Neural networks and spatial topology

Neuro-mathematician Carina Curto has recently published a fascinating paper, 'What can topology tell us about the neural code?' The centrepiece of the paper is a simple and profound exposition of the method by which the neural networks in animal brains can represent the topology of space.

As Curto reports, neuroscientists have discovered that there are so-called place cells in the hippocampus of rodents which "act as position sensors in space. When an animal is exploring a particular environment, a place cell increases its firing rate as the animal passes through its corresponding place field - that is, the localized region to which the neuron preferentially responds." Furthermore, a network of place cells, each representing a different position, is collectively capable of representing the topology of the environment.

Rather than beginning with the full topological structure of an environmental space X, the approach of such research is to represent the collection of place fields as an open covering, i.e., a collection of open sets $\mathcal{U} = \{U_1,...,U_n \}$ such that $X  = \bigcup_{i=1}^n U_i$. A covering is referred to as a good cover if every non-empty intersection $\bigcap_{i \in \sigma} U_i$ for $\sigma \subseteq \{1,...,n \}$ is contractible. i.e., if it can be continuously deformed to a point.

The elements of the covering, and the finite intersections between them, define the so-called 'nerve' $\mathcal{N(U)}$ of the cover, (the mathematical terminology is coincidental!):

$\mathcal{N(U)} = \{\sigma \subseteq \{1,...,n \}: \bigcap_{i \in \sigma} U_i \neq \emptyset \}$.

The nerve of a covering satisfies the conditions to be a simplicial complex, with each subset $U_i$ corresponding to a vertex, and each non-empty intersection of $k+1$ subsets defining a $k$-simplex of the complex. A simplicial complex inherits a topological structure from the imbedding of the simplices into $\mathbb{R}^n$, hence the covering defines a topology. And crucially, the following lemma applies:

Nerve lemma: Let $\mathcal{U}$ be a good cover of X. Then $\mathcal{N(U)}$ is homotopy equivalent to X. In particular, $\mathcal{N(U)}$ and X have exactly the same homology groups.

The homology (and homotopy) of a topological space provides a group-theoretic means of characterising the topology. Homology, however, provides a weaker, more coarse-grained level of classification than topology as such. Homeomorphic topologies must possess the same homology (thus, spaces with different homology must be topologically distinct), but conversely, a pair of topologies with the same homology need not be homeomorphic. 

Now, different firing patterns of the neurons in a network of hippocampal place cells correspond to different elements of the nerve which represents the corresponding place field. The simultaneous firing of $k$ neurons, $\sigma \subseteq \{1,...,n \}$, corresponds to the non-empty intersection $\bigcap_{i \in \sigma} U_i \neq \emptyset$ between the corresponding $k$ elements of the covering. Hence, the homological topology of a region of space is represented by the different possible firing patterns of a collection of neurons.

As Curto explains, "if we were eavesdropping on the activity of a population of place cells as the animal fully explored its environment, then by finding which subsets of neurons co-fire, we could, in principle, estimate $\mathcal{N(U)}$, even if the place fields themselves were unknown. [The nerve lemma] tells us that the homology of the simplicial complex $\mathcal{N(U)}$ precisely matches the homology of the environment X. The place cell code thus naturally reflects the topology of the represented space."

This entails the need to issue a qualification to a subsection of my 2005 paper, 'Universe creation on a computer'. This paper was concerned with computer representations of the physical world, and attempted to place these in context with the following general definition:

A representation is a mapping $f$ which specifies a correspondence between a represented thing and the thing which represents it. An object, or the state of an object, can be represented in two different ways:

$1$. A structured object/state $M$ serves as the domain of a mapping $f: M \rightarrow f(M)$ which defines the representation. The range of the mapping, $f(M)$, is also a structured entity, and the mapping $f$ is a homomorphism with respect to some level of structure possessed by $M$ and $f(M)$.

$2$. An object/state serves as an element $x \in M$ in the domain of a mapping $f: M \rightarrow f(M)$ which defines the representation. 

The representation of a Formula One car by a wind-tunnel model is an example of type-$1$ representation: there is an approximate homothetic isomorphism, (a transformation which changes only the scale factor), from the exterior surface of the model to the exterior surface of a Formula One car. As an alternative example, the famous map of the London Underground preserves the topology, but not the geometry, of the semi-subterranean public transport network. Hence in this case, there is a homeomorphic isomorphism.

Type-$2$ representation has two sub-classes: the mapping $f: M \rightarrow f(M)$ can be defined by either (2a) an objective, causal physical process, or by ($2$b) the decisions of cognitive systems.

As an example of type-$2$b representation, in computer engineering there are different conventions, such as ASCII and EBCDIC, for representing linguistic characters with the states of the bytes in computer memory. In the ASCII convention, 0100000 represents the symbol '@', whereas in EBCDIC it represents a space ' '. Neither relationship between linguistic characters and the states of computer memory exists objectively. In particular, the relationship does not exist independently of the interpretative decisions made by the operating system of a computer.

In 2005, I wrote that "the primary example of type-$2$a representation is the representation of the external world by brain states. Taking the example of visual perception, there is no homomorphism between the spatial geometry of an individual's visual field, and the state of the neuronal network in that part of the brain which deals with vision. However, the correspondence between brain states and the external world is not an arbitrary mapping. It is a correspondence defined by a causal physical process involving photons of light, the human eye, the retina, and the human brain. The correspondence exists independently of human decision-making."

The theorems and empirical research expounded in Curto's paper demonstrate very clearly that whilst there might not be a geometrical isometry between the spatial geometry of one's visual field and the state of a subsystem in the brain, there are, at the very least, isomorphisms between the homological topology of regions in one's environment and the state of neural subsystems.

On a cautionary note, this result should be treated as merely illustrative of the representational mechanisms employed by biological brains. One would expect that a cognitive system which has evolved by natural selection will have developed a confusing array of different techniques to represent the geometry and topology of the external world.

Nevertheless, the result is profound because it ultimately explains how you can hold a world inside your own head.

Trump and Brexit

One of the strangest things about most scientists and academics, and, indeed, most educated middle-class people in developed countries, is their inability to adopt a scientific approach to their own political and ethical beliefs.

Such beliefs are not acquired as a consequence of growing rationality or progress. Rather, they are part of what defines the identity of a particular human tribe. A particular bundle of shared ideas is acquired as a result of chance, operating in tandem with the same positive feedback processes which drive all trends and fashions in human society. Alex Pentland, MIT academic and author of 'Social Physics', concisely summarises the situation as follows:

"A community with members who actively engage with each other creates a group with shared, integrated habits and beliefs...most of our public beliefs and habits are learned by observing the attitudes, actions and outcomes of peers, rather than by logic or argument," (p25, Being Human, NewScientistCollection, 2015).

So it continues to be somewhat surprising that so many scientists and academics, not to mention writers, journalists, and the judiciary, continue to regard their own particular bundle of political and ethical ideas, as in some sense, 'progressive', or objectively true.

Never has this been more apparent than in the response to Britain's decision to leave the European Union, and America's decision to elect Donald Trump. Those who voted in favour of these respective decisions have been variously denigrated as stupid people, working class people, angry white men, racists, and sexists.

To take one example of the genre, John Horgan has written an article on the Scientific American website which details the objective statistical indicators of human progress over hundreds of years. At the conclusion of this article he asserts that Trump's election "reveals that many Americans feel threatened by progress, especially rights for women and minorities."

There are three propositions implicit in Horgan's statement: (i) The political and ethical ideas represented by the US Democratic party are those which can be objectively equated with measurable progress; (ii) Those who voted against such ideas are sexist; (iii) Those who voted against such ideas are racist.

The accusation that those who voted for Trump feel threatened by equal rights for women is especially puzzling. As many political analysts have noted, 42% of those who voted for Trump were female, which, if Horgan is to be believed, was equivalent to turkeys voting for Christmas.

It doesn't say much for Horgan's view of women that he thinks so many millions of them could vote against equal rights for women. Unless, of course, people largely tend to form political beliefs, and vote, according to patterns determined by the social groups to which they belong, rather than on the basis of evidence and reason. A principle which would, unfortunately, fatally undermine Horgan's conviction that one of those bundles of ethical and political beliefs represents an objective form of progress.

In the course of his article, Horgan defines a democracy "as a society in which women can vote," and also, as an indicator of progress, points to the fact that homosexuality was a crime when he was a kid. These are two important points to consider when we turn from the issue of Trump to Brexit, and consider the problem of immigration. The past decades have seen the large-scale migration of people into Britain who are enemies of the open society: these are people who reject equal rights for women, and people who consider homosexuality to be a crime.

So the question is as follows: Do you permit the migration of people into your country who oppose the open society, or do you prohibit it?

If you believe that equal rights for women and the non-persecution of homosexuals are objective indicators of progress, then do you permit or prohibit the migration of people into your country who oppose such progress?

It's a well-defined, straightforward question for the academics, the writers, the journalists, the judiciary, and indeed for all those who believe in objective political and ethical progress. It's a question which requires a decision, not merely an admission of complexity or difficulty.

Now combine that question with the following European Union policy: "Access to the European single market requires the free migration of labour between participating countries."

Hence, Brexit.

What unites Brexit and Trump is that both events are a measure of the current relative size of different tribes, under external perturbations such as immigration. It's not about progress, rationality, reactionary forces, conspiracies or conservatism. Those are merely the delusional stories each tribe spins as part of its attempts to maintain internal cohesion and bolster its size. It's more about gaining and retaining membership of particular social groups, and that requires subscription to a bundle of political and ethical ideas.

However, the thing about democracy is that it doesn't require the academics, the writers, the journalists, the judiciary, and other middle-class elites to understand any of this. They just need to lose.