Graph Theory and Connectivity

Question 1

Why is univariate analysis problematic?

Answer 1

1. Multiple functions: There is no 1-to-1 correspondence between a brain region and function
2. Context dependence: activation of different areas when the seemed context is similar, but the communication with the context is different
3. Co-activation vs. Functional integration: co-activation sometimes despite no functional connection

Question 2

What options are there for analysis of the brain?

Answer 2

Univariate vs. Bivariate vs. Multivariate = Process vs. Interaction vs. Pattern

Question 2

What is (rest) functional connectivity?

Answer 2

Functional connectivity = statistical dependencies among spatially remote neurophysiologic events
interaction between X and y: y = Xβ+ e, where β is how much of y can be explained by X and e is how much cannot be explained

Question 3

What does seed-voxel correlation do?

Answer 3

Seed-voxel correlation is a statistical analysis technique used to examine the correlation between a “seed” region of interest (ROI) and other voxels in the brain.
1. extract time series y from a seed region
2. conduct voxel-wise test for correlations
=> determine the degree of similarity or synchrony between the neural activity of the seed region and other brain regions => identify regions that are functionally connected or exhibit similar patterns of activity to the seed region

Question 4

What are the pros and cons of functional connectivity

Answer 4

Pros:
- no a priori hypothesis needed (i.e., data driven)
- useful when we have no model of what caused the data

Cons:
- no mechanistic insight into the neural system of interest
- inappropriate for situations where we have a priori knowledge and experimental control about the system of interest

Question 5

What is Psycho-Physiological interaction

Answer 5

Psycho-Physiological Interaction (PPI) is a statistical analysis technique used in neuroimaging to investigate how psychological or cognitive processes modulate the connectivity or interaction between brain regions.

Question 6

How to compute general linear model (GLM) for 2x2 factorial design?

Answer 6

y = (Ta - Tb)β1 + (S1 - S2)β2+ (Ta - Tb)(S1 - S2)β3 + e
y = main effect of task + main effect of stimulus type + interaction + e

Question 7

What two interpretations of PPI are possible?

Answer 7

There is modulation from one brain area to another by a certain process
There is modulation of the impact of a certain process on one brain area by another

Question 8

What are the pros and cons of PPI?

Answer 8

Pros:
* given a single source region, we can test for its context-dependent connectivity across the entire brain
* complies with the KISS (Keep it simple, stupid) principle

Cons:
* very simplistic model: only allows to model contributions from a single area
* not easily used with event-related data

=> limited causal interpretability in neural terms => need for more powerful models

Question 9

What is effective connectivity? What models of effective connectivity are there?

Answer 9

Effective connectivity = the influence that the elements of a neuronal system exert over another

Models of effective connectivity
* Granger causality: determines whether the past values of one time series help predict the future values of another time series
* Dynamic Causal Modelling (DCM): combines neural and hemodynamic models to study effective connectivity and investigate casual influences and interactions between brain regions
* Structural Equation Modelling (SEM): estimates the strength and directionality of connections between brain regions based on observed data
* Multivariate autoregressive models (MAR): capture the dependencies and causal relationships between mulyiple variables over time

Question 10

What is Granger causality? What limitations does it have?

Answer 10

Given two timeseries y1 and y2, y1 is considered to be caused by y2 if its dynamics can be predicted better using past values from y1 and y2 as opposed to using past values of y1 alone.
It focuses more on statistical dependencies than on underlying physiological processes =>
- +causal interpretability
- -not in neural terms

Question 11

What types of patterns are distinguished in the network analysis?

Answer 11

• Dense — really good information flow, but cost of wiring is very high =>not optimal
• Sparse — only the most important connections
• Fragmented — connected central parts, disconnected outskirts
• Connected — connected via the centre
• Ordered — very high clustering,
• Random — same connections, shorter paths, less clustering

Question 12

How can we quantitatively assess these patterns?

Answer 12

Using Graph Theory and Complex Network Theory

Question 13

What is an alternative representation of graphs and networks?

Answer 13

Matrices

Question 14

What constitues a graph?

Answer 14

Nodes and edges (links between the nodes)

Question 15

What is the degree of a node?

Answer 15

It is the number of its edges

Question 16

How are weighted and binary networks different?

Answer 16

For weighted networks, matrix elements are continuous values and can range from strong to weak
For binary networks, matrix elements are either 0 (no connection) or 1 (connection)

Question 17

How to assess a network?

Answer 17

• Network measures relate to nodes and links
• Describe how these features are embedded in the network
• Measures can be characterized on a local basis and a global basis

Classification into measures of:
* functional segregation
* functional integration
* centrality
* resilience

Question 18

What is functional segregation?

Answer 18

• Functional segregation in the brain is the ability for specialized processing to occur within densely interconnected groups of brain regions
• Quantifies the presence of such groups (clusters or modules) within the network
• Suggests an organization of brain regions indicative of segregated neural processing

Question 19

How to measure segregation?

Answer 19

Simple measures of segregation are based on the number of triangles in the network (i.e., a high number of triangles imply segregation)

Clustering Coefficient:
* measure of local connectedness, measuring the proportion of how many nearest neighbors of node i are connected to each other as well
* All nodes that are connected with node i are neighbors of that node

More sophisticated measures of segregation not only describe the presence of densely interconnected groups of regions, but also find the exact size and composition of these individual groups
Modular Structure (Community Structure):
* subdividing the network into groups of nodes, with a maximally possible number of withingroup links, and a minimally possible number of between-group links

Question 20

Give an example of the study where clustering coefficient was used as a measure?

