Learning from experience depends at least in part on changes in neuronal connections. We present the largest map of connectivity to date between cortical neurons of a defined type (layer 2/3 L2/3 pyramidal cells in mouse primary visual cortex), which was enabled by automated analysis of serial section electron microscopy images with improved handling of image defects (250 × 140 × 90 μm volume). We used the map to identify constraints on the learning algorithms employed by the cortex. Previous cortical studies modeled a continuum of synapse sizes by a log-normal distribution. A continuum is consistent with most neural network models of learning, in which synaptic strength is a continuously graded analog variable. Here, we show that synapse size, when restricted to synapses between L2/3 pyramidal cells, is well modeled by the sum of a binary variable and an analog variable drawn from a log-normal distribution. Two synapses sharing the same presynaptic and postsynaptic cells are known to be correlated in size. We show that the binary variables of the two synapses are highly correlated, while the analog variables are not. Binary variation could be the outcome of a Hebbian or other synaptic plasticity rule depending on activity signals that are relatively uniform across neuronal arbors, while analog variation may be dominated by other influences such as spontaneous dynamical fluctuations. We discuss the implications for the longstanding hypothesis that activity-dependent plasticity switches synapses between bistable states.