An open problem in neuroscience is to explain the functional role of oscillations in neural networks, contributing, for example, to perception, attention, and memory. Cross-frequency coupling (CFC) is associated with information integration across populations of neurons [1]. Impaired CFC is linked to neurological disease [2]. It is unclear what role CFC has in information processing and brain functional connectivity. We construct a model of CFC which predicts a computational role for observed $\theta - \gamma$ oscillatory circuits in the hippocampus and cortex [3]. Our model predicts that the complex dynamics in recurrent and feedforward networks of coupled oscillators performs robust information storage and pattern retrieval. Based on phasor associative memories (PAM) [4], we present a novel oscillator neural network (ONN) model [5] that includes subharmonic injection locking (SHIL) [6] and which reproduces experimental observations of CFC. We show that the presence of CFC increases the memory capacity of a population of neurons connected by plastic synapses. CFC enables error-free pattern retrieval whereas pattern retrieval fails without CFC. In addition, the trade-offs between sparse connectivity, capacity, and information per connection are identified. The associative memory is based on a complex-valued neural network, or phasor neural network (PNN). To our knowledge, this is the first work to validate the capacity analysis of $Q$-state PAM networks [7] through simulation and provide a model of how to implement such a memory on a physical system. We show that for values of $Q$ which are the same as the ratio of $\gamma$ to $\theta$ oscillations observed in the hippocampus and the cortex, the associative memory achieves greater capacity and information storage than previous models. The novel contributions of this work are providing a computational framework based on oscillator dynamics which predicts the functional role of neural oscillations and connecting concepts in neural network theory and dynamical system theory.