Answer 20

Superkar et al. 2008:
Embedding of the hippocampus into the network in AD vs. controls: clustering coefficients for the left and right hippocampus were significantly lower (p<0.01) in the AD group

Question 21

Give examples of studies where modularity was used as a measure?

Answer 21

Braun et al. 2015:
individuals with greater network reconfiguration in frontal cortices show enhanced memory performance, and score higher on neuropsychological tests
Baum et al. 2017:
young brain — more global connections; adult brain — more modular segregation
Also correlations with executive functions — young people are better

Question 22

What is functional integration?

Answer 22

Functional integration is the ability to rapidly combine specialized information from distributed brain regions. It characterizes the ease with that brain regions communicate
* commonly based on the concept of a path
* Paths are sequences of links between distinct nodes
* represent potential routes of information flow between pairs of brain regions

Question 23

How the optimal balance between functional segmentation and integration is called?

Answer 23

Small-worldness
Features: High clustering + Low path Length
It is an attractive model for the organization of brain anatomical and functional networks:
* high global and local efficiency of parallel information processing facilitates rapid adaptive reconfiguration of neuronal assemblies in support of changing cognitive states

Question 24

What is special about brain networks in schizophrenia?

Answer 24

Liu et al., 2008: More of a random organisation in schizophrenia

Question 25

What is centrality?

Answer 25

Centrality quantifies the importance, influence, or prominence of nodes (individual entities) within a network. It provides a way to identify nodes that play critical roles in various aspects of network structure, information flow, or dynamics.
Hubs — highly connected nodes — often interact with many other regions and facilitate functional integration

Question 26

What measures of centrality are used in the network analysis?

Answer 26

• degree: number of connections or links that a node has
• betweenness: the extent to which a node lies on the shortest paths between other pairs of nodes
• connection between modules

Question 27

What are the characteristics of the hubs?

Answer 27

• 6% of all connections connect 90% of the brain
• highest edge betweenness centrality
• Highest functional connectivity (FC)
• Above average FC distance
• Organize in a “rich-club”

The distribution of lowly vs. average vs. highly linked is not normal

Question 28

What is resilience?

Answer 28

Resilience refers to the ability of a network to maintain its structural or functional integrity and continue functioning efficiently in the face of disturbances or damage.
Anatomical brain connectivity influences the capacity of neuropathological lesions to affect functional brain activity

Question 29

What is degeneracy?

Answer 29

Degeneracy:
* capacity of structurally distinct elements of a system to carry out the same function
* describes the ability of distinct neuronal systems to make overlapping contributions to the same output, offering both functional adaptability and robustness to damage

Question 30

How can nodes be defined?

Answer 30

Depends on the research hypothesis

Brain nodes should have
1. Intrinsic consistency: relatively stable and consistent properties or functions within themselves
2. Extrisic differentiation: specialized functions of different nodes
3. Spatially contrained: localized in specific regions of the brain

Defining nodes may be :
- based on cytoarchitecture (Brodmann areas)
- probabilistic
- based on chemoarchitecture
- anatomical
- functional
- random
- data-driven
- voxel-based
- based on myeloarchitecture
- multimodal

Question 31

What problems with edge definition (thresholding) may arise when building the graph?

Answer 31

1. thresholding can modify the network structure (gain of links when low connectivity and loss of lonk when high connectivity)
2. thresholding can result in different stages of „network fragmentation“ for each subject and thus in different numbers of nodes forming the network
3. low edge thresholds may result in „uniformity“ of the networks -> no possibility to detect differences between subjects
4. Results (association of network topology with independent variables) may uniquely dependent on the precise threshold value

=> Solution: construct each individual network over a range of thresholds (10%, 20%, …, 50% etc.)

Question 32

Is fine-grained approach to defining nodes bettr than coarse-grained one?

Answer 32

Fine-grained vs. Coarse-grained approach to defining nodes depends on the objective
NB! With increasing globally the granularity doesn’t matter quite soon — from 30% of most important connections the explanatory power doesn’t differ much

Question 33

Describe connectome pipeline

Answer 33

1. data acquisiton (sMRI + fMRI)
2. Preprocessing
3. Defining nodes
4. Defining edges
5. Building a graph/a matrix
6. Network Analysis

Question 34

How can edges be defined?

Answer 34

Again, depends on the hypothesis and objective
May be based on:
- correlation
- partial correlation
- mutual information

And one may map functional connectivity in:
- rest condition
- dynamic conditions: e.g. using sliding window

Question 35

Why is thresholding needed? What two approaches to thresholding are there?

Answer 35

Thresholding solves the problem of weak links, which tend to obscure the topology of strong and significant connections
absolute (everything < thr = 0) vs. proportional (% of highest correlations are considered as links) thresholding

Question 36

What is the difference between directed and undirected networks? Where are they applied?

Answer 36

Directed networks:
* the connections between nodes have a specific direction or flow associated with them
* inference of causality from functional data; tract tracing in sMRI

Undirected networks:
* the connections between nodes are bidirectional and symmetric
* fMRI, MEG, EEG; diffusion MRI, sMRI