Quantitative relations between BOLD responses, cortical energetics and impulse firing across cortical depth

The blood oxygen level-dependent (BOLD) functional magnetic resonance imaging (fMRI) signal arises as a consequence of changes in cerebral blood flow (CBF) and cerebral metabolic rate of oxygen $$ that in turn are modulated by changes in neural activity. Recent advances in imaging have achieved sub-millimetre resolution and allowed investigation of the BOLD response as a function of cortical depth. Here, we adapt our previous theory relating the BOLD signal to neural activity to produce a quantitative model that incorporates venous blood draining between cortical layers. The adjustable inputs to the model are the neural activity and a parameter governing this blood draining. A three-layer version for transient neural inputs and a multi-layer version for constant or tonic neural inputs are able to account for a variety of experimental results, including negative BOLD signals.