Course Content
What is fNIRS?
Where fNIRS comes from, what it measures in principle, and where it sits among other neuroimaging tools.
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The Optical Window
The fNIRS Signal
Interpreting fNIRS signals
fNIRS as a neuroimaging technique
Introduction to fNIRS
How does fNIRS work?
The physical logic from light entering tissue to a number quantifying the detected light in an fNIRS device
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Absorption and scattering
Modified Beer-Lambert Law (MBLL)
From Optical Signals to Hemoglobin
High-Density fNIRS & DOT
Brain Hemodynamics and Neurovascular Coupling
NIRS does not measure neural activity directly; it measures the vascular response that follows it. This unit builds the physiological bridge between the two. We start at the level of neurons and synapses, trace the cascade through metabolism and blood flow regulation, and arrive at the haemodynamic signals we actually record: changes in oxyhaemoglobin (ΔHbO) and deoxyhaemoglobin (ΔHbR). We examine what these signals represent biophysically, how to interpret them, and what the canonical Hemodynamic Response Function looks like. A recurring theme is the distinction between different physiological quantities that are often conflated: CMRO₂, cerebral blood flow, blood volume, oxygen saturation, and hemoglobin concentration are related but not equivalent, and confusing them leads to misinterpretation.
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Neurovascular Coupling
Physiological Interpretation of Hemoglobin Changes
The Hemodynamic Response Function
Collecting fNIRS Data
Not all fNIRS systems are the same. This unit provides a broad overview of the hardware landscape, from the ubiquitous Continuous Wave (CW) systems to Frequency Domain (FD) and Time Domain (TD) instruments, and the emerging class of speckle-based devices that measure blood flow rather than hemoglobin concentration. A central message is that the choice of hardware determines what you can and cannot measure: for example, TD and FD systems can independently quantify absorption and scattering, and therefore provide absolute hemoglobin concentrations, while CW systems are almost always limited to relative changes and must assume a fixed DPF. We also cover the practical side: how sources and detectors are arranged on the head (the montage), and what "good coupling" between optode and scalp actually means.
Processing fNIRS Data
Collecting fNIRS data is relatively accessible, but clean, interpretable data is not. This unit confronts the reality that the fNIRS signal is dominated by noise: physiological fluctuations from the heart, respiration, and slow vasomotion can be an order of magnitude larger than the neural signal of interest. Motion artifacts are arguably one of the defining challenges of the modality. We work through the major noise sources and introduce strategies to address each before considering statistical analysis. On the way, you will understand why simple averaging over blocks is insufficient, and why the General Linear Model (GLM) has been the standard framework. The unit closes with an overview of alternative quantification approaches for data analysis, including inter-subject correlation and functional connectivity, and the broader challenge of pipeline standardization.
fNIRS Challenges and Future Frontiers
The final unit steps back from technical details to consider the bigger picture: what fNIRS uniquely enables, and what still stands in its way. We explore three of the most exciting directions in the field: hyperscanning and social/developmental neuroscience, ecological validity through wearable deployment, and the study of populations that other neuroimaging modalities struggle to accommodate. We also return to the challenge of pipeline standardization and reproducibility as an open problem that newcomers entering the field have an opportunity (and a responsibility!) to address.
Introduction to fNIRS (English)

The Hemodynamic Response Function

🎯 as you read, try to

  • Describe the canonical shape and timing of the hemodynamic response
  • Distinguish a phenomenological model of the HRF from a physiological model 
  • Describe the balloon and Windkessel models
  • Explain why every model in this lesson is a simplification, and identify concrete cases where the canonical shape does not hold

So far, we discussed what drives the hemodynamic response and what its two signals represent. This lesson asks a more specific question: what does the response look like over time, and how has that shape been modelled? The answer to these can be done through two different kinds of model, one that fits the shape of the response, and one that explains why it has that shape, and it is worth being clear from the outset that neither kind of model is the response itself.

1. Modelling the Hemodynamic Response with a Function

The hemodynamic response function (HRF) is the time-course of the hemodynamic response evoked by a single, brief, isolated neural event. Its canonical shape is well established: a delay of one to two seconds after stimulus onset before any change appears, a rise to a peak at roughly five to six seconds, and a slow decay back toward baseline, frequently passing below it for several seconds before settling, an effect known as the post-stimulus undershoot.

🎍 MUST-READ paper

Boynton, G.M., Engel, S.A., Glover, G.H. & Heeger, D.J. (1996). “Linear systems analysis of functional magnetic resonance imaging in human V1.” Journal of Neuroscience, 16(13), 4207–4221.

This paper established that, to a good approximation, the hemodynamic response behaves as a linear, time-invariant system: the response to a stimulus can be predicted by convolving the stimulus time-course with a fixed impulse response function, and responses to longer stimuli can be predicted from responses to shorter ones. This is what makes it possible to model the HRF with a single, compact mathematical function rather than having to measure a separate response for every stimulus duration and condition. The authors modelled the impulse response itself with a gamma function, with two free parameters controlling its time constant and phase delay.

