Introduction to Functional Brain
Networks
August 26, 2008
Overview
•
Problem Statement
•
Functional Brain Networks
•
Functional Connectivity Measures:
–
Phase Synchrony
–
Directed Information Measure
•
Analysis of the Network Structure:
–
Small World Problem, clustering of the network
(discovering local networks)
–
Building Predictive Models (Bayesian Networks)
Problem Statement
•
With the advance of neuroimaging technology, it is
possible to record brain activity with higher resolution and
accuracy than before.
•
However, the current imaging modalities solely reflect the
local neural activity, rather than the large

scale
interactions.
•
Capturing these interactions is important in the study of
many neurological and psychological disorders.
•
Study the brain as a distributed information processing
system and quantify how the information is integrated
across different sites.
Functional Integration
•
Cognitive acts require the integration of numerous
functional areas distributed over the brain [Friston,
1997, Tononi & Edelman, 1998].
•
The functional integration between different parts of
the brain is established through synchronization (phase
locking) of the neuronal oscillations.
•
Quantifying the integration of neural activity could
help in identifying the underlying networks.
•
Applications:
–
Visual processing
–
Study of neurological (Parkinson’s disease) and psychological
pathologies (schizophrenia) [Uhlhaas, 2006]
Measures of Functional Connectivity
•
Linear Measures:
–
Correlation, spectral coherence, directed transfer function, partial directed coherence, Granger
causality
–
Assume stationarity of the underlying signals
–
Limited to amplitude effects an do not allow the separation of the effects of amplitude and
phase.
•
Nonlinear Measures:
–
Information

theoretic measures (mutual information)
–
Symmetric
•
Time

Varying Measures of Phase Synchrony:
–
Wavelet Coherence (Lachaux et al. 2001):
•
Complex Morlet wavelet to define W(t,f).
•
Non

uniform resolution over time and frequency.
–
Hilbert Transform Based Phase Estimation:
•
Find the analytic signal, s(t)+j
š
(t).
•
Assumes the signal in narrowband.
•
Proposed Solution: Time

Varying Phase Synchrony Measures Based on Cohen’s class
Rihaczek Distribution
•
Rihaczek Distribution is a complex energy
distribution and provides an appreciation of phase

modulated signals [Rihaczek, 1968].
•
Preserves energy
•
Uniform time

frequency resolution
•
Can separate the amplitude and the phase
information.
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Time

Varying Phase Distribution
•
For , the phase
distribution based on Rihaczek Distribution is:
•
The phase difference between two signals can be
defined as:
•
For a real

valued signal, the phase difference
between x(t) and x(t

t
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) is
–
ω
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Phase Locking Value (PLV)
•
There are different ways of computing phase
synchrony based on the phase difference (inter

trial, single trial, inter

electrode)
•
Phase Locking Value: The phase locking between
two signals averaged over N realizations/trials:
•
Measures stability of phase differences across
trials, and is always between 0 and 1.
N
k
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PLV
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Example for Bivariate Phase
Synchrony
•
Consider two linear chirp
signals in noise with uniformly
distributed random phase
difference.
•
Compare the phase locking
value for wavelet vs. Rihaczek
based phase synchrony
measures.
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Application to EEG data
•
Application 1: Bivariate phase synchrony for the study of
ERN
•
Error

Related Negativity (ERN) is an event

related
potential (ERP) peak that occurs 50

100ms after the
commission of a speeded motor response that the subject
immediately realizes to be an error.
•
Previous analysis has shown that ERN is dominated by
partial phase

locking of intermittent theta band (4

7 Hz)
EEG activity.
•
The primary neural generator of ERN is the anterior
cingulate cortex.
Spatial Localization of Average
Phase Synchrony
•
Apply to 92 subjects, 62 electrodes
•
Average phase synchrony
difference between error and
correct responses with respect to
FCz.
•
Increased phase synchrony for
error responses was observed 25

75
ms after the response, and in the 4

7 Hz range corresponding to the
ERN.
•
The phase synchrony differences
are significant for the central
electrodes proximal to the motor
cortex.
Open Problems with Phase
Synchrony
•
Are there any ‘better’ time

varying
estimates of phase?
•
Extension to multivariate case
•
Problems with volume conduction, need for
baseline correction or source separation
•
Detailed Statistical Analysis of the Measure
Directed Information Measure
•
Phase synchrony does not determine the causality
relationships between signals.
•
In determining the functional networks, it is
important to identify which neural population is
the cause of the activity.
•
Information theoretic measures are also natural for
quantifying concepts of complexity and
information flow in the brain.
Directed Information Measure
•
Directed Information measures can capture
both linear and nonlinear interactions
between signals.
•
It is not symmetric, can capture the
direction/causality of the interactions.
•
Two measures of DI:
–
Transinformation (Saito)
–
Directed Information (Masey)
Two Measures of DI
•
Measure 1 (Masey):
•
Mutual information between the sequence X
up to time n and the current value of Y
conditioned on the past n

1 samples of Y.
Two Measures of DI
•
Measure 2 (Transinformation) (Saito):
Relationship between two measures
•
The information flow is the same:
•
When the two random processes are
independent, both measures are equal to
zero.
•
When the two random processes are
completely dependent, DI2 is zero.
Implementation of the Measures
•
Computing mutual information between length N
sequences is challenging.
•
In practice, we may want to consider the
information flow between sequences of N=2 for
simplification
•
For N=2:
•
Other possible simplification: The assumption that
the processes are Gaussian.
Current Problems with DI measures
•
Implementation:
–
What should N be? Tradeoff between computation complexity and
stationarity
–
Estimating DI values requires estimating entropies which depends
on estimating pdfs from limited data
–
Can we estimate entropy directly from the data without estimating
pdfs? Yes!
•
Determining the significance of DI values
•
Normalization of DI values (standard scale)
•
Application to real data
•
Currently, DI values are computed between two time
series, no mention of frequency. However, we would like
to determine how DI changes in different frequency bands.
Functional Networks
•
After the functional connectivity between
different neural sites is established, the goal
is to investigate and characterize the
neuronal pathways.
Proposed Plan
•
Use graph theoretic methods:
–
Use functional connectivity measures to build a
connectivity matrix for the brain (values need to be
between 0 and 1)
–
Build either a unigraph/digraph depending on the
measure
–
Transform the connectivity matrix to adjacency matrix
(apply some sort of thresholding)
–
Adjacency matrix will be 0 or 1s (either there is an edge
or not)
Proposed Plan
•
Compute network attributes based on the
graph:
–
Small

world network? High degree of local
clustering with short path lengths:
•
Parameters: Average degree, path length, clustering
coefficient
–
Scale free network? Degree distribution should
follow a power law
Proposed Plan
•
Graph Clustering:
–
Partition the graph into subclusters to maximize intra

cluster
connectivity and minimize inter

cluster connectivity
–
Some existing algorithms:
•
K

medoids
•
Minimum cut
•
Markov clustering
•
Cut clustering
–
Problems:
•
Choice of a metric
•
Number of clusters
•
Validation of clusters
Extensions
•
Predictive network models:
–
Bayesian networks, factor graphs:
•
Incorporate some a priori knowledge about the
anatomy of the brain in predicting the network
models
–
Different neuroimaging methods:
•
Currently we are focusing on EEG data. However,
we have MEG, DTI, MRI and fMRI data available.
•
How to combine the different modalities in forming
network models?
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