Dendrite Growth

So far I worked with a simplified model with a single dendrite. But moving the signal to layer two with more inter-neuronal connections allowed, it led to obvious errors. Why is this important ? In my model distance matter, synapses further away from the neuronal body contribute less to the overall neuronal potential. The dendrites, with their growth, provide that variable distance. I could not find a model in literature to fit my requirements so I came up with two very different models, eventually I decided to start with the one that seems easier because it does not require any calculations (as in geometrical calculations). This model should lead to structures close enough to what is observed in literature, but regardless if it’s close or not, it should clearly link synapses based on distance. It will allow for branching, this is very important since branching allow for synapses with same distance to the neuronal body.

However there are still many questions unanswered. What happens when a synapse is removed from a dendrite ? Does it bind to a further away position ? It is removed forever ? It binds to a different dendrite ? Should I allow multiple synapses in between 2 neurons ? What happens with the vacancy left by the removed synapse ? remains empty ? is occupied by other synapses ? a further away synapse takes its place ? What should I do about far away synapse (away from the neuron), their contributions to the overall potential is insignificant even with a linear decrease in contribution with distance, I now have an exponential decrease so it’s even worse.. Sure in some cases the synaptic strength (AMPA receptors equivalent) increase and the contribution is a bit bigger, but still small. Whys so many direct connections with so small contributions ? The signal would still reach a target neuron through its neighbors, more like in a GNN network .. that would make more sense to me.

As far as I can tell, I now have a good model for :

  1. Glutamatergic synapse (kinetics of glutmate and of AMPA)
  2. GABA synapses (half baked.. is acting on the axon resulting in 100% percent inhibition, but is also affecting active synapses.. so it’s a half man half bear kind of a situation.. maybe half pig as well)

What is similar ?

I’ve been getting many unexpected results in my quest for invariance. I thought is because of a bug in coding or a bad theory. But no, everything seems to be in order, there are no errors that I’m aware of.. So I was left with the improbable.. I don’t get identical results for identical patterns, because “similar” is not what I assumed it to be. Something is similar (or identical) not only when is formed from identical components but it also needs to have the same history. When I was thinking of “context” I was usually thinking only about the “stuff” around, did not think that I need the whole history behind that event (history in context is : sequence of patterns in time).

In the meantime I have some explanations for the lack of synchronization I now encounter on a regular basis.

  1. I don’t have horizontal cells or amacrine cells, in my code this results in an out of phase state that cannot be corrected (my input cells don’t fire every frame but on a certain frequency, set every other frame at this point in time). I have a button that brings them all to frame 1 when this happens, but this is just a cheap easy fix.
  2. The input cells are not out of phase, but patterns within the same visual field, fire at different frequencies. Not sure what to do about this one, could be normal and be somewhat “fixed” within the next layer. I was thinking to link inhibitory neurons among themselves so if one is activated it it will inhibit the inhibitory neurons around( meaning it will inhibit the inhibition they were providing)

Is learning guaranteed ?

I’ve been spending a lot of time trying to reach a state in which I’d get a partial Invariance… I observed that there are at least 2 pathways that would lead to different results..

  1. The frequency and sequence of patterns – only certain sequences would lead to the desired result. There is no way to guarantee for a certain result. The only way to guarantee a result is to have a precise training set which is not what I want, but so far I could not think of a way to correct for an imbalanced training set..
  2. Timing. Learning depends on time, meaning at time t1 we can have result R1 and at time t2 (where t2>>t1), we have a result R2. There is a time limit after which there is no change, but again, there is no guarantee that I get a certain results and no way of assessing when something learned would not change with time.

Both to me seem reasonable but very annoying… I find it very hard to set up an objective function with so many unknowns..

Invariance to nowhere

I’ve already done many simulations with the new model. There’s no invariance in sight. I also have no theories that would predict this elusive invariance, so much so that I’m not so sure anymore that this is obtained at the neuronal level. So I’ve prepared plan B. Even without invariance learn as many patterns possible, very much like the regular ML used these days. Well, that also failed. When active, neurons send a call to other neurons looking for binding partners. But how are the neurons active in the first place ? To solve this problem I linked from the beginning, neuron ij from L1 to same ij neuron from L2. That is proving to be a limiting factor now since in L2 I need to be able to form more patterns than in L1. Even if L2 has more neurons, they never get activated and they never connect to anything. In my previous post I showed a 2by2 matrix learning a 2 pixel pattern, but the cross lines were missing as a pattern because I did not have enough neurons in L2 to learn those patterns.

