Translation-equivariantRepresentationinRecurrent NetworkswithaContinuousManifoldofAttractors: SupplementaryInformation

Based on the requirement of equivariant representation (Eq. Since the translation is continuous, the amount of translation can be made infinitesimally small. It is clear to see the translation operator ˆT is an exponential map of the translation generatorˆp. Wesee the left and right hand sides inaboveequations contain the same Gaussian terms with the samepositionsandwidthσ. Intheory, we consider the instantaneous stateu(x,t) is perturbed around the attractor state u(x s) in the originalCANdynamics(Eqs. 8a-8b), u(x,t)= u(x s)+δu(x,t). Theeigenvaluesλn forn 2aresmaller than1,indicating the coefficientsan (n 2) will eventually decay to zero and then they can be ignored.