Neural systems from an algebraictopology point of view

Joint project with:Pawel D lotko: INRIA

Kathryn Hess, Martina Scolamiero: EPFLHenry Markram, Eilif Muller,

Max Nolte, Michael Reimann: Blue Brain ProjectSophie Raynor: University of Aberdeen

Rodrigo Perin: Brain and Mind Institute

Workshop on Applied TopologySingapore, August 2015

August 20, 2015

The Blue Brain Project

I Digital reconstruction of the microcircuitry of the hind-limbsomatosensory cortex of a 14 days old rat, based on detailedexperimental data from five live rat brains.

I “The column”: ∼ 31, 000 simulated neurons of 55morphological types in 6 layers, ∼ 8.2× 106 connections, and∼ 36.7× 106 synapses. Simulating a cortex region 0.5mm indiameter and 2mm high.

I Data at our disposal: 42 such columns, 7 for each rat and 7based on averaged data from all of them.

I Validated against experimental data sets not used in thereconstruction.

I Key application: Study emergent properties of themicrocircuit through simulated structure and activity.

The Blue Brain Project

I Digital reconstruction of the microcircuitry of the hind-limbsomatosensory cortex of a 14 days old rat, based on detailedexperimental data from five live rat brains.

I “The column”: ∼ 31, 000 simulated neurons of 55morphological types in 6 layers, ∼ 8.2× 106 connections, and∼ 36.7× 106 synapses. Simulating a cortex region 0.5mm indiameter and 2mm high.

I Data at our disposal: 42 such columns, 7 for each rat and 7based on averaged data from all of them.

I Validated against experimental data sets not used in thereconstruction.

I Key application: Study emergent properties of themicrocircuit through simulated structure and activity.

The Blue Brain Project

I Digital reconstruction of the microcircuitry of the hind-limbsomatosensory cortex of a 14 days old rat, based on detailedexperimental data from five live rat brains.

I “The column”: ∼ 31, 000 simulated neurons of 55morphological types in 6 layers, ∼ 8.2× 106 connections, and∼ 36.7× 106 synapses. Simulating a cortex region 0.5mm indiameter and 2mm high.

I Data at our disposal: 42 such columns, 7 for each rat and 7based on averaged data from all of them.

I Validated against experimental data sets not used in thereconstruction.

I Key application: Study emergent properties of themicrocircuit through simulated structure and activity.

The Blue Brain Project

I Digital reconstruction of the microcircuitry of the hind-limbsomatosensory cortex of a 14 days old rat, based on detailedexperimental data from five live rat brains.

I “The column”: ∼ 31, 000 simulated neurons of 55morphological types in 6 layers, ∼ 8.2× 106 connections, and∼ 36.7× 106 synapses. Simulating a cortex region 0.5mm indiameter and 2mm high.

I Data at our disposal: 42 such columns, 7 for each rat and 7based on averaged data from all of them.

I Validated against experimental data sets not used in thereconstruction.

I Key application: Study emergent properties of themicrocircuit through simulated structure and activity.

The Blue Brain Project

I Digital reconstruction of the microcircuitry of the hind-limbsomatosensory cortex of a 14 days old rat, based on detailedexperimental data from five live rat brains.

I “The column”: ∼ 31, 000 simulated neurons of 55morphological types in 6 layers, ∼ 8.2× 106 connections, and∼ 36.7× 106 synapses. Simulating a cortex region 0.5mm indiameter and 2mm high.

I Data at our disposal: 42 such columns, 7 for each rat and 7based on averaged data from all of them.

I Validated against experimental data sets not used in thereconstruction.

I Key application: Study emergent properties of themicrocircuit through simulated structure and activity.

The Blue Brain Project

I Digital reconstruction of the microcircuitry of the hind-limbsomatosensory cortex of a 14 days old rat, based on detailedexperimental data from five live rat brains.

I “The column”: ∼ 31, 000 simulated neurons of 55morphological types in 6 layers, ∼ 8.2× 106 connections, and∼ 36.7× 106 synapses. Simulating a cortex region 0.5mm indiameter and 2mm high.

I Data at our disposal: 42 such columns, 7 for each rat and 7based on averaged data from all of them.

I Validated against experimental data sets not used in thereconstruction.

I Key application: Study emergent properties of themicrocircuit through simulated structure and activity.

A single Neuron

Current estimates claim that the human brain contains approximately 86 billion neurons.

A 7-neuron Microcircuit Model

A neocortical column (31,000 neurons)

Connectivity patterns in rat brains

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

The Connectome

Describe the brain network of neurons as a graph. This can bedone on several levels:

I Neurons connected to each other via synapses.

I Functional clusters within a microcircuit connected to eachother via multiple synaptic connections.

