Representing Connectivity

Question 1

What is a connectivity matrix?

Answer 1

A matrix that represents a network (also called a graph).

Connectivity matrix is commonly called an ‘adjacency’ matrix.

Question 2

What types of Brain Connectivity exist?

Answer 2

Brain connectivity refers to the network of neural connections in the brain. It includes:

*	Structural Connectivity: Physical links between neurons or brain regions, often represented by anatomical pathways.
*	Functional Connectivity: Statistical relationships between neural activities in different regions.
*	Effective Connectivity: Causal influence one neural system exerts over another.

Question 3

A 3-neuron ring is a cyclic directed network where each neuron connects to the next in ring. What’s its adjacency matrix?

assume rows are ‘from’ and columns are ‘to’

Answer 3

0 1 0
0 0 1
1 0 0

Question 4

What is an Adjacency Matrix?

Answer 4

An adjacency matrix, also called connectivity matrix, is a square matrix used to represent a graph or network. In brain connectivity:

*	Rows and columns represent neurons or brain regions.
*	Entries (0 or 1) indicate the absence (0) or presence (1) of a connection.
*	It provides a compact representation of the network’s structure.

Question 5

What’s the Adjacency Matrix representation of a 3 neuron ring?

Answer 5

A 3-neuron ring is a circular network where each neuron connects to the next. The adjacency matrix for this network is:

0 1 0
0 0 1
1 0 0

Question 6

What’s a recurrent neural network?

Answer 6

In a recurrent network, neurons have feedback loops, possibly including self-connections.

One possible adjacency matrix is:

1 1 0
0 1 1
1 0 1

Question 6

What is the Adjacency matrix of a feed forward neural network with three neurons where neuron 1 projects to 2 and 3?

Answer 6

A feedforward network involves a single neuron projecting to two others. The adjacency matrix is:

0 1 1
0 0 0
0 0 0

*	Neuron 1 projects to Neurons 2 and 3.
*	Neurons 2 and 3 have no outgoing connections.

Question 7

Describe the network in the following adjacency matrix:

1 1 0
0 1 1
1 0 1

Answer 7

• Neurons have self-connections (diagonal 1s).
  
  • Neuron 1 connects to Neuron 2.
  • Neuron 2 connects to Neuron 3.
  • Neuron 3 connects back to Neuron 1.

Question 8

What is a weighted graph?

Answer 8

It’s a network where edges have weights (usually real numbers).

Question 9

What’s a directed graph?

Answer 9

It’s a graph where edges are directed from nodes in a specific order. Directed graphs are usually not symmetrical.

Question 10

Make the adjacency matrix for this network.

Answer 10

Note: this is an undirected network

Question 11

What is the ‘in-degree’ of a node in a network?

node = neuron

Answer 11

The number of incoming-connections

Question 12

Given an adjacency matrix, how can you calculate the in-degree for all nodes?

assume the matrix is unweighted

Answer 12

By summing the matrix columnwise.

Because each column represents the incoming connections into a neuron.