It is well known that mammalian cells have water permeable membranes and that their volume is constantly challenged by both external perturbations as well as their own physiological regulation. Thus, it is critical for them to be able to control their volume in response to environmental stresses in order to maintain their expected function. Abnormal cellular volume regulation can be involved in the pathophysiology of various neuropsychiatric disorders including major depression disorders, bipolar disorders, and schizophrenia. Assessing the homeostatic capacity of cells harvested from patients in several experimental paradigms represent thus a strategy to identify cellular biomarkers of those disorders. To this end, in silico mathematical models are developed in order to describe transmembrane water fluxes and ion intracellular concentrations. Those models include many parameters such as leak conductances, pump and cotransport strengths, and water permeability. Despite their widespread use, the mathematical properties of those models are usually unknown. Here, we establish analytical results on the existence and uniqueness of steady-state for a large family of pump-leak-cotransport models.