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Let generator $G$ have parameters $\theta$ and discriminator $D$ paramenters $\psi$.
Theory requires: $$ \psi_{t+1} \leftarrow \operatorname{argmax}_{\psi} \mathbb{E}_{x\sim P} \log D\left(x;\psi\right) + \mathbb{E}_{z\sim \mathcal{N}} \log \left(1 - D\left(G(z;\theta_t);\psi\right)\right) $$
In practice gradient step only.
The first corresponds to definition. What does the second correspond to?
Nice article about GAN (not tutorial).
Theis