Real-world complex systems can often be described by changes in a much smaller set of variables; the space they parameterise is called the system’s manifold. Shifting from single-cell to population-level analysis, we see this phenomenon in neuroscience, with recent evidence for “neural manifolds”: low-dimensional, functionally-relevant descriptions of population activity. The most general way to discover manifolds is to find the simplest transform of activity that maximises the mutual information (MI) between the transformation and the relevant task variables, since MI is the most general measure of dependency between variables. However, estimating MI directly is computationally expensive and affected by limited sampling, making optimisation infeasible. We propose the use of the Quadratic Mutual Information (QMI), a related quantity that is maximal where MI is maximal. Estimators of QMI are efficient and differentiable, and we can effectively maximise MI by gradient methods on QMI. Our approach yields a new neural manifold analysis method: Quadratic Mutual Information Manifold Inference (Q-MIMI). It maximises the MI between the task variables and the manifold variables, while maximising the entropy of the latter. We apply this method to: i) A synthetic dataset of Poisson-model place-cells, and ii) a two-photon calcium imaging dataset from populations of neurons in hippocampal subfield CA1 while navigating a circular track. We show that Q-MIMI can effectively retrieve the one-dimensional manifold in both. This will help us ask more precise questions about the relationship between task-variables along the neural manifold, helping us understand the context-dependent nature of representation in the hippocampus.