A dynamically stable associative learning neural network system include a plurality of synapses (122,22-28), a non-linear function circuit (30) and an adaptive weight circuit (150) for adjusting the weight of each synapse based upon the present signal and the prior history of signals applied to the input of the particular synapse and the present signal and the prior history of signals applied to the input of a predetermined set of other collateral synapses. A flow-through neuron circuit (1110) embodiment includes a flow-through synapse (122) having a predetermined fixed weight. A neural network is formed by a set of flow-through neuron circuits connected by flow-through synapses to form separate paths between each input (215) and a corresponding output (245). In one embodiment (200), the neuron network is initialized by setting the adjustable synapses at some value near the minimum weight and setting the flow-through neuron circuits at some arbitrarily high weight. The neural network embodiments are taught by successively application of sets of inputs signals to the input terminals until a dynamic equilibrium is reached.