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Introduction to Connectivity

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### Introduction to Connectivity

Rosalyn Moran

Wellcome Trust Centre for Neuroimaging

Institute of Neurology

University College London

With thanks to the FIL Methods Group

for slides and images

SPM Course, Virginia Tech

25th Jan 2012

?

Principles of Organisation

+

Functional Integration

Functional Specialization

How do regions influence each other?

- Brain connectivity: types & definitions
- Functional connectivity
- Effective connectivity
- - Psycho-physiological Interactions
- - Structural Equation Modelling

- Brain connectivity: types & definitions
- Functional connectivity
- Effective connectivity
- - Psycho-physiological Interactions
- - Structural Equation Modelling

Structural, functional & effective connectivity

- anatomical/structuralconnectivity= presenceofaxonalconnections
- functionalconnectivity= statisticaldependenciesbetween regional time series
- effectiveconnectivity= causal (directed) influencesbetweenneuronsor neuronal populations

Sporns 2007, Scholarpedia

Anatomical connectivity

Definition:

presence of axonal connections

- Measured with - tracing techniques - Diffusion tensor imaging (DTI)

- Neuronal communication via synaptic contacts, long range connections employ glutamate
- Regions arranged hierarchically: useful prior see later
- Presence of anatomical connection a necessary but not sufficient condition for functional integration
- Though some transmitters employ diffuse mechanisms; volume transmission

Knowing anatomical connectivity is not enough...

- Context-dependent recruiting of connections :
- Local functions depend on network activity

- Connections show synaptic plasticity
- change in the structure and transmission properties of a synapse
- even at short timescales

- Look at functional and effective connectivity

- Brain connectivity: types & definitions
- Functional connectivity
- Effective connectivity
- - Psycho-physiological Interactions
- - Structural Equation Modelling

Functional connectivity

Definition: statisticaldependenciesbetween regional time series

- Seed voxel correlation analysis
- Coherence analysis
- Eigen-decomposition (PCA, SVD)
- Independent component analysis (ICA)
- Any technique describing statistical dependencies amongst regional time series

Eg. 1 Seed-voxelcorrelationanalyses

- hypothesis-driven choice of a seed voxel
- extract reference time series
- voxel-wise correlation with time series from all other voxels in the brain

seed voxel

Eg. 1 Seed-voxelcorrelationanalyses

Task Driven Activations

(finger tapping)

Identification of VxOI

Resting State

Correlations ~0.0-1 Hz

RSNs: Resting State Networks

Eg 2. Melodic Algorithm (>ICA)

- ICA separates a multivariate signal into additive subcomponents assuming independence in mixing vectors which are non-Gaussian
- Tensor ICA separates multi-subject data into sets of vectors characterizing underlying signals in the temporal, spatial and subject domain
- Bayesian Algorithm: MELODIC determines the number of independent components at rest using a Laplace approximation to the Bayesian evidence of the model order
- (FSL Christian F. Beckmann)

The time course of the DMN revealed increased activation at rest after 1-back and 2-back blocks compared to the activation after a 0-back block

Pykaet al. 2009

Summary of functional connectivity analysis

- Pros:
- useful when we have no experimental control over the system of interest and no model of what caused the data (e.g. sleep, hallucinations, resting state: DMNs)
- Large scale network characterisations available eg. Through graph theoretic metrics
- Anatomical parsellation based on resting state asymmetries

- Cons:
- interpretation of resulting pairwise patterns is difficult
- no mechanistic insight
- usually suboptimal for data with priori knowledge / experimental control
Effective connectivity

- Brain connectivity: types & definitions
- Functional connectivity
- Effective connectivity
- - Psycho-physiological Interactions
- - Structural Equation Modelling

Effective connectivity

Definition: causal (directed) influences between neurons or neuronal populations i.e. the effect one brain region has on another

- In vivo and in vitro stimulation and recording
- Models of causal interactions among neuronal populations
- explain regional effects in terms of interregional connectivity

Some models for computing effective connectivity from fMRI data

- Regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997
- Structural Equation Models (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000
- Volterra kernels Friston & Büchel 2000
- Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003
- Dynamic CausalModels (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

Some models for computing effective connectivity from fMRI data

- Regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997
- Structural Equation Models (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000
- Volterra kernels Friston & Büchel 2000
- Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003
- Dynamic CausalModels (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

Task factor data

GLM of a 2x2 factorial design:

Task B

Task A

main effect

of task

TA/S1

TB/S1

Stim 1

main effect

of stim. type

Stimulus factor

interaction

Stim 2

TB/S2

TA/S2

Psycho-physiologicalinteraction: Friston et al 1997(PPI)The idea is to explain responses in one cortical area, in terms of an interaction

between the influence of another area and some experimental (sensory or task-related) parameter. We refer to these effects as psychophysiological interactions.

