Classical cable theory uses mathematical models to calculate the flow of electric current (and accompanying voltage

Voltage

Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

) along passive neuronal fibers (neurites) particularly dendrite

Dendrite

Dendrites are the branched projections of a neuron that act to conduct the electrochemical stimulation received from other neural cells to the cell body, or soma, of the neuron from which the dendrites project...

s that receive synaptic

Synapse

In the nervous system, a synapse is a structure that permits a neuron to pass an electrical or chemical signal to another cell...

 inputs at different sites and times. Estimates are made by modeling dendrites and axons as cylinders composed of segments with capacitance

Capacitance

In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

s and resistance

Electrical resistance

The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

s combined in parallel (See Figure 1). The capacitance of a neuronal fiber comes about because electrostatic forces are acting through the very thin phospholipid bilayer

Lipid bilayer

The lipid bilayer is a thin membrane made of two layers of lipid molecules. These membranes are flat sheets that form a continuous barrier around cells. The cell membrane of almost all living organisms and many viruses are made of a lipid bilayer, as are the membranes surrounding the cell nucleus...

 (See Figure 2). The resistances in series along the fiber is due to the cytosol

Cytosol

The cytosol or intracellular fluid is the liquid found inside cells, that is separated into compartments by membranes. For example, the mitochondrial matrix separates the mitochondrion into compartments....

’s significant resistance to movement of electric charge

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

.

History

Cable theory in computational neuroscience has roots leading back to the 1850s, when Professor William Thomson from panvel(later known as Lord Kelvin) began developing mathematical models of signal decay in submarine (underwater) telegraphic cables. The models resembled the partial differential equations used by Fourier

Fourier

Fourier most commonly refers to Joseph Fourier , French mathematician and physicist, or the mathematics, physics, and engineering terms named in his honor for his work on the concepts underlying them:In mathematics:...

 to describe heat conduction in a wire.

The 1870s saw the first attempts by Hermann to model axon

Axon

An axon is a long, slender projection of a nerve cell, or neuron, that conducts electrical impulses away from the neuron's cell body or soma....

al electrotonus also by focusing on analogies with heat conduction. However it was Hoorweg who first discovered the analogies with Kelvin’s undersea cables in 1898 and then Hermann and Cremer who independently developed the cable theory for neuronal fibers in the early 20th century. Further mathematical theories of nerve fiber conduction based on cable theory were developed by Cole and Hodgkin

Hodgkin

Hodgkin may refer to:People with the surname Hodgkin who are part of an extended Quaker family:*Alan Lloyd Hodgkin , British physiologist and biophysicist*Dorothy Crowfoot Hodgkin , British chemist*Douglas Hodgkin...

 (1920s-1930s), Offner et al. (1940), and Rushton (1951).

Experimental evidence for the importance of cable theory in modeling real nerve axon

Axon

An axon is a long, slender projection of a nerve cell, or neuron, that conducts electrical impulses away from the neuron's cell body or soma....

s began surfacing in the 1930s from work done by Cole, Curtis, Hodgkin

Hodgkin

Hodgkin may refer to:People with the surname Hodgkin who are part of an extended Quaker family:*Alan Lloyd Hodgkin , British physiologist and biophysicist*Dorothy Crowfoot Hodgkin , British chemist*Douglas Hodgkin...

, Sir Bernard Katz, Rushton, Tasaki and others. Two key papers from this era are those of Davis and Lorente de No (1947) and Hodgkin

Hodgkin

Hodgkin may refer to:People with the surname Hodgkin who are part of an extended Quaker family:*Alan Lloyd Hodgkin , British physiologist and biophysicist*Dorothy Crowfoot Hodgkin , British chemist*Douglas Hodgkin...

 and Rushton (1946).

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

Later, cable theory with its mathematical derivatives allowed ever more sophisticated neuron models to be explored by workers such as Jack, Rall

Wilfrid Rall

Wilfrid Rall is a neuroscientist who spent most of his career at the National Institutes of Health. He is considered one of the founders of computational neuroscience, and was a pioneer in establishing the integrative functions of neuronal dendrites...

