Synergistic-interaction mechanism in T(K*u) configuration

Establish whether applying the nonlinear target function T after convolution (the T(K*u) operation) with a sharply peaked bell-shaped profile in Asymptotic Lenia enhances synergistic interactions between spatial regions of the pattern u, thereby driving the emergence of complex structures.

Background

The discussion compares Asymptotic Lenia to neural field models and highlights a structural difference: Lenia applies the nonlinearity after convolution (T(K*u)), whereas classical neural field models often apply the nonlinearity before convolution (K*f(u)). The authors note that Lenia typically uses ring-shaped kernels and a sharply peaked, bell-shaped target function T, which differ from common choices in neural field theory.

Within this context, the authors propose a conjecture that the convolution-first configuration combined with a sharp activation profile may enhance synergistic interactions among pattern regions, potentially explaining the rich pattern formation observed in Lenia. This mechanism, if validated, could bridge insights between Lenia and neural field analyses of traveling waves and pattern formation.