Cable theory

Classical theory cable tries the electric currents and the potential differences along passive nerve fibers ( axons ), in particular along the dendrites to describe using mathematical models. The nerve fibers are simplistically viewed as a series of similarly constructed cylindrical segments. The wall of each of said segments is formed by the lipid bilayer of the axon membrane, whose electrical properties can be described as a parallel circuit of an electric resistor and a capacitor with the capacity. The capacity of the nerve fiber is due to the electrostatic forces, which are effective on the very thin lipid bilayer of the plasma membrane due to unequal charge distribution in the extra - and intracellular space ( see Figure 2). In the longitudinal direction of the nerve fiber is composed by the resistance that the cytosol of the movement of electrically charged particles opposes the longitudinal resistance.

History

The origins of cable theory date back to the 50s of the 19th century, when William Thomson (later known as Lord Kelvin) began to develop mathematical models about the signal drop in telegraphic submarine cables. These models had similarity with the partial differential equations, which were used by Fourier to describe the heat conduction in solids.

In 1870 Hermann was the first who tried ' to develop a model of axonal electrotonus, where he was based on the analogies for heat conduction. However, it was not until JL Hoorweg in 1898 discovered the similarity with Kelvin 's undersea cables. As a result, there were Hermann and Cremer, who developed the cable theory for nerve fibers independently of each other at the beginning of the 20th century. Further mathematical theories of conduction in nerve fibers, which were based on the cable theory, were of Cole and Hodgkin (1920-1930), Offner et al. (1940 ) and Rushton developed.

Experimental evidence for the importance of cable theory in the description of real nerve fibers were supplied in the 30s by the work of Cole, Curtis, Hodgkin, Bernard Katz, Rushton, Tasaki and others. Two important articles from this period are Davis and Lorente de No, or Hodgkin and Rushton.

Through the development of techniques for measuring the electrical activity of individual nerve fibers in the 50's the cable theory became increasingly important in the analysis of intracellular microelectrode derived currents and voltages as well as in the investigation of the electrical properties of neuronal dendrites. Scientists like Coombs, Eccles, Fatt, Frank, and Other Fuertos now oriented to the design of new experiments to acquiring new knowledge of the functioning of neurons at the cable theory.

Later, the cable theory allowed with their mathematical formulas ever more sophisticated neuron models that have been studied by scientists as Rall, Redman, Rinzel, Idan Segev, Tuckwell, Bell, Poznanski and Ianella.

Several important approaches for extending the cable theory have recently led to the use of ion channels and endogenous structures, in order to investigate the effects of varying the distribution of synaptic inputs over the dendritic surface.

Derivation of the cable equation

The above introduced and sizes are measured as a function of the fiber length. Therefore, the unit [ OMEGA.m ] and has Farad per meter ( [F / m] ). In contrast, the specific resistance of the membrane and the specific membrane capacitance can be measured as a function of the membrane surface, and having the unit [ Ω m2] or [ F/m2 ]. Is the fiber radius R, which is simplified assumed constant, known and therefore of the fiber scope, and so can be calculated as follows:

The equations are understandable when one realizes that the membrane surface is proportional to the fiber scope: The membrane resistance decreases with increasing fiber scope, because the charged particles is now a larger area for the passage available. Accordingly, the membrane capacity is increasing, because of the larger circumference of the fiber increases, the surface of the capacitor formed by the plasma membrane.

Similarly, the resistivity (unit: [ OMEGA.m ] ) allows the calculation of the cytoplasm of the intracellular longitudinal resistance of the fiber with the unit [ Ω / m]:

In order to understand the derivation of the cable equation nerve fiber described in the introduction is initially further simplified: it is assumed in the following that it has a charged particle absolutely impermeable plasma membrane, so that is infinite and therefore, no loss of charge occurs outside. Similarly, the membrane capacitance was null (). A current is injected x = 0 under these assumptions, at position in the nerve fiber would run away unchanged along the interior of the fiber. With increasing distance from the injection of the current location can be calculated by means of Ohm's law, the change of the voltage:

If you let x Δ go to zero, we obtain the following partial differential equation:

Or

If one includes now the fact that it is not infinite, again in the considerations one, this is as if you make holes in a garden hose. All the contains more holes in the hose, the more water will be lost to the outside and the less water will reach a certain point of the hose. Similarly, a part of current flowing in the longitudinal direction of the current is lost through the plasma membrane of the nerve fiber.

If the current which is lost through the membrane per fiber section, then the sum over all currents that are lost along n sections, too. The current change in the cytoplasm after traveling the route can therefore be written as

Or

Taking into account that the capacity is not zero, can be expressed by means of another formula. The capacitance produces a charge flow towards the membrane, which is usually referred to as a displacement current, and is written in the following as in the cytoplasm. This current flow is only as long as the membrane voltage has not reached its final value. can then be expressed as:

Wherein the potential change describes a function of the time. The current flowing across the membrane is calculated according to the Ohm's law as follows:

Because more is that, the result for:

In this case indicates the change of the current flowing in the longitudinal direction of flow per fiber section.

Substituting equation (6 ) into ( 11) we obtain the first version of the cable equation, the following partial differential equation of second order:

By conversion of equation (12) (see below ), one obtains two major variables, namely, the membrane constant of the longitudinal and the membrane time constant, which will be described in more detail in the following sections.

Membrane length constant

The membrane constant with the longitudinal unit is a parameter indicating how much a current spread along the inside of the nerve fibers and thus influence the voltage along that route. The bigger the more spreads from the stream. From the membrane resistance and the series resistance of the longitudinal constant is calculated as follows:

This formula is appropriate because the larger the membrane resistance, the more current will remain in the cytosol, in order to migrate along the nerve fiber, leading to an increase in the membrane length constant. The same applies to a reduction in the series resistance, because it is easier for the carriers to move along the nerve fiber. Solving equation (12), one comes to the following equation is valid for a steady state, for example, when the time t goes to infinity:

Here is the potential difference across the plasma membrane at the site ( site of the injection current ), " e" is the Euler's number, and the potential difference in the distance "x" from the location of injection of current. Is, then:

And

Is the potential difference measured at the distance of, one obtains

It follows that more and 36.8 percent is from. The membrane length constant is therefore precisely the distance in meters at which the potential difference has dropped to 36.8 percent from.

Membrane time constant

Neurobiologists are often interested in knowing how quickly the membrane potential of a nerve fiber in response to a change in the current, which is injected into the cytosol, changes. This velocity is described by the so-called. " Membrane time constant" that specifies the time in seconds after which the amplitude of the potential difference has fallen to 36.8 % of the initial value. The membrane time constant is thus a direct measure of the speed of the potential change, which can be calculated as follows:

Cable equation with membrane longitudinal and time constant

Multiplying equation (12) on both sides yields:

If we replace by and by now you get the most well -known form of the cable equation:

Comments

Introduction it is noted that the cable theory describes the conditions of passive nerve fibers. Passive means, in this case, the independence of the membrane resistance of membrane potential. Recent studies with dendritic membranes have shown, however, that many of them are equipped with voltage-dependent ion channels and the membrane resistance is therefore very well depend on the membrane potential. Consequently, an adaptation of the classical cable theory is required to account for this fact.