Despite decades of meticulous classification of spinal subtypes involved in central pattern generation (CPG), the synaptic architecture and mechanisms underlying locomotion remain elusive. We recently proposed a random network model, based on balanced excitation/inhibition, termed the Balanced Sequence Generator (BSG). This model recapitulated the rhythmic sequences observed in the cord and accommodated multi-functionality critical to CPGs. However, it did not include or address the diversity of spinal cell types. Here, we present a new approach that expands the BSG-framework to reconcile network structure, cell identities, and function. First, we ask what properties of the BSG allowed it to generate oscillations from structure by disentangling the network structure according to sequence. Strikingly, a consistent spatial pattern emerged forming a distinct “Mexican-Hat”. Next, we asked if such a structure is present in the cord and whether it supports locomotor function. Using spatial transcriptomics, single-cell RNA datasets, and atlases from mice, we reconstructed densities of cardinal cell-types in the lumbo-sacral spinal cord volume. We sampled hundreds of networks from these distributions, imposing either random connectivity or literature-derived cell-type specific projections. Remarkably, the latter approach revealed a Mexican-hat structure, robustly generated rhythms and appropriate muscular coordination. Flexibility of motor output was obtained by incorporating sensory input and cellular properties. The model predicts the emergence of spatially propagating waves reminiscent of aquatic vertebrates generating undulatory locomotion, suggesting a continuation of mechanisms into tetrapod terrestrial species. Hence, we propose that genetic identities constrain spatial projection biases such that function emerges from spatially organized patterns.