Biological synaptic plasticity exhibits nonlinearities that are not accounted\nfor by classic Hebbian learning rules. Here, we introduce a simple family of\ngeneralized nonlinear Hebbian learning rules. We study the computations\nimplemented by their dynamics in the simple setting of a neuron receiving\nfeedforward inputs. These nonlinear Hebbian rules allow a neuron to learn\ntensor decompositions of its higher-order input correlations. The particular\ninput correlation decomposed and the form of the decomposition depend on the\nlocation of nonlinearities in the plasticity rule. For simple, biologically\nmotivated parameters, the neuron learns eigenvectors of higher-order input\ncorrelation tensors. We prove that tensor eigenvectors are attractors and\ndetermine their basins of attraction. We calculate the volume of those basins,\nshowing that the dominant eigenvector has the largest basin of attraction. We\nthen study arbitrary learning rules and find that any learning rule that admits\na finite Taylor expansion into the neural input and output also has stable\nequilibria at generalized eigenvectors of higher-order input correlation\ntensors. Nonlinearities in synaptic plasticity thus allow a neuron to encode\nhigher-order input correlations in a simple fashion.\n