I Microcircuits (mini columns) connected to each other.

I Brain regions connected to each other.

I Hierarchically modular graphs - several levels of connectionssimultaneously.

Example:

I Connections among brain regions can be successfully mappedusing fMRI.

I Various psychiatric and neurological conditions are detectedand sometimes distinguished by deviation from standard graphtheoretic invariants of the relevant connectome.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

Functional connectivity

Microcircuits examined in vitro can only be mapped on a verysmall scale.The Blue Brain in silico models are fully functional.

I Individual columns can be connected to each other to createlarger regions of the cortex.

I Columns can be stimulated and their reaction measured andrecorded in great detail.

I Various electro-chemical conditions in which the brainnormally functions can be simulated.

I The full connectivity matrix of a column or any cluster ofcolumns can be extracted, including information onmorphological types, strength of connection, more.

I An active column gives rise to a time series of connectivitymatrices by recording the activity in time bins.

The directed flag complex

I A directed graph G is a pair (V,E) where V is the set ofvertices and E ⊆ V × V is the set of directed edges.

I With G we associate its geometric realisation: a vertex forevery v ∈ V , and a directed edge from v to w for every(v, w) ∈ E.

I The directed flag complex of a directed graph G is theabstract simplicial complex K(G), whose n-simplices are(n+ 1)-tuples of vertices

{(v0, v1, . . . , vn) | (vi, vj) ∈ E, ∀0 ≤ i < j ≤ n}

I The basic data object: An adjacency matrix - an n× n binarymatrix A = (ai,j) with ai,j = 1 if there is connection fromneuron i to neuron j. (In Blue Brain every neuron has anumerical name or a GID.)

The directed flag complex

I A directed graph G is a pair (V,E) where V is the set ofvertices and E ⊆ V × V is the set of directed edges.

I With G we associate its geometric realisation: a vertex forevery v ∈ V , and a directed edge from v to w for every(v, w) ∈ E.

I The directed flag complex of a directed graph G is theabstract simplicial complex K(G), whose n-simplices are(n+ 1)-tuples of vertices

{(v0, v1, . . . , vn) | (vi, vj) ∈ E, ∀0 ≤ i < j ≤ n}

I The basic data object: An adjacency matrix - an n× n binarymatrix A = (ai,j) with ai,j = 1 if there is connection fromneuron i to neuron j. (In Blue Brain every neuron has anumerical name or a GID.)

The directed flag complex

I A directed graph G is a pair (V,E) where V is the set ofvertices and E ⊆ V × V is the set of directed edges.

I With G we associate its geometric realisation: a vertex forevery v ∈ V , and a directed edge from v to w for every(v, w) ∈ E.

I The directed flag complex of a directed graph G is theabstract simplicial complex K(G), whose n-simplices are(n+ 1)-tuples of vertices

{(v0, v1, . . . , vn) | (vi, vj) ∈ E, ∀0 ≤ i < j ≤ n}

I The basic data object: An adjacency matrix - an n× n binarymatrix A = (ai,j) with ai,j = 1 if there is connection fromneuron i to neuron j. (In Blue Brain every neuron has anumerical name or a GID.)

Topological Invariants

The topological invariants and metrics we discuss can obviously beassociated to any (oriented simplicial complex).

I Betti numbers (in our computations mod-2) but only forcomputational convenience.

I Euler characteristic.

Question: The column construction algorithm is based on semistochastic processes. How do we know that its connectivitystructure is not random?

Dim Random BB0 31146 311461 7764739 76480792 15492757 730366163 247176 599452054 36 65995295 0 1331156 0 529

Number of simplices by dimension in an Erdos Renyi random graph vs. typical Blue Brain graph.

Topological Invariants

The topological invariants and metrics we discuss can obviously beassociated to any (oriented simplicial complex).

I Betti numbers (in our computations mod-2) but only forcomputational convenience.

I Euler characteristic.

Question: The column construction algorithm is based on semistochastic processes. How do we know that its connectivitystructure is not random?

Dim Random BB0 31146 311461 7764739 76480792 15492757 730366163 247176 599452054 36 65995295 0 1331156 0 529

Number of simplices by dimension in an Erdos Renyi random graph vs. typical Blue Brain graph.

Topological Invariants

The topological invariants and metrics we discuss can obviously beassociated to any (oriented simplicial complex).

I Betti numbers (in our computations mod-2) but only forcomputational convenience.

I Euler characteristic.

Question: The column construction algorithm is based on semistochastic processes. How do we know that its connectivitystructure is not random?

Dim Random BB0 31146 311461 7764739 76480792 15492757 730366163 247176 599452054 36 65995295 0 1331156 0 529

Number of simplices by dimension in an Erdos Renyi random graph vs. typical Blue Brain graph.

