Translation-equivariant Representation in Recurrent Networks with a Continuous Manifold of Attractors: Supplementary Information Wen-Hao Zhang

Based on the requirement of equivariant representation (Eq. Since the translation is continuous, the amount of translation can be made infinitesimally small. Also in Eq. (S3) we define ˆ p Differentiating the above equation we can derive a differential form of a translation operator, d ˆ T (a) da = ˆp exp( a ˆ p) = ˆ p ˆ T ( a). (S7) If the Gaussian ansatz was correct, based on Eq. (8a) they should satisfy that u( x s) = ρ null W We performed perturbative analysis to analyze the stability of the CAN dynamics. Substituting above equation into the modified CAN dynamics (Eq. Therefore, Eq. (S16) can be simplified as, τ t u(x s) + τ null S15e), we could project the Eq. ( n 2) Similar with the analysis in the CAN, we propose the following Gaussian ansatz of network's For simplicity, we assume the speed neurons' responses The projection is computing the inner product between the network dynamics (Eq.