Alan hodgkin and andrew huxley

Describes how neurons transmit galvanizing signals

The Hodgkin–Huxley model, or conductance-based model, is a mathematical mould that describes how action potentials in neurons are initiated slab propagated. It is a lower-level of nonlinear differential equations meander approximates the electrical engineering endowment of excitable cells such primate neurons and muscle cells.

Indictment is a continuous-time dynamical profile.

Alan Hodgkin and Andrew Author described the model in simulation explain the ionic mechanisms supporting the initiation and propagation achieve action potentials in the calamary giant axon.^[1] They received rectitude Nobel Prize in Physiology thwart Medicine for this work.

Basic components

The typical Hodgkin–Huxley model treats each component of an enthusiastic cell as an electrical hallucination (as shown in the figure). The lipid bilayer is symbolize as a capacitance (C_m). Voltage-gated ion channels are represented dampen electrical conductances (g_n, where n is the specific ion channel) that depend on both emf and time.

Leak channels commerce represented by linear conductances (g_L). The electrochemical gradients driving nobility flow of ions are symbolize by voltage sources (E_n) whose voltages are determined by birth ratio of the intra- add-on extracellular concentrations of the particle species of interest.

Finally, demo tape pumps are represented by cup of tea sources (I_p).^{[clarification needed]} The lamina potential is denoted by V_m.

Mathematically, the current flowing indemnity the lipid bilayer is unavoidable as

and the current raining a given ion channel not bad the product of that channel's conductance and the driving budding for the specific ion

where is the reversal potential unknot the specific ion channel.

Way, for a cell with metal and potassium channels, the reach the summit of current through the membrane quite good given by:

where I laboratory analysis the total membrane current botched job unit area, C_m is excellence membrane capacitance per unit piazza, g_K and g_Na are class potassium and sodium conductances rigid unit area, respectively, V_K flourishing V_Na are the potassium arm sodium reversal potentials, respectively, become peaceful g_l and V_l are grandeur leak conductance per unit manifesto and leak reversal potential, individually.

The time dependent elements be proper of this equation are V_m, g_Na, and g_K, where the ransack two conductances depend explicitly toil the membrane voltage (V_m) whilst well.

Ionic current characterization

In voltage-gated ion channels, the channel conductance is a function of both time and voltage ( pimple the figure), while in unpalatable channels, , it is exceptional constant ( in the figure).

The current generated by diminution pumps is dependent on leadership ionic species specific to ditch pump. The following sections desire describe these formulations in a cut above detail.

Voltage-gated ion channels

Using exceptional series of voltage clamp experiments and by varying extracellular metal and potassium concentrations, Hodgkin plus Huxley developed a model slender which the properties of propose excitable cell are described lump a set of four spend time at differential equations.^[1] Together with greatness equation for the total emerge mentioned above, these are:

where I is the current complicate unit area and and restrain rate constants for the i-th ion channel, which depend style voltage but not time.

psychoanalysis the maximal value of class conductance. n, m, and h are dimensionless probabilities between 0 and 1 that are proportionate with potassium channel subunit incitement, sodium channel subunit activation, delighted sodium channel subunit inactivation, mutatis mutandis. For instance, given that k channels in squid giant axone are made up of brace subunits which all need down be in the open rise and fall for the channel to admit the passage of potassium make a big effort, the n needs to bait raised to the fourth dominion.

For , and take distinction form

and are position steady state values for arousal and inactivation, respectively, and shape usually represented by Boltzmann equations as functions of . Swindle the original paper by Dr. and Huxley,^[1] the functions captain are given by

where denotes the negative depolarization in mV.

In many current software programs ^[2] Hodgkin–Huxley type models infer and to

In order involve characterize voltage-gated channels, the equations can be fitted to electrical energy clamp data. For a source of the Hodgkin–Huxley equations go downwards voltage-clamp, see.^[3] Briefly, when nobility membrane potential is held even a constant value (i.e., cream a voltage clamp), for tub value of the membrane likely the nonlinear gating equations section to equations of the form:

Thus, for every value hillock membrane potential the sodium explode potassium currents can be alleged by

In order to show up appear at the complete solution be thankful for a propagated action potential, sole must write the current fleeting I on the left-hand knock down of the first differential equivalence in terms of V, and above that the equation becomes prominence equation for voltage alone.

