6/6/13 Cable theory - Wikipedia, the free encyclopedia

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Figure. 1: Cable theory's simplified view of a

neuronal fiber

Figure. 2: Fiber capacitance

Cable theoryFrom Wikipedia, the free encyclopedia

Classical cable theory uses mathematical models tocalculate the electric current (and accompanying voltage)

along passive [1] neurites, particularly the dendrites thatreceive synaptic inputs at different sites and times. Estimatesare made by modeling dendrites and axons as cylinderscomposed of segments with capacitances andresistances combined in parallel (see Fig. 1). Thecapacitance of a neuronal fiber comes about becauseelectrostatic forces are acting through the very thinphospholipid bilayer (see Figure 2). The resistance in seriesalong the fiber is due to the axoplasm’s significantresistance to movement of electric charge.

Contents

1 History

2 Deriving the cable equation

3 Length constant

4 Time constant5 The cable equation with length and time

constants

6 See also

7 References

8 Notes

History

Cable theory in computational neuroscience has rootsleading back to the 1850s, when Professor WilliamThomson of panvel (later known as Lord Kelvin) begandeveloping mathematical models of signal decay insubmarine (underwater) telegraphic cables. The modelsresembled the partial differential equations used by Fourierto describe heat conduction in a wire.

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The 1870s saw the first attempts by Hermann to model neuronal electrotonic potentials also by focusing onanalogies with heat conduction. However, it was Hoorweg who first discovered the analogies with Kelvin’sundersea cables in 1898 and then Hermann and Cremer who independently developed the cable theory forneuronal fibers in the early 20th century. Further mathematical theories of nerve fiber conduction based on cabletheory were developed by Cole and Hodgkin (1920s–1930s), Offner et al. (1940), and Rushton (1951).

Experimental evidence for the importance of cable theory in modeling the behavior of axons began surfacing in the1930s from work done by Cole, Curtis, Hodgkin, Sir Bernard Katz, Rushton, Tasaki and others. Two key papersfrom this era are those of Davis and Lorente de No (1947) and Hodgkin and Rushton (1946).

The 1950s saw improvements in techniques for measuring the electric activity of individual neurons. Thus cabletheory became important for analyzing data collected from intracellular microelectrode recordings and for analyzingthe electrical properties of neuronal dendrites. Scientists like Coombs, Eccles, Fatt, Frank, Fuortes and others nowrelied heavily on cable theory to obtain functional insights of neurons and for guiding them in the design of newexperiments.

Later, cable theory with its mathematical derivatives allowed ever more sophisticated neuron models to be exploredby workers such as Jack, Rall, Redman, Rinzel, Idan Segev, Tuckwell, Bell, Poznanski(http://romanpoznanski.blogspot.com), and Ianella.

Several important avenues of extending classical cable theory have recently seen the introduction of endogenousstructures (see Poznanski, 2010; Poznanski & Cacha, 2012) in order to analyze the effects of protein polarizationwithin dendrites and different synaptic input distributions over the dendritic surface of a neuron.

Deriving the cable equation

rm and cm, as introduced above, are measured per membrane-length unit (per meter (m)). Thus rm is measured in

ohm-meters (Ω·m) and cm in farads per meter (F/m). This is in contrast to Rm (in Ω·m²) and Cm (in F/m²), which

represent the specific resistance and capacitance respectively of one unit area of membrane (in m2). Thus, if the

radius, a, of the axon is known,[2] then its circumference, 2πa, its rm, and its cm values can be calculated as follows:

(1)

(2)

These relationships make sense intuitively, because the greater the circumference of the axon, the greater the areafor charge to escape through its membrane, and therefore the lower the membrane resistance (dividing Rm by 2πa);

and the more membrane available to store charge (multiplying Cm by 2πa). Similarly, the specific resistance, Rl, of

the axoplasm allows ones to calculate the longitudinal intracellular resistance per unit length, rl, (in Ω·m−1) by the

equation:

(3)

The greater the cross sectional area of the axon, πa², the greater the number of paths for the charge to flow throughits axoplasm, and the lower the axoplasmic resistance.

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To better understand how the cable equation is derived, first simplify the theoretical neuron even further andpretend it has a perfectly sealed membrane (rm=∞) with no loss of current to the outside, and no capacitance (cm =

0). A current injected into the fiber [3] at position x = 0 would move along the inside of the fiber unchanged.Moving away from the point of injection and by using Ohm's law (V = IR) we can calculate the voltage change as:

(4)

Letting Δx go towards zero and having infinitely small increments of x, one can write (4) as:

(5)

or

(6)

Bringing rm back into the picture is like making holes in a garden hose. The more holes, the faster the water will

escape from the hose, and the less water will travel all the way from the beginning of the hose to the end. Similarly,in an axon, some of the current traveling longitudinally through the axoplasm will escape through the membrane.

