5th HLF – Lecture: Where Computer Science Meets Neuroscience
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Transcript: English(auto-generated)
00:00
And it's my great pleasure and honor to introduce the next speaker, is Leslie Weiland. And he received the Navalina Prize in 1986 for his work on complexity theory, in particular for finding problems which are easy to show solutions,
00:24
but to count the solutions is very difficult. And later in 2010, he received also the Turing Award for his, partly because of his work on approximately correct learning problems.
00:43
And now he will talk about where computer science meets neuroscience and we're looking forward to this interesting topic. So it's a great pleasure to be here again.
01:05
And I do have to again thank the organizers for making this possible for all of us. So as the title suggests, I'm really interested in the brain, in the real brain. And I'm also interested in explanations which computer science offers,
01:21
algorithms, concrete explanations of states, change of states, the next step. So in some sense, while many have remarked here that the boundaries between mathematics and computer science are pretty hard to discern now, so I'm hoping in the future the boundaries between biology and computer science will be hard to discern also, because once one gets mechanistic explanations
01:45
to the precision of algorithms, then the two are the same field. So just to position what I'm going to say, so I'll start by saying what I'm not going to say. So this talk isn't about machine learning or inductive learning.
02:06
And that's not because I don't think that's a good field. I think the question of inductive learning is a secret of many things and I have worked a lot on that. But in fact, while working on that, I thought that I would have some advantage
02:22
to understanding how the brain works. But then when I started to think about the brain, I discovered that it wasn't very helpful, because there are some much more basic unresolved problems about the brain than that. So there are things about the brain which are so simple that no theories of generalization can help you.
02:43
And to be specific, so we've all come here and we've all met some new people. So the question is, how have our brains changed to memorize these new people we've met? So maybe you remember five people from here. But for each person, will you assign one neuron or ten neurons
03:04
or ten million neurons? And so amazingly, no one has the foggiest answer to this question. So I think one neuron, a few people believe it's one neuron. But it could be anywhere otherwise in that range.
03:22
And it's a kind of rather astonishing state of ignorance. So I suppose summarizing my talk very simply, so the most obvious fact about the brain, which is different from other devices we have, is that as far as we know, it's got no addressing. So in a computer, you can go to address seven.
03:42
In fact, on the internet, you can also go to address seven. But as far as we know, in the brain, there's no such thing. So how do we store things? And of course, that's compounded with the fact that the brain does many more things than just memorize. And I'll go through that. And so to make a scientific problem out of this,
04:01
we have to, or I try to make a maximally complex problem out of it, something which is difficult to realize and show that that constrains what the solutions may be. And of course, when I say how do we memorize the five people, we'll remember from here.
04:20
The problem I'm really implying is that every week, for all your life, you've got this capability. We can keep putting more memories in our brains. And there's some interference with previous memories, but the amazing thing is that it's not that bad. So just the quantity of information which we can realize
04:42
in a certain way is how we build our constrained system. So just more broadly, so what impresses me about the human brain, so what impresses me is symbolic processing. So I can tell you a story. I know what green means. I mean, I know what fox means. So now I tell you that in a certain place,
05:01
there's a restaurant called the Green Fox. You can handle this information. I went there. It was a bad choice, okay? So you can handle this information, too. It was recommended by Joe. I was there when I heard the election results. So I can go and carry on telling you a long story
05:20
of relationships and facts and combinations, and you can recall this in your brain for a lifetime. So this is what I'll kind of make into the difficult task, which we have to explain. And, of course, psychologists have studied this also, as we heard earlier in the week. So the idea of chunking is important for psychologists,
05:40
which is the idea that you can combine two things into more complex things, and the more complex things becomes as simple as the original simple things. So, for example, in mathematics, we learn more and more complicated definitions. They pile up on each other, but at the end, if you're an expert, then the compound, the definition,
06:00
becomes as simple for you as the original ones. So that's a very basic factor of cognition. And so, as I said, what's impressive is not that you can do this once, but you can do this for hundreds of thousands of such tasks. And basically, there's no theory of this, which even in principle, everyone explains is right.