Topological Invariants

The topological invariants and metrics we discuss can obviously beassociated to any (oriented simplicial complex).

I Betti numbers (in our computations mod-2) but only forcomputational convenience.

I Euler characteristic.

Question: The column construction algorithm is based on semistochastic processes. How do we know that its connectivitystructure is not random?

Dim Random BB0 31146 311461 7764739 76480792 15492757 730366163 247176 599452054 36 65995295 0 1331156 0 529

Number of simplices by dimension in an Erdos Renyi random graph vs. typical Blue Brain graph.

Betti numbers

I Work of Matthew Kahle puts strong restrictions on Bettinumbers of (non-directed) flag complexes of random graphswith a given number of vertices and connection probability.

I Kahle’s theorem implies that the flag complex X of a randomgraph with our parameters (31000 vertices and p = 0.008)satisfies w.h.p H2(X,Z) 6= 0, and that w.h.p Hi(X,Z) = 0for i > 2.

I Explicitly computing homology for a complex this size isbeyond the capacity of a computer with 0.5TB RAM.

I We considered the 5-coskeleta of our directed flag complexes.This allowed us to compute Hi(−,F2) for all 42 columns fori ≥ 0. In all cases H6 = 0, but H5 6= 0.

I By comparison, for a random graph of a comparable size anddensity H4 = 0 and H3 6= 0.

Betti numbers

I Work of Matthew Kahle puts strong restrictions on Bettinumbers of (non-directed) flag complexes of random graphswith a given number of vertices and connection probability.

I Kahle’s theorem implies that the flag complex X of a randomgraph with our parameters (31000 vertices and p = 0.008)satisfies w.h.p H2(X,Z) 6= 0, and that w.h.p Hi(X,Z) = 0for i > 2.

I Explicitly computing homology for a complex this size isbeyond the capacity of a computer with 0.5TB RAM.

I We considered the 5-coskeleta of our directed flag complexes.This allowed us to compute Hi(−,F2) for all 42 columns fori ≥ 0. In all cases H6 = 0, but H5 6= 0.

I By comparison, for a random graph of a comparable size anddensity H4 = 0 and H3 6= 0.

Betti numbers

I Work of Matthew Kahle puts strong restrictions on Bettinumbers of (non-directed) flag complexes of random graphswith a given number of vertices and connection probability.

I Kahle’s theorem implies that the flag complex X of a randomgraph with our parameters (31000 vertices and p = 0.008)satisfies w.h.p H2(X,Z) 6= 0, and that w.h.p Hi(X,Z) = 0for i > 2.

I Explicitly computing homology for a complex this size isbeyond the capacity of a computer with 0.5TB RAM.

I We considered the 5-coskeleta of our directed flag complexes.This allowed us to compute Hi(−,F2) for all 42 columns fori ≥ 0. In all cases H6 = 0, but H5 6= 0.

I By comparison, for a random graph of a comparable size anddensity H4 = 0 and H3 6= 0.

Betti numbers

I Work of Matthew Kahle puts strong restrictions on Bettinumbers of (non-directed) flag complexes of random graphswith a given number of vertices and connection probability.

I Kahle’s theorem implies that the flag complex X of a randomgraph with our parameters (31000 vertices and p = 0.008)satisfies w.h.p H2(X,Z) 6= 0, and that w.h.p Hi(X,Z) = 0for i > 2.

I Explicitly computing homology for a complex this size isbeyond the capacity of a computer with 0.5TB RAM.

I We considered the 5-coskeleta of our directed flag complexes.This allowed us to compute Hi(−,F2) for all 42 columns fori ≥ 0. In all cases H6 = 0, but H5 6= 0.

I By comparison, for a random graph of a comparable size anddensity H4 = 0 and H3 6= 0.

Betti numbers

I Work of Matthew Kahle puts strong restrictions on Bettinumbers of (non-directed) flag complexes of random graphswith a given number of vertices and connection probability.

I Kahle’s theorem implies that the flag complex X of a randomgraph with our parameters (31000 vertices and p = 0.008)satisfies w.h.p H2(X,Z) 6= 0, and that w.h.p Hi(X,Z) = 0for i > 2.

I Explicitly computing homology for a complex this size isbeyond the capacity of a computer with 0.5TB RAM.

I We considered the 5-coskeleta of our directed flag complexes.This allowed us to compute Hi(−,F2) for all 42 columns fori ≥ 0. In all cases H6 = 0, but H5 6= 0.

I By comparison, for a random graph of a comparable size anddensity H4 = 0 and H3 6= 0.

Activity

The Blue Brain column can be stimulated in a variety of ways, andthe reaction recorded in great detail.

