Introduction to Functional Brain

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.

t
j
e
X
t
x
t
C





)
(
)
(
)
,
(
*
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
0
) is
–
ω
t
0
.


t
t
t
C
t
C
t


















)
(
)
(
)
,
(
)
,
(
arg
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
2
*
2
1
1
2
,
1






t
C
t
C
t
C
t
C
t

)
(
)
(
)
(
)
(
,
)
(
)
(





j
t
j
e
X
X
e
t
x
t
x


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
k
t
j
N
t
PLV
1
2
,
1
))
,
(
exp(
1
)
,
(



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.

))
(
exp(
)
(
))
(
exp(
)
(
2
2
2
1










t
t
j
t
x
t
t
j
t
x
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?