A fundamental question is how the cerebral neocortex operates functionally, computationally. The cerebral neocortex with its superficial and deep layers and highly developed recurrent collateral systems that provide a basis for memory-related processing might perform somewhat different computations in the superficial and deep layers. Here we take into account the quantitative connectivity within and between laminae. Using integrate-and-fire neuronal network simulations that incorporate this connectivity, we first show that attractor networks implemented in the deep layers that are activated by the superficial layers could be partly independent in that the deep layers might have a different time course, which might because of adaptation be more transient and useful for outputs from the neocortex. In contrast the superficial layers could implement more prolonged firing, useful for slow learning and for short-term memory. Second, we show that a different type of computation could in principle be performed in the superficial and deep layers, by showing that the superficial layers could operate as a discrete attractor network useful for categorisation and feeding information forward up a cortical hierarchy, whereas the deep layers could operate as a continuous attractor network useful for providing a spatially and temporally smooth output to output systems in the brain. A key advance is that we draw attention to the functions of the recurrent collateral connections between cortical pyramidal cells, often omitted in canonical models of the neocortex, and address principles of operation of the neocortex by which the superficial and deep layers might be specialized for different types of attractor-related memory functions implemented by the recurrent collaterals.