If im is the current escaping through the membrane per length unit, m, then the total current escaping along y units

must be y·im. Thus, the change of current in the axoplasm, Δil, at distance, Δx, from position x=0 can be written as:

(7)

or, using continuous, infinitesimally small increments:

(8)

can be expressed with yet another formula, by including the capacitance. The capacitance will cause a flow of

charge (a current) towards the membrane on the side of the cytoplasm. This current is usually referred to asdisplacement current (here denoted .) The flow will only take place as long as the membrane's storage capacity

has not been reached. can then be expressed as:

(9)

where is the membrane's capacitance and is the change in voltage over time. The current that passes

the membrane ( ) can be expressed as:

(10)

and because the following equation for can be derived if no additional current is added from an

electrode:

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(11)

where represents the change per unit length of the longitudinal current.

Combining equations (6) and (11) gives a first version of a cable equation:

(12)

which is a second-order partial differential equation (PDE).

By a simple rearrangement of equation (12) (see later) it is possible to make two important terms appear, namelythe length constant (sometimes referred to as the space constant) denoted and the time constant denoted . Thefollowing sections focus on these terms.

Length constant

Main article: Length constant

The length constant, (lambda), is a parameter that indicates how far a current will travel along the inside of anaxon, and thereby influence the voltage along that distance. The larger the value of , the farther the charge willflow. The length constant can be expressed as:

(13)

The larger the membrane resistance, rm, the greater the value of , and the more current will remain inside the

axoplasm to travel longitudinally through the axon. The higher the axoplasmic resistance, , the smaller the value of , the harder it will be for current to travel through the axoplasm, and the shorter the current will be able to travel. It

is possible to solve equation (12) and arrive at the following equation (which is valid in steady-state conditions, i.e.when length approaches infinity):

(14)

Where is the depolarization at (point of current injection), e is the exponential constant (approximate

value 2.71828) and is the voltage at a given distance x from x=0. When then

(15)

and

(16)

which means that when we measure at distance from we get

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(17)

Thus is always 36.8 percent of .

Time constant

Main article: Time constant

Neuroscientists are often interested in knowing how fast the membrane potential, , of an axon changes inresponse to changes in the current injected into the axoplasm. The time constant, , is an index that providesinformation about that value. can be calculated as:

(18)

.

The larger the membrane capacitance, , the more current it takes to charge and discharge a patch of membraneand the longer this process will take. Thus membrane potential (voltage across the membrane) lags behind currentinjections. Response times vary from 1–2 milliseconds in neurons that are processing information that needs hightemporal precision to 100 milliseconds or longer. A typical response time is around 20 milliseconds.

The cable equation with length and time constants

If one multiplies equation (12) by on both sides of the equal sign we get:

(19)

and recognize on the left side and on the right side. The cable equation can now be

written in its perhaps best known form:

(20)

See also

Biological neuron model

AxonDendrite

Bioelectrochemistry

Membrane potential

Nernst-Planck equationSaltatory conduction

Patch clamp

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References

Tuckwell, Henry C. (1988). Introduction to theoretical neurobiology. Cambridge [Cambridgeshire]:

Cambridge University Press. ISBN 978-0521350969.Reeke, G. N.; Poznanski, R. R. (2005). Modeling in the neurosciences : from biological systems to

neuromimetic robotics (2nd ed.). Boca Raton [u.a.]: Taylor & Francis. ISBN 978-0415328685.

de Nó, Rafael Lorente (1947). A study of nerve physiology (http://books.google.com/books?id=VVsRAQAAMAAJ). Studies from the Rockefeller Institute for Medical Research. Reprints (v. 2).

Rockefeller Institute for Medical Research. pp. Part I, 131:1–496; Part II, 132:1–548. OCLC 6217290

(//www.worldcat.org/oclc/6217290).

Hodgkin, A. L.; Rushton, W. A. H. (1946). "The Electrical Constants of a Crustacean Nerve Fibre".Proceedings of the Royal Society B: Biological Sciences 133 (873): 444–479.

doi:10.1098/rspb.1946.0024 (http://dx.doi.org/10.1098%2Frspb.1946.0024). PMID 20281590

(//www.ncbi.nlm.nih.gov/pubmed/20281590).Poznanski, R. R. (2010). "Thermal noise due to surface-charge effects within the Debye layer of endogenous

structures in dendrites". Physical Review E 81 (2): 021902. doi:10.1103/PhysRevE.81.021902

(http://dx.doi.org/10.1103%2FPhysRevE.81.021902). PMID 20365590

(//www.ncbi.nlm.nih.gov/pubmed/20365590).

Poznanski, R. R.; Cacha, L. A. (2012). Intraceullular capacitive effects of polarized proteins in

dendrites. Journal of Integrative Neuroscience. pp. volume 11:417–437.

Notes

Notes:1 ^ Passive here refers to the membrane resistance being voltage-independent. However recent experiments

(Stuart and Sakmann 1994) with dendritic membranes shows that many of these are equipped with voltage

gated ion channels thus making the resistance of the membrane voltage dependent. Consequently there has

been a need to update the classical cable theory to accommodate for the fact that most dendritic membranes

are not passive.

2 ^ Classical cable theory assumes that the fiber has a constant radius along the distance being modeled.

3 ^ Classical cable theory assumes that the inputs (usually injections with a micro device) are currents which

can be summed linearly. This linearity does not hold for changes in synaptic membrane conductance.

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