# Connectome

1. Structure:
1. Directed, weighted, dynamic graph
2. Simple model: a random directed graph $D_{n,p}$
2. Dynamics:
• Edge weights change over time
• New connections form and existing connections break

# Simple random model

A random, directed graph $D_{n,p}$ where n = # of neurons, p = probability that a connection exists between two neurons.

Question: what can we predict using this model?

1. What is the probability of having a path of length 2 between a and b a direct connection?
-> $\Pr$(having a length 2 path without a direct connection) = $(1-p)(1-(p(1-p) + (1-p)p + (1-p)^2)^{n-2}) = (1-p)(1-(1-p^2)^{n-2})$
2. What is the number of bidirected cycles?
-> $\mathbf{E}$(# of bidirected cycles) = ${{n}\choose{2}}p^2$
3. What is the number of paths of length k?
-> $\mathbf{E}$(# of paths of length k) = ${{n}\choose{k+1}}(k+1)!p^k \approx (\frac{ne}{k+1})^{k+1}p^k \approx n(np)^k$
4. When does the graph become connected?
-> $\mathbf{E}$(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:

1. Connections that are used more are strengthened
2. Connections that are used less are decayed
3. If there is a connection A -> B, then new connection B -> A is formed (reciprocity)
4. If there is a connection A -> B -> C, then new connection A -> C is formed (transitivity)
5. If two neurons keep firing together, the connection between them are strengthened over time:
$X(t+1) = \frac{X(0) + \alpha W(t)^T X(t)}{ \lVert X(0)^T + \alpha W(t)^T X(t) \rVert }$
$W(t+1) = \frac{1}{1+\beta} (I + \beta X(t) X(t)^T ) W(t)$