Introduction to Functional Brain

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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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t
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t
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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
0
) is

ω
t
0
.


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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
k
t
j
N
t
PLV
1
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,
1
))
,
(
exp(
1
)
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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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exp(
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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?