Key concepts#

General dynamical systems#

System concerned with movement of point over time in some relevant geometrical space
The state of a dynamical system is a complete snapshot of the system at that point in time.
The state space is the relevant geometrical space in which to model the state evolving (the set of all possible states)
Are discrete or continuous depending on how the evolution rules are defined
Discrete dyanmical systems have step-like update rules: $x_{n+1} = F(x_n)$
Continuous dyanmical systems have differential equations governing the evolution: $\frac{dx}{dt} = F(x, t)$

We can model a neural circuit as a discrete dynamical system. If $\bar{u}_t$ is the vector of neural activity at time $t$ and $W$ is the synaptic weight matrix, then: $$\bar{u}_{t+1} = W\bar{u}_t $$

An eigenvector is a vector that doesn’t change direction in a transformation ($\bar{v}$ below).
The associated eigenvalue is the factor by which the eigenvector is scaled ($\lambda$ below). \ $$A\bar{v} = \lambda\bar{v} $$
np.linalg.eig gives you the eigenvalues and eigenvectors of a matrix.
Eigenvalues are more specifically defined than eigenvectors: for an eigenvector $\bar{v}$, $-2\bar{v}$ and $6\bar{v}$ are also eigenvectors. In other words, there are an infinite number of eigenvectors for a given eigenvalue that lie along the same line. For this reason, we often use unit eigenvectors (although this does not account for flipping between negative/positive).
Additionally, sometimes all of the vectors in a plane (or more in higher D space) can be eigenvectors for a given eigenvalue. This will not always be obvious from the outputs of np.linalg.eig.
You will not necessarily be able to form a basis for a space from the eigenvectors.
We solve for the eigenvalues of matrix $B$ by solving det($B$ - $\lambda I$) = 0.
Complex eigenvalues exist and have associated complex eigenvectors - a matrix with complex eigenstuff performs some kind of rotation.
Nonsquare matrices do not have eigenvalues.
An n x n matrix has n eigenvalues but they could be real or complex and they are not necessarily distinct (meaning a matrix could have two eigenvalues of 2, with different associated eigenvectors).