219 — The connection-set algebra — a novel formalism for the representation of connectivity structure in neuronal network models.

In this work, Mikael Djurfeldt puts forward a common language for conveying the connectivity patterns of neurons in a neuronal network, a terminology that I like a lot for describing neural networks in a biological setting (versus a computer science one), but which I think didn’t really catch on except in the community that actively discusses both neurons and machine learning on a daily basis WHICH OUGHT TO BE EVERYONE but I digress

The adjacency matrix is a common notation for directed graph connectivity: If node $i$ has an edge to node $j$, then $M(i, j) = \mathcal{T}$. Otherwise, $M(i, j) = \mathcal{F}$, where $\mathcal{T}$ and $\mathcal{F}$ are the boolean “true” and “false” values, or, in matrix-talk, integers 1 and 0.

This is obviously extensible to neuronal connectivity: If neuron $i$ transmits data to neuron $j$…

Djurfeldt proposes a formal algebra for dealing with neuronal networks, which uses a few matrix primitives to express complex concepts succinctly and consistently.

For example, here is a network where every neuron is connected to every other neuron except itself:

\[\bar{\Omega} - \bar{\delta}\]

A neuronal network where every neuron connects to its peers with a probability dependent on their distance could be represented as:

\[(\rho\mathcal{D})(i, j) = X_{ij}\]

…where $\mathcal{D}$ is some distribution.

Though this is still a very simple formulation, we can now begin to play with the mathematic truths that arise from these definitions. For example, we can set:

\[\bar{\delta} = \mathcal{D} \textit{ where } \mathcal{D} = \{^{\mathcal{T} \text{ if distance is 0}}_{\mathcal{F} \text{ otherwise}}\]

Now these formulas are interchangeable and anything that we learn about a probablistic connectivity set we can now apply to our Diagonal pattern.

This feels like a very useful grammar to define, because we can use it to learn about connection sets in neuronal networks that are too large to simulate explicitly. I can see this sort of mathematical reduction being useful for reducing complex networks to tractable ones: Djurfeldt explains that this can even be used to partition a network into subpopulations that can be separately simulated in parallel on different (compute) nodes.