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Classical cable theory describes the development of mathematical models that can calculate the flow of electric current (and accompanying voltage) along passive dendrites that receive synaptic (see synapse) inputs at different sites and times. Such estimates can be achieved by regarding dendrites and axons as cylinders composed of segments with capacitances and resistances 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 (See Figure 2). The resistances in series along the fiber is due to the cytosol’s significant resistance to movement of electric charge.neuronal fibers (neurites) particularly
Additional recommended knowledge
Cable theory in computational neuroscience has roots leading back to the 1850s, when Professor William Thomson (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 to describe heat conduction in a wire.
The 1870s saw the first attempts by Hermann to model axonal 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 (1920s-1930s), Offner et.al. (1940), and Rushton (1951).
Experimental evidence for the importance of cable theory in modeling real nerve axons began surfacing in the 1930s from work done by Cole, Curtis, Hodgkin, Katz, Rushton, Tasaki and others. Two key papers from 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 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, Christof Koch, Noble, Poggio, Rall, Redman, Rinzel, Idan Segev, Shepherd, Torre and Tsien. One important avenue of research became to analyze the effects of different synaptic input distributions over the dendritic (see dendrite) surface of a neuron.
Deriving the cable equation
and introduced above are measured per fiber-length unit (usually centimeter (cm)). Thus is measured in ohms times centimeters () and in micro farads per centimeter (). This is in contrast to and , which represent the specific resistance and capacitance of the membrane measured within one unit area of membrane . Thus if the radius a of the cable is knownand hence its circumference , and can be calculated as follows:
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 ( is infinite) with no loss of current to the outside, and no capacitance (.) A current injected into the fiberat position x = 0 would move along the inside of the fiber unchanged. Moving away from the point of injection and by using ohms law () we can calculate the voltage change as:
or using continuous infinitesimally small increments:
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:
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, namely the length constant (sometimes referred to as the space constant) denoted λ and the time constant denoted τ. 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:
This formula makes sense because the larger the membrane resistance () (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 (albeit not straightforward) to solve equation (12) and arrive at the following equation:
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 from . When then
which means that when we measure at distance from we get
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:
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:
and recognize on the left side and on the right side. The cable equation can now be written in its perhaps best known form:
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Cable_theory". A list of authors is available in Wikipedia.|