Modular complexity, and reverse engineering the brain

The Forbes columnist Matthew Herper has a profile of Microsoft co-founder Paul Allen who has placed his bets on a brain institute whose goal is to to map the brain…or at least the visual cortex. His institute is engaged in charting the sum total of neurons and other working parts of the visual cortex and then mapping their connections. Allen is not alone in doing this; there’s projects like the Connectome at MIT which are trying to do the same thing (and the project’s leader Sebastian Seung has written an excellent book about it) .

Well, we have heard echoes of reverse engineered brains from more eccentric sources before, but fortunately Allen is one of those who does not believe that the singularity is near. He also seems to have entrusted his vision to sane minds. His institute’s chief science officer is Christof Koch, former professor at Caltech and longtime collaborator of the late Francis Crick who started at the institute this year. Just last month Koch penned a perspective in Science which points out the staggering challenge of understanding the connections between all the components of the brain; the “neural interactome” if you will. The article is worth reading if you want to get an idea of how simple numerical arguments illuminate the sheer magnitude of mapping out the neurons, cells and proteins that make up the wonder that’s the human brain.

Koch starts by pointing out that calculating the interactions between all the components in the brain is not the same as computing the interactions between all atoms of an ideal gas since the interactions are between different kinds of entities and are therefore not identical. Instead, he proposes, we have to use something called Bell’s number B(n) which reminds me of the partitions that I learnt when I was sleepwalking through set theory in college. Briefly for n objects, B(n) refers to the number of combinations (doubles, triples, quadruples etc.) that can be formed. Thus, when n=3 B(n) is 5. Not surprisingly, Bn scales exponentially with n and Koch points out that B(10) is already 115,975. If we think of a typical presynaptic terminal with its 1000 proteins or so, B(n) already starts giving us heartburn. For something like the visual cortex where n= 2 million B(n) would be prohibitive. And as the graph demonstrates, for more than 10^5 components or so the amount of time spirals out of hand at warp speed. Koch then uses a simple calculation based on Moore’s law in trying to estimate the time needed for “sequencing” these interactions. For n = 2 million the time would be of the order of 10 million years.

And this considers only the 2 million neurons in the visual cortex; it doesn’t even consider the proteins and cells which might interact with the neurons on an individual basis. Looks like we can rapidly see the outlines of what Allen himself has called the “complexity brake”. And this one seems poised to make an asteroid-sized impact.

So are we doomed in trying to understand the brain, consciousness and the whole works? Not necessarily, argues Koch. He gives the example of electronic circuits where individual components are grouped separately into modules. If you bunch a number of interacting entities together and form a separate module, then the complexity of the problem reduces since you now have to only calculate interactions between modules. The key question then is, is the brain modular? Commonsense would have us think it is, but it is far from clear how we can exactly define the modules. We would also need a sense of the minimal number of modules to calculate interactions between them. This work is going to need a long time (hopefully not as long as that for B(2 million) and I don’t think we are going to have an exhaustive list of the minimal number of modules in the brain any time soon, especially since these are going to be composed of different kinds of components and not just one kind.

Any attempt to define these modules are going to run into problems of emergent complexity that I have occasionally written about. Two neurons plus one protein might be different from two neurons plus two proteins in unanticipated ways. Nevertheless this goal seems far more attainable in principle than calculating every individual interaction, and that’s probably the reason Koch left Caltech to join the Allen Institute in spite of the pessimistic calculation above. If we can ever get a sense of the modular structure of the brain, we may have at least a fighting chance to map out the whole neural interactome. I am not holding my breath too hard, but my ears will be wide open.

Image source: Science magazine