Anyway I need to make neurons activate “spontaneously”, till they make their first connection at least, see if that solves my problem.. Why not make fully connected layers ? Theoretically, that should work too but is a bit impractical because training takes a lot of time. Also this random activation is not as simple as it seems, because it will result very fast in a fully connected network.

on the right track….

After many trials and errors, the four pairs are now forming as expected, the priority mechanism seems to be working also:

When 2 neurons from the input layer are firing, they should be recorded by a single neuron in the first hidden layer, without overlap.

The priority mechanism has to be able to separate patterns correctly, scroll to 0:12 to see what I’m talking about. Basically when a new pattern is started the initial response is not always specific (more than 1 neuron is active in the first hidden layer), but after couple of firing events only the correct neuron is firing.

Training data for upper left neuron

The mechanism is still fragile but at least I know I’m on the right track… Hopefully next update will be more consistent.

I’m stuck… again

I ran through 4 models already and nothing seems to be working. I can only learn 3 patterns out of 4 at one single time. The fact that same problem shows up even when I tried to correct it 4 times already makes me believe there must be something else that is actually not right, something that I’m not aware of… And the 4 models are not just changing some numbers each model has a different ways of forming and breaking synapses, different rules. Last model is the most complex allowing for every conceivable permutation and still gets stuck in timing issues. I still have couple of more ideas to try out, but not many, I’ll be running out of ideas soon… And no idea is earth shattering … so I don’t have much hope that it will solve the problem… Still my main problem remains that models are too complex and I can’t predict what should happen, there are always unintended consequences. So either I’m going to make an unexpected progress soon or I’ll stop working on this project…

What is learning ?

This seems like a simple question for a ML algorithm .. train some variables to some values and the combination uniquely identifies something. But those variables can also change when something new is learned and you lose the initial meaning..

From my previous post, I concluded that learning must be somewhat random, but I was not happy with that conclusion. After all we can all tell a circle is a circle, so it can’t be that random. So I eventually came up with a middle ground theory.. Assume 4 patterns with equal probability, there has to be a network configuration that would be stable and have all 4 patterns “memorized” as long as no additional information (additional patterns) are entering the network. I managed to prove there are such states in a 3 then 4 neurons matrix. But in a 4 neuron matrix there is more than one such stable configuration and also there are also states that are stable but not specific, meaning a neuron will learn 2 patterns and another one will learn the remaining 2 patterns in a symmetrical configuration. That was to be expected but still depressing 🙁 . So the new theory is simplistic and incomplete, but I’m sure is the basis for “learning”. Still have to find a way to make learning more stable (perhaps permanent through additional synaptic variable).

I’ve also started treating the neuron more like an atom with electrons, the electrons being the synapses. So I’m actually moving away from the “fitting” hypothesis (and synaptic strength). This theory lead me to believe that there must be “empty” places on a dendrite where there is no synapse, but a synapse could have been there, a forbidden energetic location, used to separate related patterns. So all synapses have defined energies that can be perturbed by input data, but perturbation will still lead to a defined configuration or will just get back to initial state with no change. I just don’t see how a synapse could be a continuous function.

In conclusion I’m still far far away from any meaningful progress 🙁

What is true for a neuron ?

Whatever method we use to extract meaning from data, truth seems to be elusive. Assume two event A and B are 90% correlated, meaning in 90 % of the time A precedes B. Is there a cause and effect between A and B ? What does it mean “precede” ? Day precedes night and summer precedes fall. So without a time frame, we cannot correlate event A and B. Assume the correlation is 100%. Does that mean cause and effect relationship ? We don’t know, maybe we did not pool enough data, or the time frame is wrong…

How is this related to my neuron ? I can’t decide when events are correlated or not, therefore I cannot move the data upwards in the next layer for further processing. How is this problem solved in the biological neuron ? I inferred from the 1960, cat experiment of Torsten Wiesel, that synaptic plasticity must stop in the lower layers of the primary visual cortex , and must stop rather early in life. I also assumed that the same process must happen on upper layers as well, but within a longer time frame. I also found a recent article (Rejuvenating Mouse Brains …) talking about a perineuronal net, which prevents neurons from forming new synapses. This tells me that we decide rather randomly about what truth is, base on available data, up to some time period, and the subsequent data is classified base on this old data, that cannot be changed anymore (or perhaps if extremely difficult to change but not impossible ).

