Circuit Elements

Your brain and central nervous system constitute a complex network of electrical pathways. On most scales larger than that of the individual neuron, it seems that computer science provides the most appropriate models for its study. With the electrical circuit, however, physics allows us to model the propagation of nerve impulses along the axon. We must therefore begin by understanding the elements with which we construct circuits.

We think of wires as conductors of electricity which are covered by an insulator. The insulator, of course, is necessary to prevent wires from accidentally making contact, causing a short circuit: an unintended electrical connection which bypasses part of the circuit. In fact, most materials will conduct electricity to a greater or lesser extent. Metals make excellent conductors because their valance electrons are essentially free to move from one atom to another under the influence of an external electric field. We call these free valence electrons charge carriers; in an axon, the charge carriers are sodium and potassium ions. Plastics make good insulators because their valence electrons are tightly bound. In a similar fashion, the cell membrane acts as an insulator for the ions inside, although ion channels allow some leakage through the axon wall into the surrounding interstitial fluid. Mammalian motor neurons are coated with a myelin protein sheath, which improves the insulation by preventing the ions from leaking.

When we connect the terminals of a battery to an electrical circuit, we say that current flows from the positive terminal of the battery through the circuit to the negative terminal. This is a conventional definition of current, and does not actually reflect what physically occurs. When the circuit is connected to the battery, an electrical potential difference exists between the terminals. The charge carriers in the circuit then begin drifting toward the positive terminal of the battery, which for an electron is of course a position of less energy. The speed at which they drift is typically around 4 * 10-5 m/s in the wires of the circuit, although this depends on the conductivity of the wire. This seems very slow, but because all of the charge carriers move essentially simultaneously, changes in electrical energy propagate at nearly the speed of light (2.998 * 108 m/s in a vacuum). A similar thing occurs in the axon: while the ions move at less than 1 cm/s, nerve impulses can propagate at speeds up to 120 m/s!

A metallic conductor such as a wire is an equipotential volume: V has the same value everywhere in the wire. If this were not so, an electric field would exist in the wire and the charge carriers would redistribute themselves until V was again constant. But since the wire has a finite conductivity σ (the Greek letter sigma), it must also have a finite resistivity (denoted by the Greek letter rho)

ρ = 1 / σ.
Because the charge carriers in an axon are so much larger than those in a wire, the axon has a correspondingly larger resistivity.

Resistors and Capacitors

A resistor is a circuit element whose purpose is to dissipate electrical energy. This is manifested as a drop in the value of the electrical potential from one end of the resistor to the other. The size of the voltage drop depends on the amount of current flowing through the resistor. Current (denoted I) is measured in Amperes (1 A = 1 C/s), and the resistance R of a resistor is the ratio of the voltage drop to the current. R is measured in Ohms (denoted by the capital Greek letter omega): 1 Ω = 1 V/A. Typical resistors in electrical circuits range from several hundred to many thousands of Ω. Appliances often draw currents measured in A, but typical currents in circuits are of the order of mA or even μA. 20 μA is enough current to induce ventricular fibrillation if it is applied directly to the heart (ie., accidentally through faulty equipment used during a surgical procedure). Normal household current (120 V, 60 Hz alternating current) produces the following reactions for 1 second of skin contact:

I (mA)Effect
1awareness threshold
5maximum harmless current
10-20sustained muscular contraction
50pain, possibly fainting and exhaustion
100-300ventricular fibrillation
6000temporary respiratory paralysis, possibly burns

(from table 14-1 of Physics for the Health Sciences, 3rd ed., C. R. Nave and B. C. Nave, Saunders 1985)

Like every cell in your body, neurons pump sodium ions out of the cell in order to maintain an electrical potential difference between the inside and the outside of the cell membrane. The intracellular potential relative to the interstitial fluid is called the membrane potential; for human motor neurons, it is about -70 mV. In an electrical circuit, the function of separating charge carriers on either side of a barrier is performed by a capacitor.

The canonical capacitor consists of an insulating substance sandwiched between two parallel conducting plates. The insulator is characterized by its dielectric constant, and each plate has an equal but opposite charge. The capacitance C measures the amount of charge which is stored for a given potential difference across the plates. The unit of capacitance is the Farad (denoted F): 1 F = 1 C/V. It is a very large unit; typical capacitances are measured in microfarads or picofarads. Note that the S.I. units of electrical permittivity are F/m as well as C2/N m2.

The capacitors used in electrical circuits are usually parallel plate capacitors which have been rolled up and encased in a cylindrical container. It is also possible to construct a truly cylindrical capacitor using concentric cylinders separated by a dielectric substance, or to construct a spherical capacitor using concentric spheres. In each case, the capacitance depends only on the dielectric constant and the geometry of the capacitor. For a membrane, the capacitance is measured per unit length; biological membranes typically have a dielectric constant of about 9 and a capacitance of from 0.5 to 1.3 * 10-10 F/m.

You are now ready to construct the equations you will need in the next section.

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The case of the parallel plate capacitor provides a simple illustration of the relationships between a number of the concepts we have been examining. We expect the capacitance to be proportional to the dielectric constant because the units of capacitance are F and those of the electrical permittivity are F/m. Since the geometrical parameters are the areas of the plates (assumed to be equal) and the distance between them, unit analysis tells us that the capacitance must be

C = ε A / d
It turns out that the magnitude of the electric field between the plates of the charged capacitor is approximately constant. Since the electric field is the gradient of the electric potential, a simple relationship exists between them in this case:
E = V / d
Using the general equation for capacitance as a function of charge and voltage drop (which you just built), you can show that the charge per unit area on the plates (which must be a constant since they are conductors) is related to the electric field by
σ = ε E,
where σ now stands for the charge per unit area.

In the next section, we will build and analyze a circuit model for nerve impulse propagation.

©2013, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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