Neurocognitive Graph Theory - Amazon S3 · Neurocognitive Graph Theory Joshua Vogelstein Johns...

Neurocognitive Graph Theory

Joshua Vogelstein

Johns Hopkins University

March 3, 2009

Joshua Vogelstein (JHU) Neurocognitive Graph Theory March 3, 2009 1 / 16

Outline

1 Introduction

2 Theory

3 Applications

Introduction

Outline

1 Introduction

2 Theory

3 Applications

Introduction

Motivation

Brains are well represented by graphs

If we believe in “mind-brain-supervenience”, then a person’s brainmust contain all the information stored in that person’s memory

My previous work has been devoted to inferring graphs from neuralactivity

Thus, given such graphs for a particular human, we could, inprinciple, decode her graph, to learn what she knows

This obviates the need to query people, as, instead, we could simplyquery the graph

This technology could even be employed posthumously

Introduction

Preliminary Plans

Build a universally consistent classifier for neurocognitive graphs

Classify behaviorally distinct populations of drosophila

Identify humans with highly developed skills in certain areas (likemusical ability)

Theory

Outline

1 Introduction

2 Theory

3 Applications

Theory

What is a graph?

Basic terminology

Nodes — neurons, voxels, etc.

Edges — synapses, axonaltracts, etc.

Theory

A neurocognitive classifier

Formal definitions

The space of graphs is defined by {G = (V ,E ) : G ∈ G}

The space of actions is defined by {a : a ∈ A}The space of stimuli is defined by {s : s ∈ S}The decision policy of the agent is defined by δ : S × G → AOur goal is to find a classifier, gs(·) = δ(s, ·), s.t. we can predict theaction of an agent in response to any particular stimulus, s

Theory

Formal definitions

The space of graphs is defined by {G = (V ,E ) : G ∈ G}The space of actions is defined by {a : a ∈ A}

The space of stimuli is defined by {s : s ∈ S}The decision policy of the agent is defined by δ : S × G → AOur goal is to find a classifier, gs(·) = δ(s, ·), s.t. we can predict theaction of an agent in response to any particular stimulus, s

Theory

Formal definitions

The space of graphs is defined by {G = (V ,E ) : G ∈ G}The space of actions is defined by {a : a ∈ A}The space of stimuli is defined by {s : s ∈ S}

The decision policy of the agent is defined by δ : S × G → AOur goal is to find a classifier, gs(·) = δ(s, ·), s.t. we can predict theaction of an agent in response to any particular stimulus, s

Theory

Formal definitions

The space of graphs is defined by {G = (V ,E ) : G ∈ G}The space of actions is defined by {a : a ∈ A}The space of stimuli is defined by {s : s ∈ S}The decision policy of the agent is defined by δ : S × G → A

Our goal is to find a classifier, gs(·) = δ(s, ·), s.t. we can predict theaction of an agent in response to any particular stimulus, s

Theory

Formal definitions

The space of graphs is defined by {G = (V ,E ) : G ∈ G}The space of actions is defined by {a : a ∈ A}The space of stimuli is defined by {s : s ∈ S}The decision policy of the agent is defined by δ : S × G → AOur goal is to find a classifier, gs(·) = δ(s, ·), s.t. we can predict theaction of an agent in response to any particular stimulus, s

Theory

Building a universally consistent classifier

The probability of an error is: L(g) = P(g(X ) 6= Y )

A Bayes optimal classifier satisfies: g∗ = argming L(g)

A universally consistent classifier converges to g∗, given infinite data,i.e., g → g∗ as n→∞k-nearest neighbor (kNN) classifier is universally consistent

but X is assumed to be in Rd

our X is a graph, and therefore in G

Thus, we must prove the existence of a universally consistent classifieron graphs

Note that this classifier could potentially be a very general tool, foranybody classifying graphs

Theory

A universally consistent classifier converges to g∗, given infinite data,i.e., g → g∗ as n→∞

k-nearest neighbor (kNN) classifier is universally consistent

Theory

our X is a graph, and therefore in GThus, we must prove the existence of a universally consistent classifieron graphs

Theory

Universally consistent classifiers

k-nearest neighbor classifier

Theory

Building a universally consistent classifier on graphs

Three potential ways forward

Define a generative model for graphs (randomly grown graphs)

Define the distribution of naıve (initial) graphs P(G0)Define the transition distribution for our graphs, P(Gt |Gt−1)From this Markov process, we can compute the likelihood oftransitioning from any one graph to any other

Define a mapping from graphs to strings, and compute the Hammingdistance between the two strings

Uses the stationary distribution, π = Pπ, where P is the |V | × |V |matrix that defines the graphThis is very general, and could apply in theory to any graph

Project the graph onto a lower dimensional subspace, count thenumber of edges in each dimension, and then use the typical kNNclassifier in Rd

Theory

Define the distribution of naıve (initial) graphs P(G0)

Define the transition distribution for our graphs, P(Gt |Gt−1)From this Markov process, we can compute the likelihood oftransitioning from any one graph to any other

Theory

Define the distribution of naıve (initial) graphs P(G0)Define the transition distribution for our graphs, P(Gt |Gt−1)

From this Markov process, we can compute the likelihood oftransitioning from any one graph to any other

Theory

Uses the stationary distribution, π = Pπ, where P is the |V | × |V |matrix that defines the graph

This is very general, and could apply in theory to any graph

Theory

Applications

Outline

1 Introduction

2 Theory

3 Applications

Applications

Application to a small tractable system

Drosophila olfactory behavior

Determine the distribution of naıve drosophila, P(G0)

Train a group of drosophila on a simple olfactory task, i.e., flap wingsupon detecting a particular odor

Provide another (control) group with the same stimuli, but do notreward the appropriate behavior

Infer the graphs (or low dimensional representation of graphs) for thetwo populations

Given a new drosophila, classify based on her k nearest neighbors

Applications

Application to a human beings

Classifying piano virtuosos

Determine the distribution of naıve children, P(G0)

Infer the graph of adult piano virtuosos

Infer the graph of adult people with no formal musical training

Given a new human, classify based on her k nearest neighbors

Applications

More applications to human beings

Possible future queries

Where is Osama bin Laden hiding?

What is the combination?

How does this work?

Did you commit this crime?

Applications

How does this work?

Applications

How does this work?

Applications

How does this work?

Applications

Questions

Neurocognitive Graph Theory - Amazon S3 · Neurocognitive Graph Theory Joshua Vogelstein Johns...

Documents

Transcript of Neurocognitive Graph Theory - Amazon S3 · Neurocognitive Graph Theory Joshua Vogelstein Johns...