The balloon model revisited

Functional MRI based on the blood oxygenation level-dependent (BOLD) effect infers neural activity from hemodynamic changes through a hemodynamic response function (HRF). We revisit the balloon model of the BOLD effect and show that Grubb's empirical flow-volume power law can be used to determine how vascular transit time varies with flow and is governed by a single index of venous distensibility. Linearizing the resulting equations yields an explicit closed-form HRF, expressed in physiologically recognizable parameters, that reproduces the canonical response. The initial dip, time to peak and poststimulus undershoot are each traced to identifiable terms, and the relations assumed in calibrated BOLD signals emerge as outputs. When oxygen consumption is specified independently of flow, as in earlier generalizations, the analytic solution quantifies the specific relationships between the latency and depth of an initial dip and a neurovascular delay. The new model will prove useful for explaining variations in HRF in white matter, across the brain, and between different populations.