Navigating without landmarks relies on path integration (PI), the ability to estimate position by integrating self-motion cues. Grid cells in the entorhinal cortex (EC) are thought to perform this computation by combining speed and direction signals. Classical models fall into two camps: continuous attractor neural networks (CANNs), which assume fixed connectivity and externally supplied velocity inputs, and self-organizing feedforward models, which explain grid formation but not PI. In this work, we introduce a self-organizing CANN model for unidirectional PI, which, unlike other self-organizing approaches, makes PI an intrinsic property of the system’s stable solution rather than an externally imposed feature. From structured spatial input, the network learns feedforward weights via Hebbian plasticity and competition, forming a representation of a one-dimensional manifold. Stacking multiple feedforward layers yields predictive coding of future positions—predicted analytically, confirmed in simulations, and consistent with reports of predictive grid cells in superficial medial EC. The same principles allow recurrent connectivity to self-organize, enabling continuous attractor dynamics. With an independent additive speed-modulated input current, the model adapts its dynamics to running speed, achieving PI. Analytical tractability offers mechanistic insight and a unified framework that bridges learning, prediction, and navigation, paving the way for two-dimensional extensions.