Storing covariance with nonlinearly interacting neurons

A time-dependent, nonlinear model of neuronal interaction which was probabilistically analyzed in a previous article is shown here to be a natural generalization of the Hartline-Ratliff model of the Limulus retina. Although the primary physical variables in the model are the membrane potentials of neurons, the equations which govern the means and covariances of the membrane potentials are coupled through the average firing rates; as a consequence, the average firing rates control the selective storage and retrieval of covariance information. Motor learning in the cerebellar cortex is treated as a problem of covariance storage, and a predicition is made for the underlying synaptic plasticity: the change in synaptic strength between a parallel fiber and a Purkinje cell should be proportional to the covariance between discharges in the parallel fiber and the climbing fiber. Unlike previous proposals for synaptic plasticity, this prediction requires both facilitation and depression to occur (under different conditions) at the same synapse.