- Directed, weighted, dynamic graph
- Simple model: a random directed graph
- Edge weights change over time
- New connections form and existing connections break
Simple random model
A random, directed graph where n = # of neurons, p = probability that a connection exists between two neurons.
Question: what can we predict using this model?
- What is the probability of having a path of length 2 between a and b a direct connection?
-> (having a length 2 path without a direct connection) =
- What is the number of bidirected cycles?
-> (# of bidirected cycles) =
- What is the number of paths of length k?
-> (# of paths of length k) =
- When does the graph become connected?
-> (degree(i)) = p(n-1).When p > log(n)/n, the graph quickly becomes connected:
Every monotone property (i.e. connectivity, perfect matching, Hamiltonian cycle) has such a sharp transition point.
How different is a human brain from a random network?
Study of rat visual cortex (Song et al., 2005) revealed several nonrandom features in synaptic connectivity: bidirectional connections and several other three-neuron connectivity patterns (“motifs”) are more common than expected in a random network.
The study also suggests that the strong connections are more clustered than the weak ones, which can be viewed as “a skeleton of stronger connections in a sea of weaker ones”.
Connectomes also change dynamically:
- Connections that are used more are strengthened
- Connections that are used less are decayed
- If there is a connection A -> B, then new connection B -> A is formed (reciprocity)
- If there is a connection A -> B -> C, then new connection A -> C is formed (transitivity)
- If two neurons keep firing together, the connection between them are strengthened over time: