exciting new evidence supports Scruffy's brain model

[scruffy]

And... finally provides a reason for the consolidation of memory in human beings.

You'll recall that short term memory is eventually consolidated into long term memory.

And, you'll remember the earring model, where the outer hoops represent long time constants and the inner hoops go all the way to dt.

And you'll remember all the noise I made about predictive coding.

Well, look here - look what happens while memories are being consolidated:

Hippocampus Predicts Rewards by Reorganizing Memories - Neuroscience News

A new study finds the hippocampus reorganizes memories to predict rewards. This discovery explains how the brain learns and why Alzheimer's affects decision-making.

neurosciencenews.com

According to the earring model, the memories are moving straight up, along a line segment connecting all the points at infinity.

And you'll recall I said two very specific things, predicted by this model:

1. Memory lives at the point at infinity. Because this is where the ends of the compactified timeline meet. Memory is the ultimate destination of all information, and the ultimate source of all information. The only way a sensory configuration turns into a motor action is by going "through" the point at infinity.

2. Awareness lives slightly ahead of "now". It's a prediction, is what it is. A prediction in the limit as dt => 0. It is a prediction that is mapped into the timeline, so it can be treated just like any other piece of information.

The amazing speed (resolution) of the network is what makes it work. Calculated at .05 picoseconds for 20.billion neurons, 5 ps if we only do cortical mini-columns, and a fraction of a nanosecond if we take full cortical columns.

That thing that you experience to be about a second wide, isn't. It's a critical state that lasts a few nanoseconds at most. The smoothness around state changes is due to the asynchronous nature of the updates, and is supported by an embedding into a sea of relatively long time constants.

The areas that glue the compactified timeline together are exactly the hippocampus and the dorsolateral prefrontal cortex - the two areas most closely related to memory.

And guess what? This also explains why phase coding is necessary! It's the only way to represent an entire pattern of activity along the timeline, as a single spike train in a single neuron.

And it also explains the sparse coding in the hippocampus.

I think, ladies and gentlemen, we have a winner. Now that the underlying mechanism is established, we can turn to more fundamental questions like "why is red red" and where am "I".

I'm pretty sure this last piece of evidence clinches the deal on the model. I'm not aware of any contradictory evidence. (Like, "any").

It makes perfect sense, too. The outer hoop is the width of the whole network, maybe a second in each direction from "now". It corresponds with "biological time frames", in other words this is how fast signals typically change when you're, say, pursuing prey or trying not to get eaten. To access that as quickly as possible the information has to move down into the inner hoops where the radius is narrower, and there, at the end of the journey in the limit as dt => 0, you get molecular memory which equates with electrical memory.

 

[scruffy]

Okay, Eureka. Here's what we have to do. You guys can help, we can do it together.

We have to extend the Kuramoto model to multiple frequencies.

Here is the existing Kuramoto model (play the video in the Definition section):

Kuramoto model - Wikipedia

en.wikipedia.org

As you can see, there is only one frequency. The circle is the frequency. To get from here to an earring, we need multiple frequencies.

(The model as is handles coupled pendulums, if they're all the same length - which means they're swinging at the same frequency, and all the model does is describe the phase relationships. Which is GOOD, because it addresses a lot of things like synchronization and such, but we need more).

Put on your engineering hat for a minute, because whenever two frequencies interact we get the sum and difference frequencies too, right? (The "sidebands", as it were). But the starting point is clear - we take two circles of different sizes (which means different frequencies) and do the math. Then we go to three circles, do some more math, then the general case.

How about the math guys take a look at this?

talanum1
Wise47

Off the top of my head I can suggest one place to look for a little research - nonlinear nonequilibrium thermodynamics, of the kind that got Prigogine the Nobel Prize in the 70's. He was dealing with chemical reactions but it's all the same, right? Maybe ("maybe") two different reaction rates somehow relate to two different frequencies? The common element is the coupling constants.

 

[scruffy]

Wow. trevorjohnson83 , you might want to take a look at this too. AI says:

In the context of Ilya Prigogine’s work on non-equilibrium thermodynamics, dissipative structures, and statistical mechanics, the concept of
coupling refers to the interaction between different, irreversible physical or chemical processes (e.g., heat flow and diffusion) or between a system and its environment.

• Coupling in Dissipative Structures: Prigogine showed that when systems are far from equilibrium, nonlinear interactions (coupling) between chemical reactions and transport processes can lead to the emergence of ordered, self-organized patterns.
• Weak Coupling Limit: In his studies on statistical mechanics and the foundation of irreversibility, Prigogine analyzed the "weak coupling limit", which explores how interaction terms in a Hamiltonian (usually denoted by a coupling parameter
  λV
  ) lead to the destruction of correlations and the emergence of macroscopic irreversibility.
• Coupling to Environment: Prigogine emphasized that dissipative structures are open systems that maintain their ordered state through energy and matter coupling with their external environment.
• Coupling and Irreversibility: His work demonstrated that coupling allows for a "resonance" between degrees of freedom, which breaks time-reversal symmetry at the microscopic level, leading to irreversible, diffusive processes.

In summary, Prigogine's "coupling" does not refer to a single, specific constant in the way an NMR coupling constant (
J
) does, but rather to the interaction strength (
λ
) between system components or between the system and its surroundings, which drives the transition from chaos to order.

---

Coupling between chemical reactions and transport processes.

I'm not really sure what that means. Transport processes are like fluid flows, right? If you're in aqueous solution that would be convection currents and such?

Anyway, parts of this are usable.

 

[scruffy]

Maybe this is easier than I thought.

Coupling between frequencies makes that term lambda look like a polynomial, yes?

So if you have two frequencies f1 and f2, and a coupling constant k between them, then f1 looks like f1 + k*f1*f2, something like that? And, if you go cos(kx-wt+phase) then you can conveniently assign (different) coupling constants to each parameter. And then multiply out. Is it that easy?

 

[scruffy]

Hm. AI says this about coupling two pendulums of different length:

Coupling pendulums of different lengths, typically via a shared string, causes complex, non-resonant, and often irregular motion because their natural frequencies differ (

f = (1/2π) sqrt(g/L)

)

Unlike identical, resonant pendulums, they do not transfer energy completely or smoothly, resulting in jerky, uneven movement rather than perfect, alternating, or in-phase, in-phase, and anti-phase synchronization.


Key Characteristics of Differing Length Coupled Pendulums:

• Irregular Energy Transfer: Due to mismatched natural frequencies, energy is not fully transferred between the two pendulums. The "beating" behavior seen with identical lengths is altered, resulting in less predictable, chaotic-looking, or "jerky" motion.
• Reduced Synchronization: While identical pendulums can synchronize in phase or anti-phase (a phenomenon noted by Christian Huygens), different lengths limit this, often leading to complex, non-periodic behavior.
• Frequency Differences: The overall motion is governed by the beat frequency, which is the difference between the frequencies of the normal modes.
• Application: When used in systems like a "pendulum wave machine," pendulums of varying lengths produce complex, shifting, wave-like patterns because each has a different, independent natural frequency.

If the coupling is strong (a tight string), the pendulums will still affect each other, but if the lengths are very different, they will largely behave as two nearly independent oscillators rather than a single, combined resonant system.

 

[scruffy]

Okay, here's some references:

Coupling through an external medium:

Kuramoto model with coupling through an external medium - PMC

Synchronization of coupled oscillators is often described using the Kuramoto model. Here, we study a generalization of the Kuramoto model where oscillators communicate with each other through an external medium. This generalized model exhibits ...

pmc.ncbi.nlm.nih.gov

Bimodal frequency distribution:

https://link.aps.org/doi/10.1103/PhysRevE.79.026204

Interesting real world application of Kuramoto dynamics:

Millennium Bridge, London - Wikipedia

en.wikipedia.org

 

[scruffy]

Bimodal (click "view PDF"):

Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution

We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained...

arxiv.org

 

[scruffy]

The bimodal stuff is good till about their equation (7). Then we get the Ott simplification. What we need is an equation like (10) that describes the whole system.

This is doable. It's not easy, but it's doable.

 

[scruffy]

Okay, I went through their bimodal derivation. The assumptions are that the frequencies don't change and that all the coupling constants are the same. If you look at their Figure 2 that's their solution, which is conveniently visualized.

We expect that our solutions will be more intricate than this one. For one thing, we can take their global coupling constant K, and expand it into a matrix of pairwise coupling constants k(i,j)(f) where f is the frequency. In this case the Kuramoto formula still describes the coupling within a frequency, and we can expand this by adding terms representing the coupling between frequencies.

It makes little sense to talk about the phase relationships of oscillators with different frequencies, except in the neighborhood of t=0, when all the waveforms are somehow synchronized. If we have an anchor point like this, we can use the angular velocity of the oscillators to calculate the implied phase relationships.

And, multiplication is pretty key to any kind of modulation, and we can definitely do this synaptically, you can assume we have all four basic algebraic operations available.

 

[scruffy]

Here is an attempt along these lines. Oscillators of different frequencies with matrix coupling.

Matrix coupling and generalized frustration in Kuramoto oscillators - PubMed

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the...

pubmed.ncbi.nlm.nih.gov

https://pubs.aip.org/aip/cha/article/32/9/093130/2836001/Matrix-coupling-and-generalized-frustration-in

It looks like we'll end up exponentiating a matrix. That could be a special treat, we'll see.

 

[scruffy]

Here's an interesting one. It says oscillators driven by noise exhibit the same behavior as oscillators of different frequencies.

https://link.aps.org/doi/10.1103/PhysRevE.108.064124

 

[scruffy]

Oh look, I have a stalker.

Well hang around son, you might learn something. (In spite of being a leftie).

So far PyTorch is telling me I can use the same mechanism underlying contrast gain control in the visual system.

Which makes perfect sense because the timeline of electrical activity is just a retina. It's no different than having a pattern of moving lights on your eye, you can bring any of them into focus by simply rotating your eyeball.

I've taken a first pass at the math. It turns into a very complicated optimization problem, even when it's expressed as coupled ODE's. You have derivatives of the Hamiltonian (cost function) with respect to time, phase, and frequency. The "conservation of oscillators" equation takes a different form, because oscillators change frequencies.

It turns out the discrete version of this is a LOT easier to solve than the continuous version. I'm not optimistic about getting an exact solution in the continuous case (nor is it really necessary) - but with the right simplifying assumptions we can still visualize activity in the phase plane.

The neuroscientists don't know how to do this. In the brain we have two frequencies we can very conveniently measure: alpha and theta. The biologists are making a big deal about alpha-gamma coupling but that's not really a big deal since it's predicted by Hodgkin-Huxley. But aloha-theta coupling would be a big huge deal. Nobel Prize material, if you can figure out how it works.

I can tell you exactly where in the brain to look: retrosplenial cortex. Betcha 20 bucks right now if you stick some electrodes in there you'll see alpha theta coupling.

Unfortunately this can't be tested with MRI, we have to do it invasively with electrodes, because retrosplenial activity is very hard to pick up on the scalp with EEG.

But you could use the bimodal model to investigate it, your two frequencies are 7 and 11 Hz meaning the midline is 9 and your dw is 2. I already ran it through PyTorch, it works in principle but I haven't been able to assign brain areas to it yet.

 

[scruffy]

Okay, I have something useful. I have to figure out a way to share the math.

You guys know why we're doing this, right?

Top-down control of the phase of alpha-band oscillations as a mechanism for temporal prediction - PubMed

The physiological state of the brain before an incoming stimulus has substantial consequences for subsequent behavior and neural processing. For example, the phase of ongoing posterior alpha-band oscillations (8-14 Hz) immediately before visual stimulation has been shown to predict perceptual...

pubmed.ncbi.nlm.nih.gov

Binding Mechanisms in Visual Perception and Their Link With Neural Oscillations: A Review of Evidence From tACS - PMC

Neurophysiological studies in humans employing magneto- (MEG) and electro- (EEG) encephalography increasingly suggest that oscillatory rhythmic activity of the brain may be a core mechanism for binding sensory information across space, time, and ...

pmc.ncbi.nlm.nih.gov

In the bimodal (two-frequency) version there are only 5 regions in the phase plane. In the general case there can be hundreds, even thousands or millions.

We're looking for meta-relationships in the phase space as they may pertain to regions of criticality. Specifically we want to know under what conditions a visual image can drive the brain into criticality, and what happens once it does.

The reason we're interested in criticality is because it depends on long range coupling. Transitions to and from critical states occur instantly, they don't depend on the time constants of the underlying processes.

 

[scruffy]

This is Kuramoto, as seen from the standpoint of nonlinear thermodynamics.

This is actual nonlinear thermodynamics, a chemical reaction in aqueous solution. It's called the Belousov-Zhabotinsky reaction.

This is the brain. These are orientation columns in the primary visual cortex.

 

[scruffy]

In this image you can see how an applied stimulus affects the oscillators. So this would be, like, starting with a bunch of oscillators at different frequencies and phases, flash a light on the retina and then watch as the oscillators synchronize.

At the bottom is the applied stimulus, which corresponds with the size of the arrow inside the circle. At the top are the phases of 5 coupled Kuramoto oscillators. You can see, that the input drives them into synchrony.

And the trick is, the bottom doesn't have to be a flash of light. It could be a nerve cell firing, elsewhere in the brain. The long range coupling under conditions of criticality strongly suggests that a nerve cell firing in one part of the brain, can directly and immediately affect the synchronization of oscillators in a different part.

This is how you get the weirdo spatial patterns. What ends up happening is the spike train from the forcing neuron creates a pattern of frequencies in the rest of the brain. Which will then entrain other local processes, creating finer grained patterns with better resolution.

What is working for me so far is to "bin" the frequencies into buckets. Kind of like the alpha-beta-theta game. The number of buckets is less than or equal to the number of oscillators. Each bucket is then treated as "a" frequency, and individual oscillators move from one bucket to another as the frequencies change. Within a bucket, it's all Kuramoto. I had to learn what a Lorentzian distribution is and why it works, but Gaussians work too.

This model is presently working in Python, and tonight I did most of the graphics work for visualization. There's still a couple of bugs but basically it works. To visualize this I actually want to plot this on an earring. If I can get that I'm done and it's going to the editors. All I'm trying to show is a straight line of points at infinity, where the oscillator phase is 180 degrees away from the point at T=0.

You'll notice that the input (a bigger arrow) makes the oscillators cluster, and in this diagram the origin (0 phase) is on the right, so they're clustering around the origin. And that, ladies and gentlemen, is where the singularity is that requires all the prediction and Kalman filtering.

You'll notice something else too. Sometimes the green ball stays at the point at infinity (the left side), even when the other oscillators are clustering around the origin. And, sometimes oscillators spontaneously move to the other side of the circle, and they do so very quickly.

 

[scruffy]

In the chemical case, the equation becomes nonlinear reaction-diffusion in three spatial dimensions. Each point in space, each point in the aqueous solution, is an oscillator. That's because of the chemical reaction, it's just the nature of it. (Boron, malonic acid, some sulfuric, I forget what else is in it - not much though, it's pretty simple). Each reaction point gets both excitatory and inhibitory influences from its neighbors ("coupling constants").

But the wave pattern that you see in the beaker, doesn't move. It just stays there, it's stable. Those are "standing" waves. The configuration that you see, is a local energy minimum for the coupled reaction. The only way to get that pattern to change is to find a better energy minimum, and in that case the rules are well known, you need the activation energy to get out of the local minimum, and then you get it back ("plus a little") when you find the new minimum.

The stability of the standing wave pattern is not guaranteed, you have to make sure the reaction "converges" to a stable point, where the coupling energy balances the reaction energy. When this balance is in place, it's like having Kuramoto oscillators 180 degrees apart.

 

[scruffy]

And - you see the red yellow green and blue colors - those are phases of oscillators.

Remember what we learned from the Ising model - when the phases line up you get a magnet, and when they don't you get bupkis.

Those colored regions are actually phases in the phase-of-matter sense. And between each pair of them, between each pair of colors, is a region of criticality. You have a sea of phases of matter, with little critical areas between them. The critical areas are what correlate via the coupling constants. They actually end up stabilizing the standing waves.

 

[Hafar1014]

And... finally provides a reason for the consolidation of memory in human beings.

You'll recall that short term memory is eventually consolidated into long term memory.

And, you'll remember the earring model, where the outer hoops represent long time constants and the inner hoops go all the way to dt.

And you'll remember all the noise I made about predictive coding.

Well, look here - look what happens while memories are being consolidated:

Hippocampus Predicts Rewards by Reorganizing Memories - Neuroscience News

A new study finds the hippocampus reorganizes memories to predict rewards. This discovery explains how the brain learns and why Alzheimer's affects decision-making.

neurosciencenews.com

According to the earring model, the memories are moving straight up, along a line segment connecting all the points at infinity.

View attachment 1212837

And you'll recall I said two very specific things, predicted by this model:

1. Memory lives at the point at infinity. Because this is where the ends of the compactified timeline meet. Memory is the ultimate destination of all information, and the ultimate source of all information. The only way a sensory configuration turns into a motor action is by going "through" the point at infinity.