As opposed to interactions based solely on experimental factors (i.e., psychological interactions)

Task factor data

GLM of a 2x2 factorial design:

Task B

Task A

main effect

of task

TA/S1

TB/S1

Stim 1

main effect

of stim. type

Stimulus factor

interaction

Stim 2

TB/S2

TA/S2

Psycho-physiological interaction (PPI)Interactions based solely on experimental factors (i.e., psychological interactions)

Task factor data

GLM of a 2x2 factorial design:

Task B

Task A

main effect

of task

TA/S1

TB/S1

Stim 1

main effect

of stim. type

Stimulus factor

interaction

Stim 2

TB/S2

TA/S2

Psycho-physiological interaction (PPI)The idea is to explain responses in one cortical area, in terms of an interaction

between the influence of another area and some experimental (sensory or task-related) parameter.

Replace one main effect in the GLM by the deconvolved time series of an area that shows this main effect. E.g. let's say V1 showed a main effect of stimulus type

Task factor data

GLM of a 2x2 factorial design:

Task B

Task A

main effect

of task

TA/S1

TB/S1

Stim 1

Stimulus factor

Stim 2

TB/S2

TA/S2

Psycho-physiological

interaction

Psycho-physiological interaction (PPI)The idea is to explain responses in one cortical area, in terms of an interaction

between the influence of another area and some experimental(sensory or task-related) parameter.

Replace one main effect in the GLM by the deconvolved time series of an area that shows this main effect. E.g. let's say V1 showed a main effect of stimulus type

V1 time series main effect

of stim. type

Task factor data

GLM of a 2x2 factorial design:

Task B

Task A

main effect

of task

TA/S1

TB/S1

Stim 1

Stimulus factor

Stim 2

TB/S2

TA/S2

Psycho-physiological

interaction

Psycho-physiological interaction (PPI)The idea is to explain responses in one cortical area, in terms of an interaction

between the influence of another area and some experimental (sensory or task-related) parameter.

Test using reconstructed Design Matrix as usual

V1 time series main effect

of stim. type

Task factor data

Task factor

Task B

Passive

Task A

Attend

TA/S1

TA/S1

TB/S1

TB/S1

Static

Stim 1

Dots Stimulus factor

Moving

TB/S2

TB/S2

TA/S2

TA/S2

Example: Attention to motion in the visual system

Büchel & Friston 1997, Cereb. Cortex

Büchel et al.1998, Brain

Task factor data

Task factor

Task B

Passive

Attend

Task A

TA/S1

TA/S1

TB/S1

TB/S1

Static

Stim 1

Dots Stimulus factor

Moving

TB/S2

TB/S2

TA/S2

TA/S2

Example: Attention to motion in the visual system

Psychological Main Effects

main effect

of attention: V5 (and SPC)

main effect

of motion in V1 and V5

Does any region in the brain exhibit a modulation of motion related activity in V1, dependent on attention? Eg V5? (Can Mask)

? data

?

Hypothesis: ‘Bottom-up’ attentionalmodulation of V1 output to V5V1→V5Results β3:

SPM{Z}

V5 activity

V1 x Att.

time

V5

attention

V5 activity

no attention

V1 activity

Friston et al. 1997, NeuroImage 6:218-229

Büchel & Friston 1997, Cereb. Cortex 7:768-778

V5 exhibits a modulation of motion related V1 activity dependent on attention

V1 data

V5

V5

V1

V5

attention

V5

V5

attention

PPI: interpretationTwo possible interpretations of the PPI term:

V1

V1

Modulation of V1V5 by attention

Modulation of the impact of attention on V5 by V1.

