Neural feedback-triads consisting of two feedback loops with a nonreciprocal lateral connection from one loop to the other are ubiquitous in the brain. We show analytically that the dynamics of this network topology are determined by algebraic combinations of its five synaptic weights. Exploration of network activity over the parameter space demonstrates the importance of the nonreciprocal lateral connection and reveals intricate behavior involving continuous transitions between qualitatively different activity states. In addition, we show that the response to periodic inputs is narrowly tuned around a center frequency determined by the effective synaptic parameters.