# Connectome

- Structure:
- Directed, weighted, dynamic graph
- Simple model: a random directed graph

- Dynamics:
- 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.

# Nonrandom Features

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:

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