A striking aspect of cortical neural networks is the divergence of a relatively small number of input channels from the peripheral sensory apparatus into a large number of cortical neurons, an over-complete representation strategy. Cortical neurons are then connected by a sparse network of lateral synapses. Here we propose that such architecture may increase the persistence of the representation of an incoming stimulus, or a percept. We demonstrate that for a family of networks in which the receptive field of each neuron is re-expressed by its outgoing connections, a represented percept can remain constant despite changing activity. We term this choice of connectivity REceptive FIeld REcombination (REFIRE) networks. The sparse REFIRE network may serve as a high-dimensional integrator and a biologically plausible model of the local cortical circuit.