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Neural activity in the brain is accompanied by changes in cerebral blood flow (CBF) and blood oxygenation that are detectable with functional magnetic resonance imaging (fMRI) techniques. In this paper, recent mathematical models of this hemodynamic response are reviewed and integrated. Models are described for: (1) the blood oxygenation level dependent (BOLD) signal as a function of changes in cerebral oxygen extraction fraction (E) and cerebral blood volume (CBV); (2) the balloon model, proposed to describe the transient dynamics of CBV and deoxy-hemoglobin (Hb) and how they affect the BOLD signal; (3) neurovascular coupling, relating the responses in CBF and cerebral metabolic rate of oxygen (CMRO₂) to the neural activity response; and (4) a simple model for the temporal nonlinearity of the neural response itself. These models are integrated into a mathematical framework describing the steps …