06:24
So if there's anyone here looking for big open problems, there are big open problems here. So I'll describe to you my journey through this space. So I'll just describe seven requirements, which are my general description
06:41
of what we do in cognitive processes we have to acknowledge. So one is that we can do one-shot learning. I say something once to you, you remember it. It's not like typical machine learning where you have to retrain a million times. We have to account for that. We can do a large number of tasks.
07:01
Not quite clear what number that should be. Maybe it should be ten to the sixth. I'm not sure. So a bit more technical. So what is learning a task? So a typical task would be like an association. And so what I am working towards is the idea
07:22
that it's not that you store some information and then when you use it, some processor comes along and interprets it, but that just for efficiency, you'll modify a part of your brain to make a sub-circuit which does something efficiently.
07:41
So and one requirement is that the sub-circuits should be able to compose themselves on the fly. That if you give me a name of someone, I can form an association and then make a deductive association in a very, very fast chain of associations and react fast. So in some sense, to relate this to Manuel's talk,
08:02
so there's always in any theory of cognition, there's a kind of a mind's eye, the stage, and then there's the rest. And so these sub-circuits would be in the rest and I'm not only interested in some general Turing machine like theory, I'm interested in how the brain actually works
08:21
and so I have to be pretty efficient. So the sub-circuits will live in the rest so it can communicate with your mind's eye, your working memory, efficiently enough. Okay, so the main thing is this will be quantitative that we are short of resources in our heads.
08:42
We have a finite number of neurons, a finite number of connections, and the connection strengths which I'll come to are also very important and these are all pressures, they take up volume and we have to account for why quantitatively we have enough for what we need.
09:02
Also there's some architecture involved, so one very simple issue is that the connections in your brain, the axons, are one directional, so theory has to respect that. And lastly, so much work traditionally in this theoretical neuroscience area
09:20
has been to study one task at a time. So these, for example, these Hopfield nets, they address one particular task. So here I'm interested in trying to address a set of tasks which among them may possibly form a basis for building high-level cognitive functions.
09:40
So just doing one task and not explaining why that leads anywhere is somehow not enough. So I'll say more about that. Okay, so just to summarize quickly what we have to know about the brain. So there's no addressing mechanism, it's slow, maybe in a hundred steps you can do amazing things.
10:01
It's sparsely connected, most neurons, but all neurons are connected to a small fraction of the others. But there are some simplifications, so long-distance communication is very stylized. There's something very simple, essentially it's one kind of communication, these spikes. And what I'll deal with are quantitative resources, so I'll need four parameters.
10:22
N is the number of neurons, D is the average connections from each neuron to every other. I'll have random graphs. The third physical parameter is the strength of a synapse, and so I'll come back to that. Okay, okay, so this is it. So there's some things we don't know about the brain,
10:43
and these will be parameters for me. And most important for me, which I don't know, is how strong synapses are. So how strong an influence does one neuron have on another? So for example, if we could have a very democratic society where for any neuron to do something,
11:00
10,000 others would have to agree and contribute a little bit, or it could be much more Boolean where one neuron can tell the other what to do. So K is this parameter. So if K is large, then that means that if K is 100, then it means that you want 100 neurons to be active before the neuron they're connected to is made active.
11:24
Whereas K equals one is a very Boolean world, where a single neuron has a very strong synapse, a neuron can fire because of a single neuron causing it to fire. And in general, strong neurons are more powerful than weak ones, as we'd expect.
11:41
And no doubt, throughout the brain, the brain is also a variety of things going on, so there isn't a single measurement anyone can take which gives all the answers. Just two other comments. So many biologists emphasize that there are many different kinds of neurons and even of synapses,
12:01
and so my interpretation is simply that these are all executing the different algorithms, and they're necessary. But this also implies that it's a bit naive to say that, you know, ask the biologists what the real neuron is like, because the answer is that they come in all kinds of variety, and for good reason.