Activity - a homological approach

I As the column is stimulated in time intervals of 50ms for awhole second. The reaction is recorded in time binst = 0 . . . 199 of 5 ms each. Let A denote the structuralconnectivity matrix for the given column.

I In each time bin t consider the “successful transmission”connectivity matrix At where Ati,j = 1 if and only if thefollowing three conditions are satisfied:

I Ai,j = 1, i.e., there is a structural connection from neuron i toneuron j,

I neuron i fired in time bin t, andI neuron j fired in time bin t or t+ 1.

I The result is a sequence of matrices which are the adjacencymatrices for subcomplexes of the flag complex for the column.

I We compute the homology and Euler Characteristic of eachsuch subcomplex and obtain sequences of betti numbers,creating a pattern of evolution of the activity complexes.

Activity - a homological approach

I As the column is stimulated in time intervals of 50ms for awhole second. The reaction is recorded in time binst = 0 . . . 199 of 5 ms each. Let A denote the structuralconnectivity matrix for the given column.

I In each time bin t consider the “successful transmission”connectivity matrix At where Ati,j = 1 if and only if thefollowing three conditions are satisfied:

I Ai,j = 1, i.e., there is a structural connection from neuron i toneuron j,

I neuron i fired in time bin t, andI neuron j fired in time bin t or t+ 1.

I The result is a sequence of matrices which are the adjacencymatrices for subcomplexes of the flag complex for the column.

I We compute the homology and Euler Characteristic of eachsuch subcomplex and obtain sequences of betti numbers,creating a pattern of evolution of the activity complexes.

Activity - a homological approach

I As the column is stimulated in time intervals of 50ms for awhole second. The reaction is recorded in time binst = 0 . . . 199 of 5 ms each. Let A denote the structuralconnectivity matrix for the given column.

I In each time bin t consider the “successful transmission”connectivity matrix At where Ati,j = 1 if and only if thefollowing three conditions are satisfied:

I Ai,j = 1, i.e., there is a structural connection from neuron i toneuron j,

I neuron i fired in time bin t, andI neuron j fired in time bin t or t+ 1.

I The result is a sequence of matrices which are the adjacencymatrices for subcomplexes of the flag complex for the column.

I We compute the homology and Euler Characteristic of eachsuch subcomplex and obtain sequences of betti numbers,creating a pattern of evolution of the activity complexes.

Activity - a homological approach

I As the column is stimulated in time intervals of 50ms for awhole second. The reaction is recorded in time binst = 0 . . . 199 of 5 ms each. Let A denote the structuralconnectivity matrix for the given column.

I In each time bin t consider the “successful transmission”connectivity matrix At where Ati,j = 1 if and only if thefollowing three conditions are satisfied:

I Ai,j = 1, i.e., there is a structural connection from neuron i toneuron j,

I neuron i fired in time bin t, andI neuron j fired in time bin t or t+ 1.

I The result is a sequence of matrices which are the adjacencymatrices for subcomplexes of the flag complex for the column.

I We compute the homology and Euler Characteristic of eachsuch subcomplex and obtain sequences of betti numbers,creating a pattern of evolution of the activity complexes.

Activity - a homological approach - log(β2)

Activity - a homological approach - log(β2)

Activity - a homological approach - log(β2) Statistics

Activity - a homological approach - log(EC)

Activity - a homological approach - log(EC)

Activity - a homological approach - EC statistics

Activity - Dot vs. Circle stimulation

Same setup as before, except the column is stimulated in apattern. Points within the pattern are given a signal at random insome fixed frequency for an full second.

Activity - Dot vs. Circle stimulation

Activity - Dot vs. Circle stimulation

Activity - Dot vs. Circle stimulation

Segregation, Clustering, Integration, Small worldness

Based on a survey paper by Rubinov and Sporns: There are manygraph theoretic invariants which proved useful in neuroscience. Werestrict to a few such invariants.

I The degree ki of a node i: the number of nodes connected toi.

I A basic measure of segregation at a node i: The number ti oftriangles with i as a vertex.

I The clustering coefficient of a node i: Ci =2ti

ki(ki−1) = thenumber of triangles divided by the number of possibletriangles. The clustering coefficient of the network:C = 1

n

∑iCi.

I Measure of integration: L = 1n

∑i Li, where Li is the average

path length from i to any other node.

I Small worldness: Higher than random segregation, close torandom integration.

Segregation, Clustering, Integration, Small worldness

Based on a survey paper by Rubinov and Sporns: There are manygraph theoretic invariants which proved useful in neuroscience. Werestrict to a few such invariants.

I The degree ki of a node i: the number of nodes connected toi.

I A basic measure of segregation at a node i: The number ti oftriangles with i as a vertex.

