18-07-19 Multi-scale brain networks¶

Betzel, R. F., & Bassett, D. S. (2017). Multi-scale brain networks. Neuroimage, 160, 73-83.

@article{betzel2017multi,
  title     =   {Multi-scale brain networks},
  author    =   {Betzel, Richard F and Bassett, Danielle S},
  journal   =   {Neuroimage},
  volume    =   {160},
  pages     =   {73--83},
  year      =   {2017},
  publisher =   {Elsevier}
}

Original | Mendeley

Overview

• 1 Intro
• 2 Functional and structural brain networks
• 3 Multi-scale network analysis
  • 3.1 Multi-scale community structure
    • 3.1.1 Multi-scale community structure in the neuroimaging literature
    • 3.1.2 Implementation and practical considerations
    • 3.1.3 Selecting the resolution parameter
    • 3.1.4 Consensus community structure and communities of interest
  • 3.2 Multi-scale rich club and core–periphery organization
  • 3.3 Multi-scale temporal networks
    • 3.3.1 Multi-scale, multi-layer network analysis
    • 3.3.2 Practical considerations
  • 3.4 Multi-scale spatial networks
• 4 Conclusion and future directions

1. Intro¶

"scale":

1. spatial
   • cell /synapse
   • brain region / large-scale fiber tract
2. temporal
   • sub-millisecond
   • entire lifespan / evolutionary changes across different species
3. topological
   • individual nodes
   • network

focus on...

• networks algorithms (e.g., community detection)
• multi-scale temporal networks & multi-layer techniques

2. Functional and structural brain networks¶

connectivity:

• structured / anatomical connectivity (SC)
• functional (FC)

connectivity mat: $\mathbf{A}$

element: $A_{i,j}$

3. Multi-scale network analysis¶

Simpson 2016

topological

degree Takeuchi 2015

spectrum

path length : average # of steps Santernecchi 2014

mesoscale: - community structure Fortunato 2010 - cores & peripheries - rich clubs

3.1 Multi-scale community structure¶

$$\begin{align*} \tag{eq.1} Q = \sum_{ij} \big[ A_{ij} - P_{ij} \big] \delta(\sigma_i\sigma_j) \end{align*}$$

• $Q$: modularity,
• $i$, $j$: node index
• $A_{ij}$: weight of observed connection between nodes $i$ & $j$
• $P_{ij}$: weight of expected connection
• $\sigma_i \in [1, \cdots, K]$: communities node $i$ is assigned
• $\delta(\sigma_i\sigma_j)$: Kronecker delta fn

$$\begin{align*} \tag{app.1} \delta(\sigma_i \sigma_j) = \begin{cases} 1 \quad \text{if arguments are the same}\\ 0 \quad \text{if NOT} \end{cases} \end{align*}$$

$$\begin{align*} \tag{eq.2} Q(\gamma) = \sum_{ij} \big[ A_{ij} - \gamma P_{ij} \big] \delta(\sigma_i \sigma_j) \end{align*}$$

4. Conclusion and future directions¶