The Topology of Neural SystemsOr alternatively, Neuroscience Inspired Topology

Joint project withKathryn Hess, Henry Markram, Sophie Raynor

Bedlewo, July 2013 - Conference on Applied Topology

July 29, 2013

A single Neuron

Close-up on the Soma

A 7-neuron Microcircuit Model

Three clusters of Neurons

A Neocortical Column Model

A Neocortical Column Model - Close-up

Flying through the neocortical columnBlue Brain Project simulation

Flying through the column0:301:45

Philosophy

There are many models of neural system behaviour usingtraditional applied mathematical methods, such as PDE,Probability and Statistics, combinatorics, graph theory etc.Topology (as opposed to network topology) is new to the game.

In most existing research, mathematicians receive data frombiological observations, such as spike timing, and interpret thatdata mathematically to create a model.

The Blue Brain model is constructed using biological data andstatistical information and aims to reverse engineer the brain.We aim to:

I Develop theory, based on structural and flow data producedby Blue Brain, and feed back into the model by means ofexperimentation based on our theories, where possible.

I Use the tools of algebraic topology to interpret data emergingfrom the Blue Brain model

The Common Neighbour Rule

The common neighbour rule: If two neurons are connected to athird, then there is a higher probability that they will be connectedto each-other.

The Common Neighbour Rule

This suggests a higher prevalence of 2-simplex structures in anorganised system than in a random one. But why stop at 2?

The higher dimensional common neighbour rule: If n 1neurons are connected to an n-th, then there is a higher probabilitythat they will be connected to each-other.

If this holds, one should expect a higher prevalence of n-simplexstructures in an organised system, than in a random one.

The Common Neighbour Rule

First attempts: Count the number of simplices.

Using a simple Matlab programme ran on a laptop on a 250 250incidence matrix (a size 250 minor of a size 1000 matrix) gavewhen applied to a random matrix 1 and 2 dimensional simplices,and very few 3 dimensional simplices. In Blue Brain sample matrixof the same density n dimensional simplices for all n 8 occurred.

Sample calculation of homology (rational coefficients). Networkssize 400. Same numbers of vertices and edges.

I PHrnd(t) = 1 + 2511t+ 27t2.

I PHbb(t) = 1 + 241t+ 1747t2 + 67t3.

A Categorical Model

What do we want to model?

I Network connectivity (undirected and directed).

I Flow in a system over time.

I Plasticity: structural changes in a system over time.

The Naive approach:

I Consider the category S, whose objects are the naturalnumbers N. Morphisms S(m,n) are equivalence classes ofdirected graphs with m labeled inputs and n labeledoutputs.

I In S one has vertical composition (by label), and monoidalproduct, which gives it the structure of a strict symmetricmonoidal category.

A Categorical Model

A monoidal category is a category C with a monoidal product onits object set, and a compatible monoidal product on itsmorphisms sets. The product is associative and has a unit object.The monoidal product is usually denoted by .A monoidal category C is said to be symmetric if for any pair ofobjects A,B C there is an isomorphism in C

sAB : AB B A,

which satisfies some obvious coherence relations:

I Unit coherence

I Associativity coherence

I Inverse Law

A Categorical Model

Problems with the naive approach:

I Identity morphisms are not easy to deal with (possible, butnot straight forward).

I Much more serious: Loops are irreducible. It is impossibleto obtain a loop by composition.

DefinitionA directed graph with leaves is a finite directed graphG = (V,E, s, t), where s, t : E V are source and targetfunctions, and where is a disjoint point. A leaf in G is an edgee E, whose source or target are .No sink or source vertices: For any v V , s1(v) 6= andt1(v) 6= .

A Categorical Model

Encoding extra information on a graph is possible by attachingweights to edges.

DefinitionLet W be a topological space with two distinguished base pointsdenoted 0 and . A W -weighted directed graph (G,) is adirected graph with leaves G, together with a map : E(G)Wsatisfying the following two properties.

1. For any bivalent vertex v V (G), and edges e, e E(G)which are incident on v and which are not leaves,(e) = (e).

2. The graph G obtained from G by removing all edgese E(G) with (e) = 0 remains a valid graph.

A Categorical Model

An equivalence relation is needed.

Definition(G1, 1) (G2, 2) if G2 can be obtained from G1 by a finitesequence of the following moves:

1. add or remove bivalent vertices whose incident edges have thesame weight.

2. add an edge of weight separating one vertex into two, orremove such an edge, identifying the incident vertices

3. add or remove edges of weight 0

A graph is said to be minimal if it is not equivalent to any othergraph with fewer edges and/or vertices.

Notice that every graph as above is equivalent to a unique minimalgraph. The equivalence class of a graph G will be denote by [G].

A Categorical Model

Fix a topological space W with two distinguished points {0, }.The category of W -weighted directed graphs BW consists of:Objects: Words c = (c1, c2, . . . , cn) {, } W Morphisms: BW (c, d) consist of equivalence classes [G] of graphsG with input and output leaves labeled in a way that correspondsto c and d.

