Category Archives: science

The Brain Operates By Frequencies

By: John H. Heibel

Background
The book “Rhythms of the Brain” by Gyorgy Buszaki, lists several comments that raise questions that perplex many neuroscientists today. We respond to these comments using our new theory of brain operations. The comments and our answers thereto are listed next. The answers are fully discussed in our new theory of brain operations, below.
1) Buszaki comment: “Once an action potential is initiated, it can propagate through the entire axon tree of the neuron and signal this event to all downstream targets”, pg 87.
Our answer: “We will use this feature to great advantage in our theory below”.
2) Buszaki comment: “In the brain, specific behaviors emerge from the interaction of its constituents, neurons and neuronal pools, Studying these self-organized processes requires the simultaneous monitoring of the activity of large numbers of individual neurons in multiple brain areas”, pg 110.
Our answer: “In our theory below, we will describe how an unlimited number of these self-organized processes, each from different areas of the brain, can be studied simultaneously”.
3) Buszaki comment:” Many other methods — provide insights into the operations of the brain, but in the end, all of these indirect observations should be re-converted into the format of neuronal spike trains to understand the brain’s control of behavior”, pg 110.
Our answer: “Our theory below does just that, except that it goes one step further. It converts them into the result of these spike trains.” This result is easier to study than the spike trains themselves.
4) On pages 134 and 135, Buszaki covers a range of research that is worthy of several related doctorate theses. His last statement therein “When needed, neuronal networks can shift quickly from a highly complex state to act as predictive cohenent units due to the deterministic nature of oscillatory order”.
Our answer: “Our theory shows exactly how this oscillatory order is achieved”.
5) Buszaki comment: “Although this temporal window is in the tens of milliseconds range for single neurons, oscillatory coalitions of neurons can expand the effective window of synchronization from hundreds of milliseconds to many seconds” pg 174.
Our answer: “ – – and we show exactly how this happens”.
6) Buszaki comment: “ Gamma oscillations have been – – ‘binding problem’ – – unique objects and concepts”. Pg 260.
Our answer: “we show exactly how down-stream neuron groups keep adding to the composite signal, finally ending up with a completed picture”.
7) Buszaki comment: “From this perspective, – – perpetually evolving network pattern in the brain’s landscape”, pg 276.
Our answer: “This is a result of a continual ‘learning’ process. We show exactly how this occurs”.

In this paper:

We will show that the brain operates on frequencies.
We will show that cognitive actions are represented by frequency pairs or triplets.
We will show that brain waves can be generated by neuronal phase locked loops.
We will show that learning is a lifelong process.
We will show that learning can be executed via neuronal phase locked loops.
We will describe a solution to the ‘Binding Problem’ (see ‘fist’, pg6).
We will demonstrate that current EEG test data can be used to map both the frequency primitives of the brain, and their processing locations. In other words, currently published test data is probably sufficient to identify the majority of primitive frequency sets in the brain.

Page 1

How Action Potentials Create a Harmonic Series Containing an Unlimited Number of Components


If you Google ‘Frequency coding in the nervous system’ and go to figure 2, ‘Supra-threshold stimulus’, you will see how action potentials can create a very large number of possible input waveforms to a neuron. These input functions represent the time-variable movement of ions in and out of a neuron. The waveform shown in figure 1 looks deceptively simple; 5/6ths of the signal has a value of ‘1’ and 1/6sth of it has a value of ‘0’; very simply specified. The information content is fourfold; 1) time at maximum, 2) time at minimum, 3) amplitude at maximum, and 4) amplitude at minimum.
However, when it is expanded into its Fourier equivalent, the information content becomes unlimited, as shown below.

First, let’s look at the input function (figure 1).
fig1
Figure 1 – Input Signal
Notice that 5/6ths of the signal has a value of ‘1’ and 1/6sth of it has a value of ‘0’.

As shown in figure 2, this produces a harmonic series where every 11th harmonic is a maximum.
fig2
Figure 2 – Harmonic Series of Figure 1.

Page 3
Only about 330 harmonics are shown here for the sake of simplicity. In the digital world, the number of harmonics is related to the sampling rate. But the brain is analog, and it works as an integral process does, where the number of harmonics is unlimited. That explains why when we start measuring brainwaves and cataloging all these waveforms, we need an EEG instrument with at least 1000 times the sampling rate of currently available equipment.
Another feature of brain operation that needs explanation before we begin is that the brain is a linear system. In a linear system, we could excite it with one frequency at a time and measure the output for each input, and the sum of the outputs will represent what the total output would be if all inputs were simultaneously applied. This characteristic of a linear system allows us to analyze the brain one-frequency-at-a-time, or in groups, or with all inputs applied simultaneously.