PPIs data

- Pros:
- given a single source region, we can test for its context-dependent connectivity across the entire brain

- Cons:
- very simplistic model: only allows to model contributions from a single area
- ignores time-series properties of data
- operates at the level of BOLD time series

Some models for computing effective connectivity from fMRI data

- Regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997
- Structural Equation Models (SEM) McIntosh et al. 1991, 1994; Büchel & Friston 1997; Bullmore et al. 2000
- Volterra kernels Friston & Büchel 2000
- Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003
- Dynamic CausalModels (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

Structural Equation Modelling (SEM) data

- Static, linear model of imaging dependencies: (MacIntosh and Gonzalez-Lima, 1991)
- Parameters are estimated in structural equation modelling by minimizing the difference between the observed covariances and these implied by a structural or path model
- The parameters of the model are connection strength or path coefficients and correspond to an estimate of effective connectivity

y data

1

y

2

y

3

SEM Generative Modely2 = b12 y1 + b32 y3+ z2

y1 = z1

b12

b13

b32

includes only paths of interest

y3 = b13 y1+ z3

After normalisationassume

Zero mean innovations drive each

region stochastically

Penny et al, 2004

y data

1

y

2

y

3

SEM Generative Model

y1 = z1

y2 = b12y1 + b32y3 + z2

includes only paths of interest

b12

b13

b32

y3 = b13y1 + z3

Regression Model where y appears on both sides.

Rearranging see dependency of vector y on path coefficients (This is repeated for each t)

y data

1

y

2

y

3

SEM Generative Model

y1 = z1

y2 = b12y1 + b32y3 + z2

includes only paths of interest

b12

b13

b32

y3 = b13y1 + z3

Regression Model where y appears on both sides.

Rearranging see dependency of vector y on path coefficients (This is repeated for each t)

Assume data are normally generated and innovations are zero mean with

Covariance R (iid)

y data

1

y

2

y

3

SEM Generative Model

y1 = z1

y2 = b12y1 + b32y3 + z2

includes only paths of interest

b12

b13

b32

y3 = b13y1 + z3

Regression Model where y appears on both sides.

Rearranging see dependency of vector y on path coefficients (This is repeated for each t)

Then the modelled covariance of y, is a function of connection paths

y data

1

y

2

y

3

SEMInversion: Estimate B

y1 = z1

y2 = b12y1 + b32y3 + z2

includes only paths of interest

b12

b13

b32

y3 = b13y1 + z3

Given Sample Covariance

From Real Data

Can estimate the connection paths & error variance

y data

1

y

2

y

3

- SEM Inversion: Estimate B

y1 = z1

y2 = b12y1 + b32y3 + z2

includes only paths of interest

b12

b13

b32

y3 = b13y1 + z3

Given Sample Covariance

Can estimate the connection paths & error variance

Using Gradient Ascent

on likelihood

y data

1

2

y

3

y

Introduction | Theory | Application | Limitations | Conclusions

Alternative modelsModel comparison: likelihood ratio (chi-squared test)

Example: Experimental Effect of Attention: 3 regions data

Basic Model

SPC

V1

Refined Hypothesis:

- Does attention effect connectivity in three region network?
Approach:

- Partition the data set into
(i) periods in which the subject was attending to moving stimuli and

(ii) periods in which stimuli were moving but the subject did not attend to that movement

V5

Penny et al, 2004

Example: Experimental Effect of Attention data

Null model in which path coefficients are fixed between conditions

Alternative Model can change between attention conditions

- Attention does significantly change the value of this connection (χ2 = 8.6, df = 1, p = 0.003)

Penny et al, 2004

Introduction data | Theory | Application | Limitations | Conclusions

SEM- Pros:
- Multivariate causal model: Can test for >2 region connectivity

- Cons:
- Assumes stochastic input
- operates at the level of BOLD time series

Some models for computing effective connectivity from fMRI data

- Regression models (e.g. psycho-physiological interactions, PPIs)Friston et al. 1997
- Volterra kernels Friston & Büchel 2000
- Time series models (e.g. MAR, Granger causality)Harrison et al. 2003, Goebel et al. 2003
- Dynamic CausalModels (DCM)bilinear: Friston et al. 2003; nonlinear: Stephan et al. 2008

A Causal Model data

Input u(t)

System = a set of elements which interact in a spatially and temporally specific fashion

connectivity parameters

systemstate z(t)

- State changes of a system are dependent on:
- the current state
- external inputs
- its connectivity
- time constants

Neural Dynamics

State Space Model data

Input u(t)

System = a set of elements which interact in a spatially and temporally specific fashion

connectivity parameters

Neural Dynamics

systemstate z(t)

- State changes of a system are dependent on:
- the current state
- external inputs
- its connectivity
- time constants

Hemodynamic Response

More dataon DCM later … Thank you

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