12:21
And then the other comment is that because of the sparse connections and on the average weak synapses, the brain is a communication challenge, like most prior computers, and for that reason, that's the real obstacle we have to fight against,
12:41
and the advantage of that is that the actual computational power of the neurons isn't so critical, so we don't have to understand that fully before we can make some conclusions. So even if you put a supercomputer at each, instead of each of your neurons, your brain will still be very limited because of the sparse connections.
13:03
Okay, so as far as the general strategy, so again, I'm saying this is a problem, it's not a problem where there are thousands of competing theories, we just have to figure out which one is right, it's a problem where there's no viable theories, so how do we proceed?
13:21
Okay, so I've got some model, I'll just sketch it, it'll underestimate the power of the brain, so I don't want anyone to complain that the brain can't do this, if the brain can do more, that's fine. I do want it to do several tasks, so it's like in between them, they should account for a possible basis of cognition.
13:43
I want to show that this simple model can execute my interesting tasks, and to make it quantitative, I really emphasize that I want to show that this model can execute large numbers of these tasks in succession,
14:04
so just doing a single memorization doesn't and shouldn't impress anyone, but doing numbers comparable to what humans can do, then maybe that should. Okay, so this is what I'm interested in. Okay, so now I have to describe to you some tasks, so what are interesting tasks?
14:23
So I've got two basic tasks I want to describe today, so first there has to be a task which assigns storage, how do you put a new item in, how do you store your new friend? So who's your new friend? So the first time you heard of Donald Trump, I'm using this only because I expect most of you have heard of him.
14:45
So what happens? Okay, so this will be an instance of hierarchical memorization, and it will have a definition, and the definition is that you assume that you already got two items, and they could be parts of his name,
15:01
so maybe you already have heard of the word Donald and Trump, or it could be parts of his appearance, it could be anything, it just depends, the first time you memorized this concept, it depends what you are doing, let's suppose A is Donald and B is Trump, then the task which your brain has to do is to allocate neurons
15:22
to a new item C, which will be this Donald Trump, and you'll change some synaptic weights, so that in the future when A and B are both active, then it will cause C to be active also. So you have to find some storage for C,
15:41
so this is again chunking, and you have to have some operational capability that in the future, under what conditions do you wake up the C? It's when you find A and B. It's a definition. Okay, so this just reflects what psychologists call chunking.
16:05
So here I haven't assumed anything about the brain, but I have, and the basic assumption is that there's some discreteness. So I'm associating a concept like Donald with a set of neurons. So this is in contrast with some sort of holographic representation
16:23
across the brain where you can't locate it. So that's an assumption of the model. So there is some evidence for this, so these are some famous experiments done, which became feasible about 15 years ago, these are done on humans with electrodes in a certain part of the brain,
16:42
the hippocampus, and these patients had electrodes in certain neurons, and they were shown pictures on a laptop, and they were shown lots of pictures, several of which were of Bill Clinton, and this particular, so this shows that this neuron, I think these columns are the responses,
17:03
electrical responses of this one particular neuron. This is a one-second interval, and so it turned out that this is all from the same neuron, all the pictures, all the parts of this thing are from the same neuron, the patient being seen in different pictures, and this neuron, whenever it was shown Bill Clinton in various positions,
17:23
sometimes even just his name, it responded rather heavily, whereas if shown different things like different American presidents or something similar, it didn't. Okay, so this is some evidence of locality, what's surprising is that these were fairly surprisingly easy to find,
17:42
so the rough calculation is that maybe in this part of the cortex, if you were looking for Bill Clinton cells, that maybe one in a thousand were Bill Clinton cells. Of course, these results have to be treated with caution, because if you poke around in a computer, most likely when you see activity, it's not where the object is stored,
18:03
but some information in transit. But there's very strong evidence that this discreteness is a characteristic cortex. So the other relationship I want to deal with is, so once you've assigned neurons to your various concepts,
18:24
then you want to describe relationships. So in this task, no allocation of new neurons, you just form relationships between them. So this will be the task of association,
18:42
and again, in my definition for association, you already know the concept Donald Trump, if you have a concept for elected president, these already exist in your brain, but you want to change your brain so that in the future, when you think of Donald Trump, you know he's been elected president. Again, similar definition,
19:00
you've got two sets of neurons representing this and this, and you want to change the weight so that whenever this is active, you'll be reminded that he's president. So this second task does have a long history, so Will Shaw, back in the 60s,
19:20
investigated nets with this particular kind of association. So the word association is used in a hundred different senses. This is a very particular one, and it's the same as Will Shaw. So okay, so I do need a model. So very roughly, it's like a,
19:42
you can start off with McCulloch pits in 1943, but you want to make it more programmable, expressing timing and things like that, and so I do have a model where these numbers are important to me, numbers of neurons, strength of synapses, and one makes light decisions about what can depend on what.