Now, if we select only the peaks of the Fourier series, we will get the series shown in figure 3.
fig3
Figure 3 – Peaks of the Full Fourier Series.
Now to see what this would look like if turned back into a time domain signal:

Page 4

fig4
Figure 4 – Time series of waveforms (EEGs) created by figure 3
The waveform shown in figure 4 starts to look suspiciously like an 11 hz sleep mode. We could now start to tweak the input waveform of figure 1 to produce a sleep mode signal, or we could start with a measured sleep mode EEG and analyze the frequency set needed to make it. This whole subject would require a book to contain detailed descriptions of everything we’ve shown and its contents would be understood only by an electrical engineer/mathematician. A neuroscientist is not schooled in this art and since this presentation is targeted to neuroscientists, we will not delve further into design theory.
Instead, we will first describe in layman’s terms, how a phase locked loop works and then turn our attention to the mechanism in neurons that makes all this happen.

Phase lock loops – Definition (the linear systems rule lets us consider one signal at a time).

fig5
The multiplier produces sum and difference frequencies of w1/2*pi and w2/2pi (pi is the irrational number 3.1416 – – -) When w1 is approximately equal to w2, the multiplier produces an approximate second harmonic and a voltage term whose frequency approaches zero (DC). The magic of a phase lock loop is that the loop drives the VCO frequency into exact synchronism with the input frequency, at which time the loop locks the two frequencies together. If the input frequency changes (within certain limits) the

Page 5
VCO follows. Without further analysis, we have to assume that the phase error between input sinewave and VCO sinewave is non-zero, but the frequency difference is zero.
fig6
The force on the ions by the interaction of the EEG wave and the ion charges = mass*Accel.
Mass = K1 + A1Sinw2t, where K1 = the mass of ions already inside the neuron.
Accel = Sinw1t
When w1 = w2 + Δ, where Δ is a small number approaching zero, a beat frequency (Δ) develops. This beat frequency is assymetric about the vertical zero axis, resulting in a non-zero DC voltage. This DC voltage is offset by more ESPS’s (Excitatory Post Synaptic Potentials)or ISPS’s (Inhibitory Post Synaptic Potentials), depending on its polarity. This change in SPS’s causes a corresponding change in the action potential frequency, causing a change in the drive frequency, w2. When w1 = w2, the net DC voltage goes to zero and the two frequencies (w1 and w2) are locked together.
Since the driving function (A1SinW1t) is really a harmonic (Fourier) series, the EEG can actually lock onto any harmonic whose amplitude is sufficiently high.

Phase Lock Loop Characteristics.
The following phase lock loop characteristics make all this possible.
1) The sketch below depicts when the loop acquires lock. If the frequency difference (w1 – w2) is within the inner frequency bounds of the figure, the loop locks within one cycle of the input frequency.
2) If the frequency difference is outside the bounds of (1) but within the bounds of the outside vertical lines, the loop may lock, depending on the time that the input
signal is applied. Marginal frequency differences may take several seconds to acquire lock.
fig7
The loop stays locked as long as the EEG is present.
The EEG frequencies can drift, but the loop remains locked as long as the drift rate remains below a specified rate.

Page 6
The EEG lock frequency can change (within limits) and the loop remains locked.

Now we can discuss how a child learns.
In the mother’s womb, all neurons within the infant’s brain oscillate at random frequencies. The signal amplitudes are likewise random.
Action potentials are formed.
Those action potentials generate an input signal of the type described at the beginning of this article. That input signal produces a Fourier series of signals. One of the series of frequencies (probably the base signal) locks onto one of the random signals. This produces an EEG whose base frequency is identical with the random one it locked onto.
All frequencies of the EEG now seek like frequencies of the random set, whose lock range is within the range described above.
This modifies the initial EEG by adding the amplitudes of each member of the random set whose frequencies were close enough to lock. The modified EEG is stored in memory by freezing the neuron firing and resting potentials and the axon and dendrite interconnections. Memory loss is primarily caused by drift in the neuron potentials.
When the learning experience is repeated, the EEG gathers more members of the random set and the EEG is ‘strengthened’. The person becomes more adept at the repeated action. This process goes on for a lifetime.

Our Concept of Frequency Primitives.
This concept is best described with an assumed example.
Assume that in the womb, the child (or the mother) decides to bend the first finger of the right hand. This action is commanded by generating a 5th (1st finger) and 7th (bend command) harmonic. The frequency primitive for this would be a frequency pair of the 5th and 7th harmonics. The amplitude of the harmonics would be proportional to the ‘strength’ of the commanded action.
Assume further that the same command for the second finger would be represented by the 6th and 7th harmonics. 6th for finger #2 and 7th for ‘bend’. Repeat for the other two fingers and thumb. After these primitives are formed, a command to clench the fist would be 5 frequency pairs, each of which contained a 7th harmonic (bend). We have just described a solution to the ‘Binding Problem’.

It is our contention that all body action commands consist of frequency primitives. These primitives consist of no more that four discrete harmonics.
If this thesis proves correct, it opens a whole new world of analysis. One could now look at the frequencies and amplitudes of a body command and; 1) identify whether the primitives are all present and 2) estimate the degree of ‘normal’ of the commanded action.

Page 7
This section consists of seven sub-sections as follows:
1) The brain operates by frequency sets.
Examples Given Below

2) The brain generates frequency sets via phase locked loops
Examples Given Below

3) Learning is via modified frequency sets through phase locked loop operation
Example Given Below

4) Learning begins in the womb
Example Given Below

5) Learning is a lifelong process
Example Given Below

6) Frequency sets are different (but similar) for each person
Example Given Below

7) Proof that optical cognition can be represented by frequency sets
Example Given Below

Page 8

1) The Brain Operates by Frequency Sets.

Alpha Waves
The reader is referred to ‘Wikipedia, Alpha Wave’ for an explanation for this waveform

The Time Domain Pattern is shown next
fig8

Page 9

fig9
fig10
This graph shows that the dominant alpha frequencies are the 12th, 13th, 14th and 15th harmonics. With a 1 sec alpha time window, this would be 12hz, 13hz, 14hz and 15hz. We consider this frequency set to be a primitive (irreducible) set.

Page 10

Beta Waves. The reader is referred to ‘Wikipedia, Beta Wave’ for an explanation for this waveform

The Time Domain Pattern is shown next
fig11

Page 11

fig12
fig13

Page 12

Delta Wave . The reader is referred to ‘Wikipedia, Deltaa Wave’ for an explanation for this waveform

The Time Domain Pattern is shown next
fig14

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fig15
fig16

Page 14

fig17

Page 15

fig18
fig19

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fig20

Page 17

Here is the Theta Wave in the frequency domain
fig21
Here are the dominant frequencies of Theta, the 9th, 10th, 11th and 12th Harmonics. These four harmonics probably comprise a primitive frequency set.
fig22

Page 18

Mu rhythm.
Time domain
fig23

Page 19

Here’s what the first 150 harmonics of the murhythm look like
fig24

Here’s what the dominate frequencies of the murhythm look like
fig25
There appear to be four or five primitive frequency sets active simultaneously, with no single frequency above 40 hz

Page 20

2) The Brain Generates Frequency Sets via Phase Locked Loops (see above)

3) Learning is via Modified Frequency Sets through Phase Locked Loop Operation (see above)

4) Learning Begins in the Womb (see above)

5) Learning is a Lifelong Process (see above)

The process described in 3) is repeated throughout one’s lifetime.
The state of learning in an individual produces a ‘fingerprint’. This fingerprint is itself subject to change as one continues to learn (or forget).

6) Frequency Sets are Different (but similar) for Each Person
The following data is repeated from above.
If you Google ‘Frequency coding in the nervous system’ and go to figure 2, ‘Supra-threshold stimulus’, you will see how action potentials, ISPS and ESPS’s can create a very large number of possible input waveforms to a neuron. These input functions represent the time-variable movement of ions in and out of a neuron. The waveform shown in figure 1 is an example. It looks deceptively simple; 5/6ths of the signal has a value of ‘1’ and 1/6sth of it has a value of ‘0’; very simply specified. The information content is fourfold; 1) time at maximum, 2) time at minimum, 3) amplitude at maximum, and 4) amplitude at minimum.
However, when it is expanded into its Fourier equivalent, the information content becomes unlimited, as shown below.

