Hodgkin–Huxley model

The Hodgkin -Huxley model is the most famous model for the simulation of neurons. It was founded in 1952 by Alan Lloyd Hodgkin and Andrew Fielding Huxley, originally developed to describe the formation of action potentials in the giant axon of the squid. Both scientists got together with Sir John Carew Eccles 1963 Nobel Prize in Medicine " for their discoveries concerning the ionic mechanism that takes place at the excitation and inhibition in the peripheral and central portions of the nerve cell membrane ".

The model simulates neurons close to the biological realities such as single action potentials ( spikes) are modeled. This model is often referred to as the basic model, model electric or cable model. The Hodgkin -Huxley model is, however, for modeling large networks poorly suited because it is expensive. Instead, then a simpler model is selected, for example, FitzHugh - Nagumo model.

Modeling by a circuit

In the Hodgkin -Huxley model, the voltage-dependent membrane resistances for the ionic currents are modeled within a membrane with memristors. Electrical membrane capacitance C is simulated by a capacitor. The total current is the sum of the membrane, potassium, sodium and leakage currents IK and IL and INa external current I ( e.g., a applicated by an external electrode current), which charges the membrane. The membrane potential U obeys the differential equation

The currents IK, INa and IL arising from the difference between the membrane voltage at the respective equilibrium potential U and the corresponding conductivity g:

As absolute values ​​for the reversal potentials the values ​​UK = - 77mV, 50mV and Una = UL = - 54mV were found for squid neurons.

By definition, an external power (for example, by an electrode ) is defined as positive inward, while a membrane current is regarded as positive outward flow. This explains the choice of the sign.

The conductivities of gK and gNa are time-dependent and thus responsible for the formation of an action potential, while the leakage conductance gL remains constant, since the resistance of the membrane can accept as immutable.

Gatingvariablen

Hodgkin and Huxley postultierten Gatingvariablen (English Gate: Gate), which the ( probabilistic ) dynamics of ion channels replicate. These variables indicate the proportion of currently open channels. Experimental results allowed the researchers to three Gatingvariablen close they n, m and h are called. Together with the maximum conductivity of an ion describes the Gatingvariable just the current flowing through the membrane current:

The three Gatingvariablen are subjected to the following dynamics:

The functions are each exponential functions whose parameters were again adjusted by experiments. The resulting coupled, nonlinear system of differential equations can not be solved analytically but only by numerical approximation methods in general.

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