There
have been different kinds of networks developed, studied and implemented such
as financial networks, social networks, bibliographic networks etc. Ever
pondered how these networks are developed?
Most
of them are developed due to interactions among human beings or impact of their
ideas on the society.
How
are humans able to create these networks, perceive them and understand them?
The
answer is obvious- human brain is the one responsible for these ideas!!
Human
brain, the natural computer is made up of billions of neurons, which are electrically
excitable cells that process and transmit information by electrochemical
signalling. These neurons interact with each other which leads in the
development of ideas, generation of movements in the body organs and what not!!
These interactions among various regions of brain can be seen as a complex
network. 1
Human brain connectomics:2
It
is the representation of structural and functional connectivity of human brain.
One way for formulation of the network is to capture the movement of the
signals from neurons of one region of brain to other. The nodes are these
anatomically separate regions and connections/edges are formed when electrical
signals are transferred from one region to other. These connections can be
interpreted in terms of correlation, mutual information, coherence etc. and are
dynamic in nature.
Figure-1.
Connectivity
study on these networks has been used to identify certain disease conditions
such as Alzheimer’s and schizophrenia. These are usually captured from
EEG/fMRI/MEG/PET/MRI.
Alzheimer’s
disease (AD) is the most common form of dementia and is closely related to the
brain’s functional network. Some people also develop dementia because of normal
aging (Normal Condition-NC). There is a transition phase between normal aging
process and disease condition which is identified as Mild Cognitive Impairment
(MCI) understanding which can be an early stage detection of AD.
Functional connectomics in
Alzheimer’s disease:
As
mentioned earlier, EEG/MEG records the electrical and magnetic field changes in
the neuronal activities during a task or during resting periods. Studies have
proven that even these connections follow a small world phenomenon in a healthy
human being. A disease condition, which is an abnormality would lead to a
difference from this behaviour both in inter-region and intra-region
interactions.
Sparse Inverse Covariance Estimation
(SICE):3
Instead
of taking the direct covariance matrix for connections among different regions
of brain, we consider inverse of covariance matrix which also accommodates for
the interactions within different regions while considering covariance.
Sparsity is used to compensate for the limited sample size.
Monotone Property of a graph:3
If
a graph G follows a property P then a subgraph G’ of G also follows the
property G.
Usage of these properties to
structure the similarity of the connectivity graph: 3
Assumptions:
{X1,X2…….Xp}
be the brain regions under study, considered as nodes of the graph and they
follow a Gaussian distribution with mean µ and covariance ∑.
Let Θ = ∑-1 be the inverse covariance matrix.
Suppose we have samples (e.g., subjects with AD) for these brain regions.
SICE can be formulated as an optimization problem with
constraint on the sparsity of the matrix as follows:
We
find the maximum likelihood of the distribution with different constraints for
the sparsity of the matrix. The larger the value of regularization λ, more
sparse the matrix is.
Monotone Property of SICE:
3
If two nodes are connected at a particular level of
sparseness then they are connected at a lower level of sparseness.
For the study of connectivity among different regions in each
condition, one model for connectivity of each of AD,NC and MCI are taken with
equal number of interconnections(50) among 42 sub-regions (anatomical variables of
interest) of brain which are of interest for AD are taken into consideration.
The Corresponding details are as follows:
Graph Partitioning: 3
Following are the dendograms that determine the order of
strength of connectivity between brain regions in AD, NC and MCI conditions
respectively:
Figure2: Dendogram for AD situation:
Figure3:
Dendogram for NC situation:
Figure4: Dendogram for MCI situation:
As the value of λ increases from small to large, we can see
that there is a split in the dendograms.
This can be attributed to the fact that the sparser the data
is, the higher the disconnectivity in the network. This is a divisive
partitioning of the regions of the graph, with similarity index dependent on
the SICE and λ.
Few conclusions from the dendograms:
The earlier a particular region gets disconnected, the
lesser its connectivity to other parts of the brain.
E.g.: In case of AD, Temporal_Sup_L gets disconnected first,
in case of NC, Cingulum_Post_R and Cingulum_Post_L are the weakest
interconnected regions which are 2nd weakest connected in case of AD
which can imply that weakest regions of the brain are affected by AD.
In case of AD, the between lobe connectivity seems to be
weaker than within lobe connectivity and frontal region has higher intra connections
as compared to NC.
Regions on the same hemisphere of brain are more connected
in NC as compared to AD.
MCI also shows weaker between lobe connectivity compared to
within lobe connectivity but they are not as distinguishable as AD which shows
that it is a transition phase from NC to AD.
Hypothesis can be made according to the requirement of the
property to be verified that distinguishes between AD, NC and MCI and tested
using the findings from Dendograms.
Assortivity
values across the four lobes of the brain can be calculated and compared across
different disease stages for testing.
Within lobe and between lobes connectivity can be measured
using SICE separately and deducing results from the block diagonals of the
matrix.
Classification of new data:
The distinguishing features from the observations can be
used to train the classifier with the available data into three classes: AD, NC
and MCI.
Any new test data can be fed into the classifier to get the
output as the patient belonging to any on the classes or probability of
belonging to each of the classes.
References:
1)
2) " Mapping the Alzheimer'sbrain with connectomics"
3) "Learning brain connectivity of Alzheimer's disease from neuroimaging data " by S Huang, J Li, L Sun, J Ye, A Fleisher, T Wu, K Chen
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