Planes, Brains and Automobiles

At a recent talk, Iain McGilchrist implied that as the human brain had evolved larger and larger, the number of connections in the corpus callosum that connects the left and right hemispheres had decreased in absolute terms. His book, ‘The Master and His Emissary’, clarifies – it is only a decrease in relative terms – presumably relative to brain mass, volume or number of neurons. So a corpus callosum of only about 300 million fibres means that only 2% of a brain’s neurons are connected to it. 2% doesn’t sound much, but I’d want to ask: what percentage would you expect?

 

Scaling laws

An aside: Consider airships and aeroplanes. If you double the length of them, their surface area goes up four-fold and their volume goes up eight-fold. In crude terms,

  • lift is proportional to surface area (of the wings) of the plane and proportional to volume of the (lighter-than-air gas within the) airship.
  • weight is proportional to volume of the plane but proportional to the surface area of the airship.

So the lift-to-weight ratio goes up two-fold for an airship but goes down two-fold for an plane. Hence, to get airships to fly, it’s easier if they’re big. And it’s easier to get model planes to fly than real passenger-carrying ones. The point here: different attributes scale at different rates.

How should the number of connections in the corpus callosum vary with brain size?

 

Comparing brains with cities

A recent Scientific American blog post highlighted the 2009 Changizi and Destefani paper (see preprint) showing how the scaling of cities matches well with that of the mammalian neocortex.

A city’s network of freeways and exits is analogous to a brain’s network of neurons and synapses. Changizi and Destefani showed that empirical scaling laws applicable to brains also apply to cities (in the U.S, at least):

ΔN is proportional to ΔS^(0.75)

ΔX is proportional to ΔS^(1.125)

where ΔN is the increase in number of neurons (or freeways), ΔX is the increase in the number of synapses (or exits) and ΔS is the increase in the surface area of the neocortex (or area of the city).

 

Rent’s rule

Surprisingly, the city/brain scaling paper doesn’t mention Rent’s Rule which relates (empirically) the number of terminals (pins) T of an silicon chip to the number of elements g (gates) within it. The equation is remarkably similar:

T is proportional to g^p

where p=0.5 for highly-ordered circuits such as a memory array and p=0.75 for random logic. For comparing against a brain, p=0.75 is appropriate – and this matches the brain and city scaling equations described above.

Maybe it is not surprising that, in the most abstract sense, arcs connecting nodes in 3-D space are subject to similar scaling laws regardless of what those arcs and nodes actually are.

Actually, in all 3 cases, we are not dealing with 3-D objects but 2.5-D objects. If you want to get from the 3rd floor of one building to the 2nd floor of another, you normally have to go via ground-level. Silicon chips are planar with only a few layers of metal connecting together transistors in the substrate. The neocortex is also plane, folded up in 3 dimensions to give the brain its distinctive wrinkles.

And maybe we shouldn’t make too much of this. After all, on a log-log scale, you can fit almost anything to anything else. Small variations in the constant exponent lead to large differences in absolute values.

 

How big should the human Corpus Callosum be?

Let us compare the human brain against that of that favourite of research animals, the Rhesus monkey:

Rhesus monkey: 97g brain mass and 45M fibres in the corpus callosum.

Human: 1300g brain mass; 300M fibres in the corpus callosum.

From monkey to human, brain mass is increased 13-fold but the number of connecting fibres has only increased by a factor of about 6. Applying Rent’s Rule to a hemisphere, an increase in mass (approximately equivalent to surface area) from 49g to 650g (ignoring the sub-cortical brain and any increase of the thickness of the neocortex) would expect the number of fibres in the corpus callosum to increase 7-fold – a fairly good match.

 

Re-engineering interconnectivity

Silicon chips are made in an extremely precisely controlled environment – but it is not perfect; the result is that we can’t make big chips with an adequate yield. Big systems must be built from many smaller chips. Rent’s rule causes congestion in electronics systems – we don’t have enough chip pins to connect the individual chips as we would like. The other main empirical law of electronics is of course Moore’s Law – the chip’s transistors will halve in size every 18 months or thereabouts. So the number of terminals (chip pins) needs to increase for a given area of silicon. Technology has not allowed the pins to shrink at anywhere near the same rate as Moore’s Law has shrunk the transistors. With the scaling laws against us, we must use design innovation. An example of such innovation is the now widespread use of SERDES (serializer-deserializers – high-speed time-multiplexed I/O). For example, we can get 8Gbps (Gigabits per second) of data across a PCI Express (Gen3) pair of differential pairs instead of, say, 4 x 200MHz = 0.8Gbps bandwidth with normal I/O – a 10-fold increase.

Similarly, there are presumably ontogenetic reasons why the brains of all mammals develop 2 hemispheres and we are genetically stuck with it. Evolution cannot rework this ‘fact’ but it can evolve ‘innovations’ such as myelination – the sheathing of the neuron’s axon to allow faster signalling. Myelin is what makes the ‘white matter’ (as opposed to ‘grey matter’) white. The Corpus callosum is the largest white matter structure in the brain. Conceivably, the communications bandwidth between the hemispheres is thereby increased.

 

Postscript

One of about every 4000 people are born without a corpus callosum — a condition called ‘agenesis of the corpus callosum’ (AgCC). Whilst there can be many resulting behavioural disorders such as autism, it is surprising that such a radical restructuring of the brain can lead to some such people having normal IQs.

Researchers at Caltech have shown that left and right hemispheres’ activity in AgCC patients is coordinated in the same way as normal people: “How do they manage to have normal communication between the left and right sides of the brain without the corpus callosum?”

A Kurzweil blog highlights a rather ambitious paper by Johnjoe McFadden (University of Surrey) which proposes an answer to that question: that electromagnetic fields permit communication between the hemispheres.

Looking for an analogy with cities, cities often evolve around a river crossing. The river divides the city, providing only a few places for traffic to cross. A major solution to overcoming this communications bottleneck – the mobile phone!

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