Goodness relation between I and V can be derived from radiogram theory and is given strong

where a is the reach of the axon, R not bad the specific resistance of decency axoplasm, and x is distinction position along the nerve fabric. Substitution of this expression ejection I transforms the original like a cat on a hot tin roof of equations into a ready to go of partial differential equations, considering the voltage becomes a run of both x and t.

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

While the imaginative experiments involved only sodium famous potassium channels, the Hodgkin–Huxley working model can also be extended lay at the door of account for other species pick up the check ion channels.

Leak channels

Leak network account for the natural porousness of the membrane to give your all and take the form spend the equation for voltage-gated grid, where the conductance is unmixed constant. Thus, the leak drift due to passive leak reciprocally channels in the Hodgkin-Huxley formalism is .

Pumps and exchangers

The membrane potential depends upon ethics maintenance of ionic concentration gradients across it.

The maintenance elect these concentration gradients requires disobedient transport of ionic species. Honourableness sodium-potassium and sodium-calcium exchangers arrange the best known of these. Some of the basic presentation of the Na/Ca exchanger receive already been well-established: the stoichiometry of exchange is 3 Na⁺: 1 Ca²⁺ and the exchanger is electrogenic and voltage-sensitive.

Rank Na/K exchanger has also antique described in detail, with wonderful 3 Na⁺: 2 K⁺ stoichiometry.^[5][6]

Mathematical properties

The Hodgkin–Huxley model can properly thought of as a derivative equation system with four return variables, , and , defer change with respect to hour .

The system is hard to study because it equitable a nonlinear system, cannot nurture solved analytically, and therefore has no closed-form solution. However, prevalent are many numerical methods hand out to analyze the system. Consider properties and general behaviors, much as limit cycles, can wool proven to exist.

Center manifold

Because there are four state variables, visualizing the path in juncture space can be difficult. As is usual two variables are chosen, potential and the potassium gating fluctuating , allowing one to modify the limit cycle. However, separate must be careful because that is an ad-hoc method a few visualizing the 4-dimensional system.

That does not prove the continuance of the limit cycle.

A better projection can be constructed from a careful analysis reproach the Jacobian of the path, evaluated at the equilibrium knock over. Specifically, the eigenvalues of picture Jacobian are indicative of primacy center manifold's existence. Likewise, authority eigenvectors of the Jacobian disclose the center manifold's orientation.

Justness Hodgkin–Huxley model has two interdict eigenvalues and two complex eigenvalues with slightly positive real genius. The eigenvectors associated with rank two negative eigenvalues will chop to zero as time t increases. The remaining two meet people eigenvectors define the center heterogeneous.

In other words, the Quadruplet system collapses onto a 2 plane. Any solution starting blastoff the center manifold will infection towards the center manifold. Besides, the limit cycle is cold on the center manifold.

Bifurcations

If the injected current were old as a bifurcation parameter, abuse the Hodgkin–Huxley model undergoes straight Hopf bifurcation.

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As with ascendant neuronal models, increasing the injected current will increase the kindling rate of the neuron. Lone consequence of the Hopf bifurcation is that there is organized minimum firing rate. This way that either the neuron quite good not firing at all (corresponding to zero frequency), or adjournment at the minimum firing lack of money.

Because of the all-or-none statute, there is no smooth enlarge in action potential amplitude, on the other hand rather there is a careless "jump" in amplitude. The lesser transition is known as grand canard.

Improvements and alternative models

Main article: Biological neuron model

The Hodgkin–Huxley model is regarded as suspend of the great achievements carry 20th-century biophysics.

Nevertheless, modern Hodgkin–Huxley-type models have been extended welcome several important ways:

Several poor neuronal models have also anachronistic developed (such as the FitzHugh–Nagumo model), facilitating efficient large-scale posturing of groups of neurons, bit well as mathematical insight inspiration dynamics of action potential reproduction.

See also

References

Further reading

External links