20:04
Okay, I do have a model. I won't spend more time on that. So the fourth parameter in this is how many neurons I use for my friend at HLF, and I call that R.
20:23
And in general, of course, with Bill Clinton, the idea that you can easily find a neuron which corresponds to Bill Clinton, at least a certain part of the brain, means that in that part of the brain, R is large. So R could be a million, 10 million. If R were 10, you wouldn't find the Bill Clinton cell.
20:46
And then the other issue is, you know, are the Bill Clinton cells really disjoined from the George Bush cells? So that would be disjoined. Are they shared? So there's a lot of evidence, at least from these kind of measurements,
21:00
is that they're shared, that there are neurons which respond to Bill Clinton and the Eiffel Tower. So it's almost like random sets for each concept, which overlap randomly. Okay, so basically I've told you everything that needs to be said. So I've got a task, like association.
21:24
I've got, you're born with a brain. You've got a certain number of neurons, a certain number of connections. I do assume that the connections are random. One can say more about what kind of random. And then we've got our parameters.
21:40
We've got random graph, n neurons, a probability of an edge being present. We're going to devote a set of r neurons to Donald Trump and to elect a president. And then we've got this synaptic strength. And then this defines what the brain needs to do, because what the brain needs to do, at least,
22:02
is that when this set of neurons fire, then you want all of these to fire. And what this means is that each of the target neurons has to have at least k connections to these neurons. So of course this is a sparse graph.
22:20
So the a could be a million, the c could be a million, k could be 20. And all that means is that for this cell, there should be at least 20 connections from the million Donald Trump neurons. Okay. So these four, you've got a random graph
22:41
with four parameters, and there are some relationships which permit this kind of task to be completed. So the main constraint here is that the random graphs should support the capability you want, actually changing the weight of the synapses to do the right thing.
23:01
That's usually easy. So that's one task, and you can analyze it. You know, you don't have to do very much. So this is Bernoulli distribution. You have to know how random graphs work. The difficulty in analysis is that, I mean, if you just had one task to do,
23:22
then you're born with your fresh random brain, and you can exploit the randomness to do whatever you like with high probability. But once you've memorized 10,000 different things, then it's not random anymore. There's lots of dependencies,
23:41
and proving that your brain can still do this large number of things, even when it's got these previous memories is what gets challenging. So the philosophy is, you're born with your random network, and then you've got these experiences which are arbitrary. So you have to be able to memorize
24:01
and deal with cognitive concepts which the world throws at you. So it's a kind of worst-case world, which your initial random graph has to be able to face. Okay, so, and hierarchical memorization, again, is a very similar graph theory problem,
24:22
just different details. Okay, so the actual conditions are a bit more complex, so the difficulties are that, well, you know, when do you declare a cell to be a Bill Clinton cell? To make this kind of theory work, you have to ensure the system where,
24:45
right, so when 95% of my Bill Clinton cells fire, I'll say, you're thinking of Bill Clinton. When less than 30%, then you're not thinking of Bill Clinton. And I want to make sure the whole system is such that you're never in between.