I The clustering coefficient of a node i: Ci =2ti

ki(ki−1) = thenumber of triangles divided by the number of possibletriangles. The clustering coefficient of the network:C = 1

n

∑iCi.

I Measure of integration: L = 1n

∑i Li, where Li is the average

path length from i to any other node.

I Small worldness: Higher than random segregation, close torandom integration.

Segregation, Clustering, Integration, Small worldness

Based on a survey paper by Rubinov and Sporns: There are manygraph theoretic invariants which proved useful in neuroscience. Werestrict to a few such invariants.

I The degree ki of a node i: the number of nodes connected toi.

I A basic measure of segregation at a node i: The number ti oftriangles with i as a vertex.

I The clustering coefficient of a node i: Ci =2ti

ki(ki−1) = thenumber of triangles divided by the number of possibletriangles. The clustering coefficient of the network:C = 1

n

∑iCi.

I Measure of integration: L = 1n

∑i Li, where Li is the average

path length from i to any other node.

I Small worldness: Higher than random segregation, close torandom integration.

Segregation, Clustering, Integration, Small worldness

Based on a survey paper by Rubinov and Sporns: There are manygraph theoretic invariants which proved useful in neuroscience. Werestrict to a few such invariants.

I The degree ki of a node i: the number of nodes connected toi.

I A basic measure of segregation at a node i: The number ti oftriangles with i as a vertex.

I The clustering coefficient of a node i: Ci =2ti

ki(ki−1) = thenumber of triangles divided by the number of possibletriangles. The clustering coefficient of the network:C = 1

n

∑iCi.

I Measure of integration: L = 1n

∑i Li, where Li is the average

path length from i to any other node.

I Small worldness: Higher than random segregation, close torandom integration.

Segregation, Clustering, Integration, Small worldness

Based on a survey paper by Rubinov and Sporns: There are manygraph theoretic invariants which proved useful in neuroscience. Werestrict to a few such invariants.

I The degree ki of a node i: the number of nodes connected toi.

I A basic measure of segregation at a node i: The number ti oftriangles with i as a vertex.

I The clustering coefficient of a node i: Ci =2ti

ki(ki−1) = thenumber of triangles divided by the number of possibletriangles. The clustering coefficient of the network:C = 1

n

∑iCi.

I Measure of integration: L = 1n

∑i Li, where Li is the average

path length from i to any other node.

I Small worldness: Higher than random segregation, close torandom integration.

Segregation, Clustering, Integration, Small worldness

Based on a survey paper by Rubinov and Sporns: There are manygraph theoretic invariants which proved useful in neuroscience. Werestrict to a few such invariants.

I The degree ki of a node i: the number of nodes connected toi.

I A basic measure of segregation at a node i: The number ti oftriangles with i as a vertex.

I The clustering coefficient of a node i: Ci =2ti

ki(ki−1) = thenumber of triangles divided by the number of possibletriangles. The clustering coefficient of the network:C = 1

n

∑iCi.

I Measure of integration: L = 1n

∑i Li, where Li is the average

path length from i to any other node.

I Small worldness: Higher than random segregation, close torandom integration.

Topological metrics

Let X be a simplicial complex.

I For v ∈ X0, let moutk (v) denote the outgoing k-valence of v -

the number of simplices σ ∈ Xk, such that v is an initialvertices in σ. Similarly, define min

k (v) - the incomingk-valence of v. Define mk(v) - the k-valence of v to be thenumber of k-simplices in X which contain v.

I Define the k-th clustering coefficient of v ∈ X0 byc0(v) = c1(v) = 1, and

ck(v) =

{mk(v)/

(m1(v)k

)k ≤ m1(v)

0 otherwise.

I Define the k-th bottleneck coefficient of v ∈ X0 by

bk(v) =moutk (v)

mink (v)

.

Topological metrics

Let X be a simplicial complex.

I For v ∈ X0, let moutk (v) denote the outgoing k-valence of v -

the number of simplices σ ∈ Xk, such that v is an initialvertices in σ. Similarly, define min

k (v) - the incomingk-valence of v. Define mk(v) - the k-valence of v to be thenumber of k-simplices in X which contain v.

I Define the k-th clustering coefficient of v ∈ X0 byc0(v) = c1(v) = 1, and

ck(v) =

{mk(v)/

(m1(v)k

)k ≤ m1(v)

0 otherwise.

I Define the k-th bottleneck coefficient of v ∈ X0 by

bk(v) =moutk (v)

mink (v)

.

Topological metrics

Let X be a simplicial complex.

I For v ∈ X0, let moutk (v) denote the outgoing k-valence of v -

the number of simplices σ ∈ Xk, such that v is an initialvertices in σ. Similarly, define min

k (v) - the incomingk-valence of v. Define mk(v) - the k-valence of v to be thenumber of k-simplices in X which contain v.

