Optimal Properties of Analog Perceptrons with Excitatory Weights

The cerebellum is a brain structure which has been traditionally devoted to supervised learning. According to this theory, plasticity at the Parallel Fiber (PF) to Purkinje Cell (PC) synapses is guided by the Climbing fibers (CF), which encode an ‘error signal’. Purkinje cells have thus been modeled as perceptrons, learning input/output binary associations. At maximal capacity, a perceptron with excitatory weights expresses a large fraction of zero-weight synapses, in agreement with experimental findings. However, numerous experiments indicate that the firing rate of Purkinje cells varies in an analog, not binary, manner. In this paper, we study the perceptron with analog inputs and outputs. We show that the optimal input has a sparse binary distribution, in good agreement with the burst firing of the Granule cells. In addition, we show that the weight distribution consists of a large fraction of silent synapses, as in previously studied binary perceptron models, and as seen experimentally. Author Summary: Learning properties of neuronal networks have been extensively studied using methods from statistical physics. However, most of these studies ignore a fundamental constraint in networks of real neurons: synapses are either excitatory or inhibitory, and cannot change sign during learning. Here, we characterize the optimal storage properties of an analog perceptron with excitatory synapses, as a simplified model for cerebellar Purkinje cells. The information storage capacity is shown to be optimized when inputs have a sparse binary distribution, while the weight distribution at maximal capacity consists of a large amount of zero-weight synapses. Both features are in agreement with electrophysiological data.