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The cerebellar cortical circuitry may support a distinct second form of associative learning, complementary to the well-known synaptic plasticity (long term depression, LTD) that has been previously shown. As the granule cell axons ascend to the molecular layer, they make multiple synapses on the overlying Purkinje cells (PC). This ascending branch (AB) input, which has been ignored in models of cerebellar learning, is likely to be functionally distinct from the parallel fiber (PF) synaptic input. We predict that AB-PF correlations lead to Hebbian-type learning at the PF-PC synapse, including long term potentiation (LTP), and allowing the cortical circuit to combine AB-PF LTP for feedforward state prediction with climbing fiber LTD for feedback error correction. The new learning mechanism could therefore add computational capacity to cerebellar models and may explain more of the experimental data.