Graph Theory and Connectivity Flashcards
Why is univariate analysis problematic?
- Multiple functions: There is no 1-to-1 correspondence between a brain region and function
- Context dependence: activation of different areas when the seemed context is similar, but the communication with the context is different
- Co-activation vs. Functional integration: co-activation sometimes despite no functional connection
What options are there for analysis of the brain?
Univariate vs. Bivariate vs. Multivariate = Process vs. Interaction vs. Pattern
What is (rest) functional connectivity?
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
What does seed-voxel correlation do?
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
What are the pros and cons of functional connectivity
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
What is Psycho-Physiological interaction
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.
How to compute general linear model (GLM) for 2x2 factorial design?
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
What two interpretations of PPI are possible?
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
What are the pros and cons of PPI?
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
What is effective connectivity? What models of effective connectivity are there?
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
What is Granger causality? What limitations does it have?
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
What types of patterns are distinguished in the network analysis?
- 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
How can we quantitatively assess these patterns?
Using Graph Theory and Complex Network Theory
What is an alternative representation of graphs and networks?
Matrices
What constitues a graph?
Nodes and edges (links between the nodes)
What is the degree of a node?
It is the number of its edges
How are weighted and binary networks different?
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)
How to assess a network?
- 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
What is functional segregation?
- 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
How to measure segregation?
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
Give an example of the study where clustering coefficient was used as a measure?
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
Give examples of studies where modularity was used as a measure?
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
What is functional integration?
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
How the optimal balance between functional segmentation and integration is called?
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
What is special about brain networks in schizophrenia?
Liu et al., 2008: More of a random organisation in schizophrenia
What is centrality?
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
What measures of centrality are used in the network analysis?
- 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
What are the characteristics of the hubs?
- 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
What is resilience?
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
What is degeneracy?
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
How can nodes be defined?
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
What problems with edge definition (thresholding) may arise when building the graph?
- thresholding can modify the network structure (gain of links when low connectivity and loss of lonk when high connectivity)
- thresholding can result in different stages of „network fragmentation“ for each subject and thus in different numbers of nodes forming the network
- low edge thresholds may result in „uniformity“ of the networks -> no possibility to detect differences between subjects
- 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.)
Is fine-grained approach to defining nodes bettr than coarse-grained one?
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
Describe connectome pipeline
- data acquisiton (sMRI + fMRI)
- Preprocessing
- Defining nodes
- Defining edges
- Building a graph/a matrix
- Network Analysis
How can edges be defined?
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
Why is thresholding needed? What two approaches to thresholding are there?
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
What is the difference between directed and undirected networks? Where are they applied?
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
