I'm learning about Hopfield networks and had a question regarding some of the mathematical equations. The concept and properties of the network are relatively straightforward, but when I look at the equations there are a few parts I don't understand.

Here's the particular portion I'm looking at:

Given $\mathbf{x} = [x_1,\ x_2,\ ...,\ x_I]$, that is $N = 1$, according to Hebbian learning the weights of the network are given as:

$$w_{ij} = x_ix_j$$

then $\mathbf{x}$ is a stable point of the Hopfield network.

$$a_i = \Sigma_jw_{ij}x_j = \Sigma x_ix_j^2 = Ix_i$$

The particular parts within this excerpt is in the first line, and also in the activation equation. First, what does the $N = 1$ mean? Does it mean that there is one layer, given the single input vector $\mathbf{x}$?

Also for the activation equation, how is it derived that $\Sigma x_ix_j^2 = Ix_i$? I believe that $I$ refers to the number of neurons in the network, and $x_i$ is the activity, so does it mean that the activation equals the activity of a particular neuron multiplied with the total number of neurons?

I'm sorry if the question seems very vague, but I'm new to the topic and have no idea where to start. I've been reading on neural networks and how they work, but the mathematical portions are proving difficult. Any advice or answers are greatly appreciated. Thank you.


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