First, let’s look at the input function (figure 1).
fig26
Figure 1 – Input Signal
Notice that 5/6ths of the signal has a value of ‘1’ and 1/6sth of it has a value of ‘0’.

As shown in figure 2, this produces a harmonic series where every 11th harmonic is a maximum.

Page 21

fig27

Figure 21 – Harmonic Series of Figure 1.
Only about 330 harmonics are shown here for the sake of simplicity. In the digital world, the number of harmonics is related to the sampling rate. But the brain is analog, and it works as an integral process does, where the number of harmonics is unlimited. That explains why when we start measuring brainwaves and cataloging all these waveforms, we need an EEG instrument with at least 1000 times the sampling rate of currently available equipment.
Another feature of brain operation that needs explanation is that the brain is a linear system. In a linear system, we could excite it with one frequency at a time and measure the output for each input, and the sum of the outputs will represent what the total output would be if all inputs were simultaneously applied. This characteristic of a linear system allows us to analyze the brain one-frequency-at-a-time, or in groups, or with all inputs applied simultaneously.

Now, if we select only the peaks of the Fourier series, we will get the series shown in figure 3.
fig28
Figure 3 – Peaks of the Full Fourier Series.
Now to see what this would look like if turned back into a time domain signal:

Page 22

fig29
Figure 4 – Time series of waveforms (EEGs) created by figure 3
The waveform shown in figure 4 starts to look suspiciously like an 11 hz sleep mode. We could now start to tweak the input waveform of figure 1 to produce a sleep mode signal, or we could start with a measured sleep mode EEG and analyze the frequency set needed to make it. This whole subject would require a book to contain detailed descriptions of everything we’ve shown and its contents would be understood only by an electrical engineer/mathematician. A neuroscientist is not schooled in this art and since this presentation is targeted to neuroscientists, we will not delve further into theory.
Instead, we have described in layman’s terms, how a phase locked loop works and then turned our attention to the mechanism in neurons that make all this happen. We have also shown that the possible set of resulting EEGs is unlimited.

All this leads is to believe that the brain operates by a series of primitive frequency sets.

Page 23

Here’s a second example.

The figure below is a series of action potentials applied to a neuron
fig30
The figure below shows what the first 150 harmonics of this waveform look like.
fig31
The next figure shows the dominant frequencies generated by the action potentials. Multiple primitives seem to be operating simultaneously.
fig32
We could go on and on with these examples, forming an unlimited number of dominant frequency sets. We can, in fact, re-generate every EEG known to man.

Page 24

7) Proof That Optical Cognition Can be Represented by Frequency Sets
This proof references the article “Single-trial classification of EEG in a visual object task using ICA and machine learning” in the Journal of Neuroscience Methods, 228 (2014) 114.
In this proof, we constructed a synthetic representation of the waveforms shown in the article. The representation is sufficiently accurate to identify the low frequency sets, but does not accurately represent the higher frequency components (above approximately 100 hz).
fig33
Sixth ERP (red) & Seventh ERP (green)
Notice that cognition is represented by a frequency pair, or triplet.
Notice also (refer to the article) that the cognition processing occurs in almost-identical portions of the brain, suggesting that operating frequency is position-dependent.
fig34
Fifth ERP (black) added
Notice that the first frequency of the 5th and 6th frequency sets are identical.

Page 25

fig35
Second ERP(magenta) and Fifth ERP (black)
fig36
First ERP (S1, magenta) and Fourth ERP (S4, blue)

Page 26

fig37
Third ERP (black) and Fifth ERP (red)

If one carefully studies the position of processing in the brain and compares it to the frequencies involved in the cognitive act, a convincing case can be made for the fact that position and frequency have a strong correlation.

We have shown that the brain operates on frequencies.
We have shown that cognitive actions are represented by frequency pairs or triplets.
We have shown that brain waves can be generated by neuronal phase locked loops.
We have shown that learning is a lifelong process.
We have shown that learning can be executed via neuronal phase locked loops.
We have described a solution to the ‘Binding Problem’ (see ‘fist’, pg6).
We have demonstrated that current EEG test data can be used to map both the frequency primitives of the brain, and their processing locations. In other words, currently published test data is probably sufficient to identify the majority of primitive frequency sets in the brain.

John Heibel
Author