Appendix 3: Some background on resonance

Electrical resonance can store up energy in an oscillation. Fig Ap3-1 shows how the voltage oscillation stored in a resonant system grows when it is driven at its resonant frequency (in magnifying phase.)


If the dendritic spines of neurons are analyzed assuming the S-K transmission equation, they will show such resonance. Resonant magnifications may be very great, and bandwidths extremely narrow, as shown in the dendritic spine resonance calculations graphed in Fig AP3-2:

Resonant systems may all be described in wave propagation terms, and many can also be treated in lumped (LRC) terms.

The International Dictionary of Applied Mathematics explains inductance-resistance-capacitance (LRC) series resonance as follows. The "coil" is a lumped inductance, the "condenser" is a lumped capacitance, and "j" is the square root of -1.

. . . In an a-c circuit containing inductance and capacitance in series ... the impedance is given by

where R is the resistance, is 2 times the frequency, L is the inductance, and C is the capacitance. It can be readily seen that at some frequency the terms in the bracket will cancel each other, and the impedance will equal the resistance alone. This condition, which gives a minimum impedance (and thus a maximum current for a fixed impressed voltage) and unity power factor is known as series resonance. Where the resistance is (relatively) small the current may become quite large. As the voltage drop across the condenser or coil is the product of the current and the impedance of that particular unit, it may also become very large. The condition of resonance may even give rise to a voltage across one of these units that is many times the voltage across the whole circuit, being, in fact, Q times the applied voltage for the condenser and nearly that for the coil. This is possible since the drops across the coil and condenser are nearly 180 degrees out of phase, and thus almost cancel one another, leaving a relatively small total voltage across the circuit . . .(1)

A neural element according to K-R will have no inductance, L. There is no LRC resonance under K-R. That same neural element according to S-K may have very large inductance. For an LRC resonator, the resonant frequency, o is:

The resonant amplification factor, Q, achieved after time to equilibrium(2), is:

For an LRC resonator, Q is

Bandwidth (the frequency difference between the half power points on a resonance curve) is:

In addition to lumped (LRC) resonance, columns can also be resonant. Columns (transmission lines) of 1/4 and 1/2 wavelength have been used as resonators in musical instruments for many centuries. More recently, column resonance has been used with precision in the radar and communication fields. Well terminated lengths of neural passage that are sharply open (short circuited) at both ends are resonant when there length is exactly 1/4 of . For column length lc:

and integer multiples of o. A well terminated length of neural

passage that sharply closed on one end will be resonant at

and integer multiples of these frequencies.

Consider a neural (dendritic) passage in terms of its defining variables, d, conductor diameter, = membrane thickness, = membrane dielectric constant and = axolemma conductivity.

Assuming S-K, conduction velocity in small neural lines (above a very low threshold frequency) is constant. v is:

For a 1/4 wave resonant column of length lc:


Q of the column resonator will be inverse with attenuation per wavelength .

According to K-R, attenuation per wavelength is enormous, Q's are very small, and resonant energy storage does not occur. According to S-K, attenuation per wavelength can be very small, Q's can be in the thousands or tens of thousands, and resonant energy storage can be important. Substituting the neural formula for attenuation per wavelength, , for a well terminated passage (having ends with 100% positive or negative reflection) Q becomes:

Note that Q increases with o. Column resonator bandwidth in terms of o, , , and is

Column resonators magnify and store repeating WAVEFORMS that fit as standing waves within them. In contrast, an LRC resonator stores a sine wave.

For spines, LRC mode Q's in the tens of thousands are possible on the assumption of S-K. Assuming K-R, there is no LRC resonance. Similarly large Q's are possible in column resonant mode assuming S-K. There is no column resonance under K-R.

1. The International Dictionary of Applied Mathematics D.Van Nostrand Company, Princeton, Toronto, New York, London 1960

2. op. cit.