physiology . tape 117 , 1952, pp. 500-544 , PMID 12991237 (English).

Web links

• Simulation of the Hodgkin-Huxley model in Python