A sample morphism in BW .

Composition is defined in the obvious way.

A Categorical Model

Identity morphisms are easy to define:

And it is possible to obtain loops by vertical composition:

The category BW is a Coloured PropBW is a strict symmetric monoidal category. The monoidal productgiven on objects by concatenation: c d = cd. On morphisms themonoidal product is given by disjoint union. The empty object andempty graph are the left and right units for the monoidal product.

The category BW is a Coloured Prop with colour set given by{, } W .

This setup suffices to describe the network as made of smallerpieces, but in order to describe processes, particularly flow, andplasticity we need more.

Enriching B over spaces and PlasticityWe define a topology on morphism sets BW (c, d).I For a W -weighted graph with leaves G, considerWE(G) =

eE(G)We.

I Consider the set of all W -weighted graphs G, which representmorphisms in BW (c, d), and the disjoint union ofcorresponding spaces WE(G).

I Introduce the defining equivalence relation on the graphs toobtain a topology on BW (c, d), which is induced from that ofW .

Given [G], [G] BW we can define paths : [0, r] BW (c, d), forsome r 0 such that (0) = [G], and (r) = [G].Composition of such paths is clearly associative (There might be a2-category lurking around).

Enriching B over spaces and PlasticityA path from [G] to [G] in BW (c, d) is called a plasticityfunction.

A path in a mapping space can be thought of as behaviour of thesystem over a time interval.

The idea of plasticity functions can be used to model plasticityphenomena in a neural system:

I Strengthening and weakening of connections.

I Creation of new connections and destruction of existingconnections.

I The nature of the weights space W (which could be thoughtof as a parameters of capacity, flow, etc) together with thedefining equivalence relation for morphism sets, impose strongrestrictions on the topology of the mapping spaces in BW .

Algebras over BWOnce we have a satisfactory model for an active neural system, weneed a device which will allow interpretation of activity.

One possibility is to consider functors from BW into some nicecategory, e.g., vector spaces over some field. Study the structure offunctor categories.

Theoretical phenomena discovered in this study may inform onwhat may be interesting for neuroscience researchers to look for.

The theory we develop may turn out to be of little immediaterelevance to neuroscience, but that will not stop us doing it Thisis the neuroscience inspired topology part

Algebras over BWExample: Consider the category (prop) G with objects the naturalnumbers, and morphisms G(m,n) equivalence classes undirectedunweighted graphs with m incoming and n outgoing leaves. Thedefining equivalence relation is generated by

I adding or removing bivalent vertices

I adding or removing pairs (v, e) where v is a univalent vertexwith incident edge e, which is not a leaf.

The category G has distinguished morphisms u G(0, 1) - the unitmorphism, represented by a single vertex with one outgoing leaf,and u G(1, 0) - the trace morphism, represented by a singlevertex with one incoming leaf.

Algebras over BWThe category G is a symmetric monoidal category, and there is anobvious forgetful functor O : BW G.The category of functors from G to vector spaces has an extremelyrich structure. Given such a functor A : G VectF, we get bycomposition a functor O(A) : BW VectF. We call theseforgetful algebras over BW .Important subcategories of G:Let A G denote the subcategory with the same objects andwith morphisms A(m,n) disjoint unions on n trees Ti with miincoming leaves and one outgoing leaf, such that

imi = m.

Algebras over A are what we normally call A algebras, i.e.differential graded algebras with multiplication which is associativeup to arbitrarily high homotopies.

Algebras over BWLet Fr G denote the subcategories with the same objects, andwhere morphisms are graphs where each vertex has valency atmost 3, with at most 2 incoming or outgoing edges.

A Frobenius algebra is a vector space V with a productm : V V V , and a coproduct m : V V V , a unit e and acount e, such that the Frobenius relation is satisfied. Frobeniusalgebras have non degenerate pairing : V V k, and coparing : k V V . A Frobenius algebra is said to be symmetric if issymmetric.

Algebras over Fr are symmetric Frobenius algebras.

Algebras over G are called symmetric A Frobenius algebras. Thisis justified (with some work) by the following:

Algebras over BW

Proposition

The category G is generated by the n-trees tn, n 0, togetherwith the morphism given by the graph p G(0, 2) with a singlevertex and two outgoing leaves.In particular an algebra : G VectF is generated by the the Astructure together with the copairing = (p) and the trace mapt = (u).

Algebras over BWCurrent aim: Study general algebras over BW .I Consider the subcategories BAW and BFrW obtained by pulling

back the forgetful functor O : BW G along the respectiveinclusions A G and Fr G.

I Under suitable assumptions on W , find generating sets for themorphisms in these subcategories. (This may be very difficultand make little sense for arbitrary W ).

I Question: Is BW generated by BAW and BFrW ? If not,then what else is required?

I What sort of W makes sense? Capacity (weight ofconnection); Charge; Flow; Presence of certain chemicals.......

THANKS FOR LISTENING