I Define the k-th clustering coefficient of v ∈ X0 byc0(v) = c1(v) = 1, and

ck(v) =

{mk(v)/

(m1(v)k

)k ≤ m1(v)

0 otherwise.

I Define the k-th bottleneck coefficient of v ∈ X0 by

bk(v) =moutk (v)

mink (v)

.

Topological metrics

Let X be a simplicial complex.

I For v ∈ X0, let moutk (v) denote the outgoing k-valence of v -

the number of simplices σ ∈ Xk, such that v is an initialvertices in σ. Similarly, define min

k (v) - the incomingk-valence of v. Define mk(v) - the k-valence of v to be thenumber of k-simplices in X which contain v.

I Define the k-th clustering coefficient of v ∈ X0 byc0(v) = c1(v) = 1, and

ck(v) =

{mk(v)/

(m1(v)k

)k ≤ m1(v)

0 otherwise.

I Define the k-th bottleneck coefficient of v ∈ X0 by

bk(v) =moutk (v)

mink (v)

.

Topological metrics

Let X be a simplicial complex.

I For v ∈ X0, let moutk (v) denote the outgoing k-valence of v -

the number of simplices σ ∈ Xk, such that v is an initialvertices in σ. Similarly, define min

k (v) - the incomingk-valence of v. Define mk(v) - the k-valence of v to be thenumber of k-simplices in X which contain v.

I Define the k-th clustering coefficient of v ∈ X0 byc0(v) = c1(v) = 1, and

ck(v) =

{mk(v)/

(m1(v)k

)k ≤ m1(v)

0 otherwise.

I Define the k-th bottleneck coefficient of v ∈ X0 by

bk(v) =moutk (v)

mink (v)

.

Topological metrics

I Define the valency polynomial of v ∈ X0 by

Mv(t) =∑k≥0

mk(v)tk,

and similarly define incoming and outgoing valencypolynomials M in

v (t) and Moutv (t) resp.

I The clustering polynomial of v is defined to be

Cv(t) =∑k≥0

ck(v)tk.

I Define the bottleneck polynomial of v ∈ X0 to be

Bv(t) =∑k≥0

bk(v)tk.

Topological metrics

I Define the valency polynomial of v ∈ X0 by

Mv(t) =∑k≥0

mk(v)tk,

and similarly define incoming and outgoing valencypolynomials M in

v (t) and Moutv (t) resp.

I The clustering polynomial of v is defined to be

Cv(t) =∑k≥0

ck(v)tk.

I Define the bottleneck polynomial of v ∈ X0 to be

Bv(t) =∑k≥0

bk(v)tk.

Topological metrics

I Define the valency polynomial of v ∈ X0 by

Mv(t) =∑k≥0

mk(v)tk,

and similarly define incoming and outgoing valencypolynomials M in

v (t) and Moutv (t) resp.

I The clustering polynomial of v is defined to be

Cv(t) =∑k≥0

ck(v)tk.

I Define the bottleneck polynomial of v ∈ X0 to be

Bv(t) =∑k≥0

bk(v)tk.

Topological metrics

I Define the valency polynomial of X by

MX(t) =1

|X0|∑v∈X0

Mk(v),

and similarly define incoming and outgoing valencypolynomials M in

X (t) and MoutX (t) resp.

I The segregation polynomial of X is defined by

SX(t) =1

|X0|∑v∈X0

Cv(t).

the bottleneck polynomial of X by

BX(t) =1

|X0|∑v∈X0

Bv(t).

Topological metrics

I Define the valency polynomial of X by

MX(t) =1

|X0|∑v∈X0

Mk(v),

and similarly define incoming and outgoing valencypolynomials M in

X (t) and MoutX (t) resp.

I The segregation polynomial of X is defined by

SX(t) =1

|X0|∑v∈X0

Cv(t).

the bottleneck polynomial of X by

BX(t) =1

|X0|∑v∈X0

Bv(t).

Topological metrics

I Define the valency polynomial of X by

MX(t) =1

|X0|∑v∈X0

Mk(v),

and similarly define incoming and outgoing valencypolynomials M in

X (t) and MoutX (t) resp.

I The segregation polynomial of X is defined by

SX(t) =1

|X0|∑v∈X0

Cv(t).

the bottleneck polynomial of X by

BX(t) =1

|X0|∑v∈X0

Bv(t).

Topological metrics

I MoutX (t) =M in

X (t) for any oriented simplicial complex X.

I MoutX (−1) = χ(X)

|X0| .

I Example: if X is the standard n-simplex (v0, . . . vn), then

MoutX (t) =

1

n+ 1

(a0 + a1t+ · · ·+ an−1tn−1 + tn

),

where

ai =

(n− i+ 1

i

)+

(n− ii

)+ · · ·

(i+ 1

i

)+ 1.

