The Brain's Schrödinger Equation
What made me change my mind isn't a new fact, but a new theory.
Although a large extent of my work is dedicated to modelling the brain, I always thought that this enterprise would remain rather limited in scope. Unlike physics, neuroscience would never create a single, major, simple yet encompassing theory of how the brain works. There would be never be a single "Schrödinger's equation for the brain".
The vast majority of neuroscientists, I believe, share this pessimistic view. The reason is simple: the brain is the outcome of five hundred million years of tinkering. It consists in millions of distinct pieces, each evolved to solve a distinct yet important problem for our survival. Its overall properties result from an unlikely combination of thousands of receptor types, ad-hoc molecular mechanisms, a great variety of categories of neurons and, above all, a million billion connections criss-crossing the white matter in all directions. How could such a jumble be captured by a single mathematical law?
Well, I wouldn't claim that anyone has achieved that yet… but I have changed my mind about the very possibility that such a law might exist.
For many theoretical neuroscientists, it all started twenty five years ago, when John Hopfield made us realize that a network of neurons could operate as an attractor network, driven to optimize an overall energy function which could be designed to accomplish object recognition or memory completion. Then came Geoff Hinton's Boltzmann machine — again, the brain was seen as an optimizing machine that could solve complex probabilistic inferences. Yet both proposals were frameworks rather than laws. Each individual network realization still required the set-up of thousands of ad-hoc connection weights.
Very recently, however, Karl Friston, from UCL in London, has presented two extraordinarily ambitious and demanding papers in which he presents "a theory of cortical responses". Friston's theory rests on a single, amazingly compact premise: the brain optimizes a free energy function. This function measures how closely the brain's internal representation of the world approximates the true state of the real world. From this simple postulate, Friston spins off an enormous variety of predictions: the multiple layers of cortex, the hierarchical organization of cortical areas, their reciprocal connection with distinct feedforward and feedback properties, the existence of adaptation and repetition suppression… even the type of learning rule — Hebb's rule, or the more sophisticated spike-timing dependent plasticity — can bededuced, no longer postulated, from this single overarching law.
The theory fits easily within what has become a major area of research — the Bayesian Brain, or the extent to which brains perform optimal inferences and take optimal decisions based on the rules of probabilistic logic. Alex Pouget, for instance, recently showed how neurons might encode probability distributions of parameters of the outside world, a mechanism that could be usefully harnessed by Fristonian optimization. And the physiologist Mike Shadlen has discovered that some neurons closely approximate the log-likelihood ratio in favor of a motor decision, a key element of Bayesian decision making. My colleagues and I have shown that the resulting random-walk decision process nicely accounts for the duration of a central decision stage, present in all human cognitive tasks, which might correspond to the slow, serial phase in which we consciously commit to a single decision. During non-conscious processing, my proposal is that we also perform Bayesian accumulation of evidence, but without attaining the final commitment stage. Thus, Bayesian theory is bringing us increasingly closer to the holy grail of neuroscience — a theory of consciousness.
Another reason why I am excited about Friston's law is, paradoxically, that it isn't simple. It seems to have just the right level of distance from the raw facts. Much like Schrödinger's equation cannot easily be turned into specific predictions, even for an object as simple as a single hydrogen atom, Friston's theory require heavy mathematical derivations before it ultimately provides useful outcomes. Not that it is inapplicable. On the contrary, it readily applies to motion perception, audio-visual integration, mirror neurons, and thousands of other domains — but in each case, a rather involved calculation is needed.
It will take us years to decide whether Friston's theory is the true inheritor of Helmholtz's view of "perception as inference". What is certain, however, is that neuroscience now has a wealth of beautiful theories that should attract the attention of top-notch mathematicians — we will need them!