25:01
So the system has some polarity that you know which concepts you're representing. Okay, and it's the big, it's a random phenomenon of a lot of random events which makes that possible. So one way of investigating this is by computer simulations.
25:22
In fact, that's a better one if you know the parameters because you get the exact answer. It's a random graph, so you just do lots of experiments. So this we did some years ago with Harley Feldman, and we found that, okay, so we had several operations, including the two which I mentioned,
25:41
and we fired these operations at this random graph and figured out at what capacity did it fill up. Okay, so eventually as you add more and more associations and other tasks, the thing breaks down. Okay, you start forgetting your earlier memories.
26:02
Okay, so one, we had two regimes. I'll quickly mention that where it worked. So we had systems with, it could be a small part of your brain with 10 to the 8 neurons, a number of connections per neuron, which is kind of realistic in some animals.
26:20
So K, as I said, K is a parameter. Different parts of the brain may use different Ks. And for these three numbers, we found that, in fact, the number of neurons you have to use to allocate the concept was 360,000, so it's a pretty big fraction of the 100 million.
26:40
And then we could support about 3,000 of these actions before behavior went subpar for our requirements. Okay, and another regime is better, and with the other regime we used, the main thing is we used strong synapses, and then we could do more human-like performance.
27:07
Okay, so the basic difference between the two regimes was that the first regime was just, as I said, you only looked at direct connections with the regime where you had a higher capacity.
27:22
You had an intermediate set of neurons so that the weights of these neurons were large. Yes, yes, yes, a small K at these neurons. So these are the things, so these are two regimes where you could do useful things
27:41
with parameters which the brain may have in this subject of investigation. And you can also prove some theorems, so if you want at least a second of a definition. So you can prove, so an association is that when this set fires, you want this set to fire, and when this set fires, you want this set to fire,
28:02
so C is the capacity, maybe C is 100,000, and the more exact definition is that, yes, when XI fires, you want all of YI to fire, but you want very little noise. You want about zero of the neurons to fire. Okay, so besides your positive intention,
28:22
you also want to make sure that there are no mistakes. And you can ask questions about what the capacity is, and you can prove theorems which track how close you can get to various ideal capacities. And one can do quite a tight analysis at least asymptotically.
28:44
Okay, so I won't give you the details of that, but I'll just finish by saying that what's special now is that there's some hope on the horizon that some of these theories can be tested. And the kind of experiments which were tested, I'd like to call in-circuit testing,
29:02
which I think is some old-fashioned word from electric engineering, where you've got some old-fashioned circuit board, and you put probes on the different components to test the thing locally, as opposed to testing the whole board as input-output. Okay, so what this would mean is that you've got some animal,
29:23
and what the hypothesis says is that if you put electrode into some random set of the right number of neurons, another random set of another right number of neurons, then if your brain has the capacity to do associations,
29:42
then by some sort of protocol of firing this set and firing this set, you can teach it so that in the future, whenever you fire just these, then these will fire. Okay, so this is a possible training thing you could do on an animal. So it's a bit strange for neuroscientists,
30:01
because neuroscientists like experiments where there's some natural input or natural output, but that's confusing. I want to do the simplest thing like this. And those are some technical difficulties, like for these neurons, when you're training and testing, you're both stimulating them and recording from them,
30:21
and that's quite kind of state-of-the-art. So these experiments are kind of just becoming feasible, so some chance of figuring out whether this capability brains have. Of course, there are quite a few parameters we don't know, like we don't know how big R is, so if R is a million,
30:42
it may be difficult to test pairs of millions of neurons. Okay, so again, just to finish, so thank you. Okay, so thank you.