So how does this help me ? I decided that I also need to stop the “learning” process at some point, otherwise data keeps changing resulting in always changing conclusions. But when to stop it ? Hard to say, it depends on the available data. If available data is enough to represent “reality” than I can confidently stop the learning process. But what does it mean “represents reality”… I’m thinking that reality is whatever sensory data is available for analysis, coupled with the environment in which the neuron evolves coupled with some evolutionary programming, objectives such as surviving. This all looks way to philosophical to be of any practical use but perhaps all is statistics underneath..

Decoding firing rates

… and it did not work… I spent close to a month on this and nothing. And this was supposed to be easy. The idea was to have neurons “learn” to respond to certain frequencies and perhaps not to other. Something like this:

Decoding firing rates

It did not work within the model I should say. Because otherwise is just training a variable to a certain value and it’s done. But within my model, I could not find a variable that would fit the purpose and that’s because no variable is truly irreversible. So the two neurons N2 and N3, initially learn pattern 1 respectively pattern 2 but then after couple of trials switching between patterns, both learn just on pattern and do not respond to the other one and if I insist with the pattern with no response then neurons will adapt and respond again to that pattern. Then I started doubting the whole idea of “learning” a certain frequency. Is the learning process for frequency at synapse level? At neuron (body) level ? Add to this the fact that Firing Rates are either faster or slower right at the beginning of a new pattern :

Firing Rates are slower or faster at the beginning of a new pattern, slower in this case.

and there you have it, doubting everything.. I read online to see how the biological neuron adapts to changing frequencies but I found nothing of interest.. vague explanations about Potassium Channels and vague descriptions of “natural” frequencies of a neuron. So I’m giving up on the 1 Neuron applications. With a single neuron I can’t really determine the impact of learning frequencies a certain way .. Also lately I started doubting the “synaptic strength ” interpretation. While that process is real (increase of AMPA receptors in the postsynaptic neuron), I don’t think it plays a role in the learning process. It may play an indirect role such as breaking or forming new synapses but by itself is not “learning”.

What’s next ? I’m going to switch to a 2 by 2 model, see if I can learn more from that.

Adding time to the fray

I’ve resisted adding time for a long time, first because I did not see the need, and second because it seemed like a very complicated variable to deal with.. But now the time has come… it seems.

As I mentioned earlier I have in my mix of variable, an equilibrium process, driving one of these variables (the equivalent to the release of glutamate). I thought I could use cycles to drive the process but it seems it should be the other way around… This variable should drive the cycles…

Why should time be important ? It seems to me that “time” is what gives importance to events. If I don’t assign importance to events, nothing is being learned or everything is. Both options are bad… You may think learning everything is good, but is not, it’s increasing processing time and is preventing generalization, in the end nothing can store or process as much data as we encounter every day..

I decided to go with synaptic plasticity to synchronize neurons… First simulations look good but I’m far from reaching a conclusion. Even with the mess created by the lack of synchronization I was still able to see some encouraging results for the learning mechanism.. the fitting functions powering the neuron are doing an ok job, adjusting all synapses to learn 2 patterns, but I’m not quite there yet. When it first sees a pattern the deviation from equilibrium is big, it adjusts, then I switch patterns.. Eventually I should see no deviations from equilibrium when switching patters. What I see instead is that the deviation is decreasing but it does not go away (as I think it should).. So I still need to adjust the fitting functions some more but I was really happy to see this working as good as it did !!! Here it is, Potential vs Time (not really time but cycles for now), for 2 totally different patters (zero overlap) with a symmetric distribution on only 2 dendrites.

Also the 2 patterns can be easily distinguished observing the firing rates:

Been thinking to take a detour to show case some 1 Neuron nonsense applications 🙂

Simplest network possible, 1 processing Neuron..

Maybe show how I can detect lines of different colors, or detect a hand written letter.. Mind you, the detection of a single letter may sound more cool than it actually is.. You change size or position … and pufff, no recognition… I’ll see… if I can do it in a single day, I’ll do it, if it takes more time… I don’t know..