The Glider Equation for Asymptotic Lenia

Lenia is a continuous extension of Conway's Game of Life that exhibits rich pattern formations including self-propelling structures called gliders. In this paper, we focus on Asymptotic Lenia, a variant formulated as partial differential equations. By utilizing this mathematical formulation, we analytically derive the conditions for glider patterns, which we term the ``Glider Equation.'' We demonstrate that by using this equation as a loss function, gradient descent methods can successfully discover stable glider configurations. This approach enables the optimization of update rules to find novel gliders with specific properties, such as faster-moving variants. We also derive a velocity-free equation that characterizes gliders of any speed, expanding the search space for novel patterns. While many optimized patterns result in transient gliders that eventually destabilize, our approach effectively identifies diverse pattern formations that would be difficult to discover through traditional methods. Finally, we establish connections between Asymptotic Lenia and neural field models, highlighting mathematical relationships that bridge these systems and suggesting new directions for analyzing pattern formation in continuous dynamical systems.