Implementation of Lenia as a Reaction-Diffusion System

The relationship between reaction-diffusion (RD) systems, characterized by continuous spatiotemporal states, and cellular automata (CA), marked by discrete spatiotemporal states, remains poorly understood. This paper delves into this relationship through an examination of a recently developed CA known as Lenia. We demonstrate that asymptotic Lenia, a variant of Lenia, can be comprehensively described by differential equations, and, unlike the original Lenia, it is independent of time-step ticks. Further, we establish that this formulation is mathematically equivalent to a generalization of the kernel-based Turing model (KT model). Stemming from these insights, we establish that asymptotic Lenia can be replicated by an RD system composed solely of diffusion and spatially local reaction terms, resulting in the simulated asymptotic Lenia based on an RD system, or "RD Lenia". However, our RD Lenia cannot be construed as a chemical system since the reaction term fails to satisfy mass-action kinetics.

Organization of a Latent Space structure in VAE/GAN trained by navigation data

We present a novel artificial cognitive mapping system using generative deep neural networks (VAE/GAN), which can map input images to latent vectors and generate temporal sequences internally. The results show that the distance of the predicted image is reflected in the distance of the corresponding latent vector after training. This indicates that the latent space is constructed to reflect the proximity structure of the data set, and may provide a mechanism by which many aspects of cognition are spatially represented. The present study allows the network to internally generate temporal sequences analogous to hippocampal replay/pre-play, where VAE produces only near-accurate replays of past experiences, but by introducing GANs, latent vectors of temporally close images are closely aligned and sequence acquired some instability. This may be the origin of the generation of the new sequences found in the hippocampus.