Inference in both brains and machines can be formalized by optimizing a shared objective: maximizing the evidence lower bound (ELBO) in machine learning, or minimizing variational free energy (F) in neuroscience (ELBO=−F). While this equivalence suggests a unifying framework, it leaves open how inference is implemented in neural systems. Here, we show that online natural gradient descent on F, under Poisson assumptions, leads to a recurrent spiking neural network that performs variational inference via membrane potential dynamics. The resulting model—the iterative Poisson variational autoencoder (iP-VAE)— replaces the encoder network with local updates derived from natural gradient descent on F. Theoretically, iP-VAE yields a number of desirable features such as emergent normalization via lateral competition, and hardware-efficient integer spike count representations. Empirically, iP-VAE outperforms both standard VAEs and Gaussian-based predictive coding models in sparsity, reconstruction, and biological plausibility. iP-VAE also exhibits strong generalization to out-of-distribution inputs, exceeding hybrid iterative-amortized VAEs. These results demonstrate how deriving inference algorithms from first principles can yield concrete architectures that are simultaneously biologically plausible and empirically effective.