Cable theory

Classical cable theory describes the development of mathematical models that can calculate the flow of electric current (and accompanying voltage) along passive neuronal fibers (neurites) particularly 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 $$c_m\ $$ and resistances $$r_m\ $$ 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 $$r_l\ $$ is due to the cytosol’s significant resistance to movement of electric charge.



History
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
$$r_m\ $$ and $$c_m\ $$ introduced above are measured per fiber-length unit (usually centimeter (cm)). Thus $$r_m\ $$ is measured in ohms times centimeters ($$\Omega \times cm\ $$) and $$c_m\ $$ in micro farads per centimeter ($$\mu F/cm\ $$). This is in contrast to $$R_m (\Omega \times cm^2)\ $$ and $$C_m (\mu F/cm^2)\ $$, which represent the specific resistance and capacitance of the membrane measured within one unit area of membrane $$(cm^2)\ $$. Thus if the radius a of the cable is known and hence its circumference $$2 \pi a\ $$, $$r_m\ $$ and $$c_m\ $$ can be calculated as follows:

$$r_m = \frac{R_m}{2 \pi a\ }$$         (1) $$c_m = C_m 2 \pi a\ $$                 (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 $$2 \pi a\ $$); and the more membrane to store charge (we multiply $$C_m\ $$ by $$2 \pi a\ $$.) In a similar vein, the specific resistance $$R_l\ $$ of the cytoplasm enables the longitudinal intracellular resistance per unit length ($$\Omega/cm\ $$) $$r_l\ $$ to be calculated as: $$r_l = \frac{R_l}{\pi a^2\ }$$          (3) Again a reasonable equation, because the larger the cross sectional area ($$\pi a^2\ $$) 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 ohms law ($$V=I R\ $$) we can calculate the voltage change as: $$\Delta V = -i_l r_l \Delta x\ $$  (4) If we let $$\Delta x\ $$ go towards zero and have infinitely small increments of $$x\ $$ we can write (4) as: $$\frac{\partial V}{\partial x} = -i_l r_l\ $$ (5) or $$\frac{1}{r_l} \frac{\partial V}{\partial x} = -i_l\ $$   (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 (cm), then the total current escaping along y units must be $$y i_m\ $$. Thus the change of current in the cytoplasm $$\Delta i_l\ $$ at distance $$\Delta x\ $$ from position x=0 can be written as: $$\Delta i_l = -i_m \Delta x\ $$   (7) or using continuous infinitesimally small increments: $$\frac{\partial i_l}{\partial x}=-i_m\ $$ (8) $$i_m\ $$ 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 $$i_c\ $$.) The flow will only take place as long as the membrane's storage capacity has not been reached. $$i_c\ $$ can then be expressed as: $$i_c = c_m \frac{\partial V}{\partial t}\ $$  (9) where $$c_m\ $$ is the membrane's capacitance and $$\frac{\partial V}{\partial t}\ $$ is the change in voltage over time. The current that passes the membrane ($$i_r\ $$) can be expressed as:

$$i_r=\frac{V}{r_m}$$ (10) and because $$i_m = i_r + i_c\ $$ the following equation for $$i_m\ $$ can be derived if no additional current is added from an electrode:

$$\frac{\partial i_l}{\partial x}=-i_m=-(\frac{V}{r_m}+c_m \frac{\partial V}{\partial t})$$ (11) where $$\frac{\partial i_l}{\partial x}$$ 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:

$$\frac{1}{r_l} \frac{\partial ^2 V}{\partial x^2}=c_m \frac{\partial V}{\partial t}+\frac{V}{r_m}$$ (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, namely the length constant (sometimes referred to as the space constant) denoted $$\lambda$$ and the time constant denoted $$\tau$$. The following sections focus on these terms.

The length constant
The length constant denoted with the symbol $$\lambda\ $$ (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 $$\lambda\ $$ is, the farther the current will flow. The length constant can be expressed as:

$$\lambda = \sqrt \frac{r_m}{r_l}$$     (13)

This formula makes sense because the larger the membrane resistance ($$r_m\ $$) (resulting in larger $$\lambda\ $$) the more current will remain inside the cytosol to travel longitudinally along the neurite. The higher the cytosol resistance ($$r_l\ $$) (resulting in smaller $$\lambda\ $$) 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:

$$V_x = \frac {V_0}{e^\frac{x}{\lambda}}$$     (14)

Where $$V_0\ $$ is the depolarization at $$x=0\ $$ (point of current injection), e is the exponential constant (approximate value 2.71828) and $$V_x\ $$ is the voltage at a given distance $$x\ $$ from $$x=0\ $$. When $$x=\lambda\ $$ then

$$\frac{x}{\lambda}=1$$    (15)

and

$$V_x = \frac{V_0}{e}$$    (16)

which means that when we measure $$V\ $$ at distance $$\lambda\ $$ from $$x=0\ $$ we get

$$V_\lambda = \frac{V_0}{e} = 0.368 V_0$$ (17)

Thus $$V_\lambda\ $$ is always 36.8 percent of $$V_0\ $$.

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

$$\tau = r_m c_m\ $$     (18)

which seem reasonable because the larger the membrane capacitance ($$c_m\ $$) 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 $$r_m\ $$ on both sides of the equal sign we get:

$$\frac{r_m}{r_l} \frac{\partial ^2 V}{\partial x^2}=c_m r_m \frac{\partial V}{\partial t}+ V$$    (19)

and recognize $$\lambda^2 = \frac{r_m}{r_l}$$ on the left side and $$\tau = c_m r_m\ $$ on the right side. The cable equation can now be written in its perhaps best known form:

$$\lambda^2 \frac{\partial ^2 V}{\partial x^2}=\tau \frac{\partial V}{\partial t}+ V$$     (20)