I Notice that the coefficient of t2 in SX(t) is the classicalclustering coefficient of the (undirected) graph correspondingto the 1-skeleton of X.

Topological metrics

I MoutX (t) =M in

X (t) for any oriented simplicial complex X.

I MoutX (−1) = χ(X)

|X0| .

I Example: if X is the standard n-simplex (v0, . . . vn), then

MoutX (t) =

1

n+ 1

(a0 + a1t+ · · ·+ an−1tn−1 + tn

),

where

ai =

(n− i+ 1

i

)+

(n− ii

)+ · · ·

(i+ 1

i

)+ 1.

I Notice that the coefficient of t2 in SX(t) is the classicalclustering coefficient of the (undirected) graph correspondingto the 1-skeleton of X.

Topological metrics

I MoutX (t) =M in

X (t) for any oriented simplicial complex X.

I MoutX (−1) = χ(X)

|X0| .

I Example: if X is the standard n-simplex (v0, . . . vn), then

MoutX (t) =

1

n+ 1

(a0 + a1t+ · · ·+ an−1tn−1 + tn

),

where

ai =

(n− i+ 1

i

)+

(n− ii

)+ · · ·

(i+ 1

i

)+ 1.

I Notice that the coefficient of t2 in SX(t) is the classicalclustering coefficient of the (undirected) graph correspondingto the 1-skeleton of X.

Topological metrics

I MoutX (t) =M in

X (t) for any oriented simplicial complex X.

I MoutX (−1) = χ(X)

|X0| .

I Example: if X is the standard n-simplex (v0, . . . vn), then

MoutX (t) =

1

n+ 1

(a0 + a1t+ · · ·+ an−1tn−1 + tn

),

where

ai =

(n− i+ 1

i

)+

(n− ii

)+ · · ·

(i+ 1

i

)+ 1.

I Notice that the coefficient of t2 in SX(t) is the classicalclustering coefficient of the (undirected) graph correspondingto the 1-skeleton of X.

Topological metrics

Some typical examples.

I Sample Valency polynomials of the (structural) column vs. acomparable random complex:Column: 1 + 470t+ 6373t2 + 6555t3 + 837t4 + 18t4.Random 1 + 496t+ 1475t2 + 31t3.

I Sample Bottleneck polynomials of the (structural) column vs.a comparable random complex:Column: 1 + 1.4t+ 3.4t2 + 8.7t3 + 7.7t4 + 0.6t5.Random: 1 + t+ t2 + 1.3t3.

I Sample segregation polynomials of the (structural) column vs.a comparable random complex:Column:1+t+0.05t2+0.0003t3+4.057e−07t4+1.4e−10t5+1.14e−14t6

Random: 1 + t+ 0.01t2 + 1.56e−06t3 + 1.84e−12t4.

Topological metrics

Some typical examples.

I Sample Valency polynomials of the (structural) column vs. acomparable random complex:Column: 1 + 470t+ 6373t2 + 6555t3 + 837t4 + 18t4.Random 1 + 496t+ 1475t2 + 31t3.

I Sample Bottleneck polynomials of the (structural) column vs.a comparable random complex:Column: 1 + 1.4t+ 3.4t2 + 8.7t3 + 7.7t4 + 0.6t5.Random: 1 + t+ t2 + 1.3t3.

I Sample segregation polynomials of the (structural) column vs.a comparable random complex:Column:1+t+0.05t2+0.0003t3+4.057e−07t4+1.4e−10t5+1.14e−14t6

Random: 1 + t+ 0.01t2 + 1.56e−06t3 + 1.84e−12t4.

Topological metrics

Some typical examples.

I Sample Valency polynomials of the (structural) column vs. acomparable random complex:Column: 1 + 470t+ 6373t2 + 6555t3 + 837t4 + 18t4.Random 1 + 496t+ 1475t2 + 31t3.

I Sample Bottleneck polynomials of the (structural) column vs.a comparable random complex:Column: 1 + 1.4t+ 3.4t2 + 8.7t3 + 7.7t4 + 0.6t5.Random: 1 + t+ t2 + 1.3t3.

I Sample segregation polynomials of the (structural) column vs.a comparable random complex:Column:1+t+0.05t2+0.0003t3+4.057e−07t4+1.4e−10t5+1.14e−14t6

Random: 1 + t+ 0.01t2 + 1.56e−06t3 + 1.84e−12t4.

Highways and Flow

I Let X be an oriented simplicial complex, and let x, y ∈ X0 beany two vertices.

