Bioelectricity: electrotonic propagation

Principle

If by (moderate) electrical stimulation a constant voltage is applied to the membrane of a nerve or muscle cell there is an exponential decay with a length constant (the lengthy over which the amplitude is diminished to the fraction 1/e), which can amount to some mm. Due to the considerable electrical resistance of the cell's interior r_interior and the high capacitance of the membrane c_m there is a slow spread of the disturbance in time. Such phenomena are named electrotonic. They play an important role in the retina and in the central and autonomous nervous system especially in the dendrites and unmyelinated axons. It is the basis of the integrative action of the nervous system making one still more wonder why and how a simple mathematical model as treated below can have such an (even predictive) success.

The propagation of a disturbance of the rest potential E_rest is electrotonic with constant permeability’s of the membrane for the various ions as long as no action potential (spike) is elicited. Electrotonic changes in membrane potential hold for dendritic, receptor and somatic potentials. They are also found in axons which axon hillock does not generate spikes. Similar potentials are found in synapses (pre- and postsynaptic potentials). All these potentials have a graded nature. During transmission they are filtered in time and place.

For the various dendrites and axons the membrane capacity per unit area and internal resistance are rather the same. Propagation speed is proportional with the square root of the diameter and reciprocal with the square root of membrane resistance. These two properties are highly variable and so speed varies a lot among dendrites and not-spiking axons. The decrement along the dendrite is proportional with the square root of membrane resistance. So, a high resistance means slow but with few losses over distance.

The electrotonic propagation is a passive phenomenon which only depends on the resistive and capacitive properties of the dendrite, axon or soma and is governed by the cable theory of physics. This is in contrast to the propagation of spikes along axons, an active process (see Bioelectricity: action potential).

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Suppose a nerve fiber or a dendrite has an axial membrane resistance r_m and capacitance c_m, both per unit length, and an internal axial, resistance of r_a (the outside resistance is neglected). The voltage difference between in- and outside is maintained by the three ionogenic equilibrium potentials (see Fig. 1 of Bioelectricity). Now we divide the dendrite in equal cylindrical pieces with a length x and we suppose that these pieces have the same properties.The current through the membrane per unit length i_m is the sum of the currents through r_m and c_m:

i_m = c_m dE/dt + E/r_m (1)

The axial current in the inside of the dendrite is proportional to the voltage difference E over a distance x. It is also proportional to the inverse of the resistance over a length x, i.e. 1/(r_ax). This means that the axial current is proportional with the second derivative of the voltage to the distance. Moreover,

this current should be equal to i_m, since there cannot flow a net current in a closed circuit. So

i_m = r_a^─1·d²E/dx². (2)

Combining (1) and (2) yields:

(1/r_a)∂²E/∂x².= c_m ∂E/∂t + E/r_m (3)

By scaling of the variable x with X = x/λ where λ = (r_m/r_a)^0.5 (λ is the length constant) and t by T = t/τ were τ = r_m c_m (τ is the time constant, see Halftime and time constant), the equation becomes:

∂²E/∂X² =∂E/∂T + E (4)

r_m, r_a and c_m can be found from the resistance and capacitance per unit of length, R_m, R_i and C_m. R_m is the specific membrane resistance times unit area, R_i the specific resistance, C_m is the specific capacitance per unit area, and d the diameter. So we obtain:

r_m = pdR_m, r_a = 4R_i/(pd²), c_m = pdC_m. (5a)

And so λ is also equal to:

λ = (dR_m/4R_i)^0.5. (5b)

Another important feature of the propagation of the electrotonic potential is the conduction velocity Θ = ∂X/∂T of its peak potential.

By using the chain rule for ∂²E/∂x².and the definition of θ it can finally be found that:

∂²E/∂x² = θ^─2·∂²E/∂T². (6)

With (4) this yields the important expression:

Θ ^─2·∂²E/∂T².= ∂E/∂T + E. (7)

From the solution of this equation it appears that for the peak of the propagated wave the conduction velocity Θ is:

Θ = ∂X/∂T = 2λ/τ = 0.5C_m^─1(d/(R_m·R_i))^0.5 (8)

For R_i = 36 Ω·cm, C_m = 1μF/cm², R_m = 10⁸ Ω·cm² and d = 0.016 mm, θ is 3.3 mm/s.

From this example it is clear that passive conduction in membranes with a high R_m takes much time. But generally dendritic pathways are short. R_m is strongly dependent on the type of membrane. Membranes without ionic channels have an R_m of the order of 10⁸ Ω·cm². This can become as small as 1000 Ω·cm², dependent on the channel density for the different ions (see Bioelectricity: action potential). For R_m =10⁸ Ω·cm² λ becomes 167 mm, so that 'the peak amplitude of the propagated wave hardly decreases within physiological distance. By applying a current pulse through an intracellular electrode τ can be estimated by measuring the voltage change with a second intracellular electrode (Fig. 1). An estimate of λ can be made by estimating the maximal potential (the asymptotic value) at various distances along the dendrite after a step function in current applied to the cell. Further r_a can be estimated. With λ, τ and d, r_m and c_m can be found (and also R_i, R_m, and C_m).

Of course the general solution of the wave equation is fairly complicated and will not be discussed here. Moreover it should be noted that the model is a strong simplification of the physiological reality. The model assumes that there is no intracellular radial resistance and that current density in the extracellular space is zero, i.e. the extracellular resistance is zero. Moreover the membrane model is not really linear, since also for small deviations from the rest potential the conductance’s for Na⁺ and K⁺ change (see next paragraph), and consequently r_m is not constant.

Fig. 1 Principle of current injection Clamping) and measuring membrane potential.

Many investigators tried to include in the model the neuronal geometry, especially the dendritic branches. Since the cell bodies are generally small and more or less globular of shape, the soma membrane is practically isopotential. At (passive) endings of dendritic and axonal branches

o∂E/∂x = 0 since there the axial internal current is 0. A bifurcation of two branches, number 1 and 2, can be substituted by a single thicker branch, number 0, such that:

d₀^3/2 = d₁^3/2 + d₂^3/2. (9)

If d₁ = d₂ then d₀ = 2^3/2 d₁ ≈ 2.8 d₁. With this formalism a whole dendritic tree of a neuron can be substituted by a single piece of dendrite with a reasonable success.