, Redman, Rinzel, Idan Segev, Tuckwell, Bell, Poznanski, and Ianella.

Several important avenues of extending classical cable theory have recently seen the introduction of ionic channels and endogenous structures (Poznanski, 2010) in order to analyze the effects of different synaptic input distributions over the dendritic (see dendrite

Dendrite

Dendrites are the branched projections of a neuron that act to conduct the electrochemical stimulation received from other neural cells to the cell body, or soma, of the neuron from which the dendrites project...

) surface of a neuron.

Deriving the cable equation

r_m and c_m introduced above are measured per fiber-length unit (per meter (m)). Thus r_m is measured in ohm

Ohm

The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

-meters (Ω·m) and c_m in farad

Farad

The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...

s per meter (F/m). This is in contrast to R_m[Ω·m²] and C_m[F/m²], which represent the specific resistance and capacitance of the membrane measured within one unit area of membrane (m²). Thus if the radius a of the cable is known and hence its circumference 2πa, r_m and c_m can be calculated as follows:

(1)

(2)
This makes sense because the bigger the circumference the larger area for charge to escape through the membrane and the smaller resistance (we divide R_m by 2πa); and the more membrane to store charge (we multiply C_m by 2πa).
In a similar vein, the specific resistance R_l of the cytoplasm enables the longitudinal intracellular resistance per unit length r_l[Ω·m⁻¹] to be calculated as:

(3)
Again a reasonable equation, because the larger the cross sectional area (πa²) the larger the number of paths for the current to flow through the cytoplasm and the less resistance.

To better understand how the cable equation is derived let's first simplify our fiber from above even further and pretend it has a perfectly sealed membrane (r_m is infinite) with no loss of current to the outside, and no capacitance (c_m = 0). A current injected into the fiber 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

Ohm's law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

 (V = IR) we can calculate the voltage change as:
(4)
If we let Δx go towards zero and have infinitely small increments of x we can write (4) as:
(5)

or

(6)
Bringing r_m back into the picture is like making holes in a garden hose. The more holes the more water will escape to the outside, and the less water will reach a certain point of the hose. Similarly in the neuronal fiber some of the current travelling longitudinally along the inside of the fiber will escape through the membrane.

If i_m is the current escaping through the membrane per length unit (m), then the total current escaping along y units must be yi_m. Thus the change of current in the cytoplasm Δi_l 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 (current) towards the membrane on the side of the cytoplasm. This current is usually referred to as displacement 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:

(11)

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

By combining equations (6) and (11) we get a first version of a cable equation:

(12)

which is a second-order partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

(PDE.)

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

The length constant

The length constant denoted with the symbol (lambda) is a parameter that indicates how far a current will spread along the inside of a neurite and thereby influence the voltage along that distance. The larger is, the farther the current will flow. The length constant can be expressed as:

(13)

This formula makes sense because the larger the membrane resistance (r_m) (resulting in larger ) the more current will remain inside the cytosol to travel longitudinally along the neurite. The higher the cytosol resistance () (resulting in smaller ) the harder it will be for current to travel through the cytosol 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 time goes to 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

(17)

Thus is always 36.8 percent of .

The time constant

Neuroscientists are often interested in knowing how fast the membrane potential of a neurite is changing in response to changes in the current injected into the cytosol. The time constant is an index that provides information about exactly that. can be calculated as:

(18)

which seem reasonable because the larger the membrane capacitance () the more current it takes to charge and discharge a patch of membrane and the longer this process will take. Thus membrane potential (voltage across the membrane) lags behind current injections. Response times vary from 1-2 milliseconds in neurons that are processing information that needs high temporal precision to 100 milliseconds or longer. A typical response time is around 20 milliseconds.

The cable equation with length and time constants

If we multiply 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)