Storage of Correlated Patterns in Standard and Bistable Purkinje Cell Models

The cerebellum has long been considered to undergo supervised learning, with climbing fibers acting as a ‘teaching’ or ‘error’ signal. Purkinje cells (PCs), the sole output of the cerebellar cortex, have been considered as analogs of perceptrons storing input/output associations. In support of this hypothesis, a recent study found that the distribution of synaptic weights of a perceptron at maximal capacity is in striking agreement with experimental data in adult rats. However, the calculation was performed using random uncorrelated inputs and outputs. This is a clearly unrealistic assumption since sensory inputs and motor outputs carry a substantial degree of temporal correlations. In this paper, we consider a binary output neuron with a large number of inputs, which is required to store associations between temporally correlated sequences of binary inputs and outputs, modelled as Markov chains. Storage capacity is found to increase with both input and output correlations, and diverges in the limit where both go to unity. We also investigate the capacity of a bistable output unit, since PCs have been shown to be bistable in some experimental conditions. Bistability is shown to enhance storage capacity whenever the output correlation is stronger than the input correlation. Distribution of synaptic weights at maximal capacity is shown to be independent on correlations, and is also unaffected by the presence of bistability. Author Summary: The cerebellum is one of the main brain structures involved in motor learning. Classical theories of cerebellar function assign a crucial role to Purkinje cells (PCs), that are assumed to perform as simple perceptrons. In these theories, PCs should learn to provide an appropriate motor output, given a particular input, encoded by the granule cell (GC) network. This learning is assumed to occur through modifications of synapses, under the control of the climbing fiber input to PCs, which is supposed to carry an error signal. In this paper, we compute storage capacity and distribution of weights in the presence of temporal correlations in inputs and outputs, which are unavoidable in sensory inputs and motor outputs. Furthermore, we study how bistability in the PCs affects capacity and distribution of weights. We find that (1) capacity increases monotonically with both input and output correlations; (2) bistability increases storage capacity, when the output correlation is larger than the input correlation; (3) the distribution of weights at maximal capacity is independent of the degree of temporal correlations, as well as the nature of the output unit (mono or bistable) and is in striking agreement with experimental data.