2. Awareness lives slightly ahead of "now". It's a prediction, is what it is. A prediction in the limit as dt => 0. It is a prediction that is mapped into the timeline, so it can be treated just like any other piece of information.

The amazing speed (resolution) of the network is what makes it work. Calculated at .05 picoseconds for 20.billion neurons, 5 ps if we only do cortical mini-columns, and a fraction of a nanosecond if we take full cortical columns.

That thing that you experience to be about a second wide, isn't. It's a critical state that lasts a few nanoseconds at most. The smoothness around state changes is due to the asynchronous nature of the updates, and is supported by an embedding into a sea of relatively long time constants.

The areas that glue the compactified timeline together are exactly the hippocampus and the dorsolateral prefrontal cortex - the two areas most closely related to memory.

And guess what? This also explains why phase coding is necessary! It's the only way to represent an entire pattern of activity along the timeline, as a single spike train in a single neuron.

And it also explains the sparse coding in the hippocampus.

I think, ladies and gentlemen, we have a winner. Now that the underlying mechanism is established, we can turn to more fundamental questions like "why is red red" and where am "I".

I'm pretty sure this last piece of evidence clinches the deal on the model. I'm not aware of any contradictory evidence. (Like, "any").

It makes perfect sense, too. The outer hoop is the width of the whole network, maybe a second in each direction from "now". It corresponds with "biological time frames", in other words this is how fast signals typically change when you're, say, pursuing prey or trying not to get eaten. To access that as quickly as possible the information has to move down into the inner hoops where the radius is narrower, and there, at the end of the journey in the limit as dt => 0, you get molecular memory which equates with electrical memory.

My explanation as trauma therapist who has pioneered therapeutic interventions for memory disorders. Trauma coherence therapy is based on active PET scans of brain activity in real time. You explanation doesnt explain how it effects behavior

Memory is organized in direct proportion to the emotion created at the time of the experience. Emotion is the underlying mechanism. The Limbic system that holds memory also creates emotion. All memories are connected to emotion and the more intense it is the strength of the memory will be greater.

Lets take trauma as an example. This is the most intense kind of memory there is. Love is also equally intense. Trauma memories are so powerful they can take over the entire brains functions. Trauma acts as if the experience never stopped or will happen again at any moment.

We can in therapy rewire the brain and modify the emotions connected to any memory.
Every recall ends with reconsolidation. The memory us created again as if new and wont fade with time.
The hippocampus works with the amygdala in storing and recalling memory.

When a memory is expressed it goers up to the prefrontal cortex which must try and accurately understand the message. The LS cant use words the PFC understands words and explicit messages. The accuracy of the understanding is called coherence. Some people have high C and others low. Thats why people do stupid things or act in ways that are damaging or self sabotaging.

 

[Hafar1014]

Okay, I have something useful. I have to figure out a way to share the math.

You guys know why we're doing this, right?

Top-down control of the phase of alpha-band oscillations as a mechanism for temporal prediction - PubMed

The physiological state of the brain before an incoming stimulus has substantial consequences for subsequent behavior and neural processing. For example, the phase of ongoing posterior alpha-band oscillations (8-14 Hz) immediately before visual stimulation has been shown to predict perceptual...

pubmed.ncbi.nlm.nih.gov

Binding Mechanisms in Visual Perception and Their Link With Neural Oscillations: A Review of Evidence From tACS - PMC

Neurophysiological studies in humans employing magneto- (MEG) and electro- (EEG) encephalography increasingly suggest that oscillatory rhythmic activity of the brain may be a core mechanism for binding sensory information across space, time, and ...

pmc.ncbi.nlm.nih.gov

In the bimodal (two-frequency) version there are only 5 regions in the phase plane. In the general case there can be hundreds, even thousands or millions.

We're looking for meta-relationships in the phase space as they may pertain to regions of criticality. Specifically we want to know under what conditions a visual image can drive the brain into criticality, and what happens once it does.

The reason we're interested in criticality is because it depends on long range coupling. Transitions to and from critical states occur instantly, they don't depend on the time constants of the underlying processes.

This is like looking at an elephants toenail and trying to figure out what kind of animal it is. Stap back and take broad perspective your lost in the minutia

 

[scruffy]

"For the Rays, to speak properly, are not coloured. In them there is nothing else than a certain power and disposition to stir up a sensation of this or that Colour."

- Sir Isaac Newton, Opticks, 1704