Now we put matrices in motion. Take two neurons wired to each other, chop time into little bins, and let each neuron's activity in the next bin depend on both neurons' activity in the current bin. That's a discrete dynamical system — and its whole future is set by one matrix.

Each neuron's next activity is a weighted mix of the current activities:

a₁(t) = w₁₁·a₁(t−1) + w₂₁·a₂(t−1)
a₂(t) = w₁₂·a₁(t−1) + w₂₂·a₂(t−1)

That's a mess of subscripts — so we pack the weights into a matrix W and the activities into a vector a, and the whole thing collapses to:

a(t) = W · a(t−1)

The activity vector at each step = the weight matrix times the activity vector at the previous step. Same matrix-vector multiply, applied over and over.

"Discrete" just means time moves in steps (bin 1, bin 2, bin 3…) rather than flowing continuously. Given a starting activity a₀, you crank the equation forward one bin at a time to see how the circuit evolves.

🤖 This is a Recurrent Neural Network (RNN). a(t) = W·a(t−1) is the skeleton of every RNN and, loosely, every transformer's sequence processing: a hidden state repeatedly multiplied by a weight matrix. Everything you learn here about stability applies directly to training them.