I A d-dimensional highway from x to y is either a d-simplex(x, x1, . . . xd−1, y) in Xd or a sequence of (d+ 1)-simplices

σ0, . . . , σm

in X, such that σi ∩ σi+1 is a back d-face of σi and a frontd-face of σi+1, for all i ≥ 0, and such that x is an initialvertex in σ0 and y is a final vertex in σm.

Highways and Flow

I Let X be an oriented simplicial complex, and let x, y ∈ X0 beany two vertices.

I A d-dimensional highway from x to y is either a d-simplex(x, x1, . . . xd−1, y) in Xd or a sequence of (d+ 1)-simplices

σ0, . . . , σm

in X, such that σi ∩ σi+1 is a back d-face of σi and a frontd-face of σi+1, for all i ≥ 0, and such that x is an initialvertex in σ0 and y is a final vertex in σm.

Example: 1-highways

Example: 1-highways

Example: 1-highway graph

Example: 1-highway - undirected

Example: 1-highway graph - undirected

Highways and Flow

I For vertices v, w ∈ X0, and d ≤ 0, let id(v, w) denote theintegration coefficient of the pair (v, w), i.e. the minimallength of a d-dimensional highway from v to w if one exists,and set Id(v, w) = 0 if it doesn’t or if v = w and d > 0. Alsoset I0(v, v) = 1.

I If the edges of a simplicial complex are weighted, then onecan assign a max flow capacity on a d-dimensional highwayfd(v, w) to each pair or vertices (v, w) (0 if such a highwaysdoes not exist).

I These are harder (more expensive in time and memory) tocompute, but there are good approximation algorithms.

Highways and Flow

I For vertices v, w ∈ X0, and d ≤ 0, let id(v, w) denote theintegration coefficient of the pair (v, w), i.e. the minimallength of a d-dimensional highway from v to w if one exists,and set Id(v, w) = 0 if it doesn’t or if v = w and d > 0. Alsoset I0(v, v) = 1.

I If the edges of a simplicial complex are weighted, then onecan assign a max flow capacity on a d-dimensional highwayfd(v, w) to each pair or vertices (v, w) (0 if such a highwaysdoes not exist).

I These are harder (more expensive in time and memory) tocompute, but there are good approximation algorithms.

Highways and Flow

I For vertices v, w ∈ X0, and d ≤ 0, let id(v, w) denote theintegration coefficient of the pair (v, w), i.e. the minimallength of a d-dimensional highway from v to w if one exists,and set Id(v, w) = 0 if it doesn’t or if v = w and d > 0. Alsoset I0(v, v) = 1.

I If the edges of a simplicial complex are weighted, then onecan assign a max flow capacity on a d-dimensional highwayfd(v, w) to each pair or vertices (v, w) (0 if such a highwaysdoes not exist).

I These are harder (more expensive in time and memory) tocompute, but there are good approximation algorithms.

Highways and Flow

Highways and Flow

a b

Highways and Flow

a b

1

1

1

1

1

1

1

111

1

0

0

0

0

0

0

00 0

0

0

Highways and Flow

a b

1

1

1

1

1

1

1

111

1

1

0

0

0

1

0

00 0

1

0

Highways and Flow

a b

1

1

1

1

1

1

1

111

1

1

1

1

0

1

0

00 0

1

1

Highways and Flow

I The integration polynomial of a (weighted) oriented simplicialcomplex X is defined by

IX(t) =1

|X0|(|X0 − 1|)·

∑(x,y)∈X0×X0

∑d≥0

id(x, y)td.

I The flow polynomial of a (weighted) oriented simplicialcomplex X is defined by

FX(t) =1

|X0|(|X0 − 1|)·

∑(x,y)∈X0×X0

∑d≥0

fd(x, y)td.

I Being computed on our data as we speak.

Highways and Flow

I The integration polynomial of a (weighted) oriented simplicialcomplex X is defined by

IX(t) =1

|X0|(|X0 − 1|)·

∑(x,y)∈X0×X0

∑d≥0

id(x, y)td.

I The flow polynomial of a (weighted) oriented simplicialcomplex X is defined by

FX(t) =1

|X0|(|X0 − 1|)·

∑(x,y)∈X0×X0

∑d≥0

fd(x, y)td.

I Being computed on our data as we speak.

Highways and Flow

I The integration polynomial of a (weighted) oriented simplicialcomplex X is defined by

IX(t) =1

|X0|(|X0 − 1|)·

∑(x,y)∈X0×X0

∑d≥0

id(x, y)td.

I The flow polynomial of a (weighted) oriented simplicialcomplex X is defined by

FX(t) =1

|X0|(|X0 − 1|)·

∑(x,y)∈X0×X0

∑d≥0

fd(x, y)td.

I Being computed on our data as we speak.

Thank you.