→ doi:10.1523/JNEUROSCI.16-13-04207.1996

A single gamma function captures the rise and peak of the response well, but not the post-stimulus undershoot, which decays on a different timescale from the rise. Friston, Josephs, Rees and Turner (1998) addressed this by modelling the HRF as a linear combination of two gamma functions: one capturing the main positive response, a second, smaller and slower one subtracted from it to capture the undershoot. This double-gamma model is the form most commonly meant today by the “canonical HRF,” and the same logic, convolving a stimulus time-course with a fixed response function, underlies the general linear model (GLM) approach used to analyse both fMRI and fNIRS data, which a later unit in this course covers directly.

2. Models for HRF Interpretation

The double-gamma model is useful precisely because it is agnostic: it fits the shape of the response without committing to any claim about why the response has that shape. It is a curve-fitting tool, not a physiological one. A separate line of models was developed specifically to explain the shape mechanistically, in terms of the vascular physiology from Lesson 1.

The balloon model (Buxton, Wong & Frank, 1998) treats the venous compartment as an elastic balloon: it inflates as blood volume increases with incoming flow, and empties passively, more slowly than it filled. This asymmetry between fast inflation and slow deflation is what produces the post-stimulus undershoot: blood flow has already returned to baseline while the venous compartment is still relaxing back to its resting volume, so deoxyhaemoglobin content briefly overshoots below baseline. The Windkessel model (Mandeville et al., 1999), developed independently and building on the same physical picture of a compliant post-arteriole compartment, has been particularly widely used in the fNIRS literature specifically, since fNIRS’s high temporal resolution is well suited to testing this kind of dynamic prediction directly.

🔬 going deeper

The Windkessel model has been extended in stages. Huppert, Allen, Benav, Jones and Boas (2007) generalised it into a three-compartment model, separating the arteriole, capillary, and venous compartments rather than treating post-arteriole blood as a single pool. This allows oxygen extraction to be modelled at each stage of transit rather than lumped into one step, and, combined with an independent blood-flow measurement, provides a route toward estimating CMRO₂ changes, something Lesson 2 established that HbO and HbR alone cannot do on their own. Mesquita, Huppert and Boas (2009) refined this further by testing whether the relationship between neural activity and arteriolar diameter change, the input driving the entire cascade, could itself be modelled as a simple linear transfer function. Establishing this tightens the link between the very first step in the neurovascular coupling chain from Lesson 1 and the vascular model built on top of it.

Huppert, T.J., Allen, M.S., Benav, H., Jones, P.B. & Boas, D.A. (2007). “A multicompartment vascular model for inferring baseline and functional changes in cerebral oxygen metabolism and arterial dilation.” Journal of Cerebral Blood Flow & Metabolism, 27(6), 1262–1279. Mesquita, R.C., Huppert, T.J. & Boas, D.A. (2009). “Exploring neuro-vascular and neuro-metabolic coupling in rat somatosensory cortex.” Physics in Medicine and Biology, 54(2), 175–185.

3. The Limits of These Models

Every model in this lesson, the single gamma, the double gamma, the balloon model, the Windkessel model and its extensions, is an attempt to compactly describe or mechanistically explain a measurement. None of them is the hemodynamic response itself, and none should be mistaken for ground truth. They differ in what they trade off: the double-gamma model fits data well without explaining it; the balloon and Windkessel models explain the shape mechanistically but rest on simplifying assumptions, elastic compartments, well-mixed blood pools, fixed relationships between flow and volume, that are approximations of a far messier vascular reality.

The “canonical” shape from Section 1 is itself an average, and it does not hold everywhere or always. Friston and colleagues, in the same 1998 paper that introduced the double-gamma model, also found that the response is not perfectly linear: responses to stimuli presented close together in time do not simply add up, they saturate and can show a reduced response to the second of two closely spaced stimuli, a form of hemodynamic refractoriness the linear model does not capture. The shape and timing of the response also varies by brain region and by task, and, as flagged at the end of the previous lesson, by age, development, and disease, since all of these models assume the same intact, mature neurovascular coupling that Lesson 1 described.

💭 pause and think

If you fit a canonical double-gamma HRF to real fNIRS data from a population or condition where the true response has a different shape, for example a shorter onset delay or a smaller undershoot, what would you expect to happen to your estimate of response amplitude? Would the fit necessarily fail, or could it succeed while quietly giving you a misleading number?

None of this is a reason to distrust these models; it is a reason to use them deliberately. Every model bakes in assumptions, and knowing which assumptions a given model makes, curve-fitting versus mechanistic, linear versus nonlinear, single compartment versus multi-compartment, is part of knowing what your analysis can and cannot tell you. The next lesson turns to independent evidence for the response this unit has been describing, comparing what fNIRS measures against electrophysiology and fMRI.

📌 Key Takeaways

  • The canonical HRF shows a one- to two-second onset delay, a peak around five to six seconds, and a post-stimulus undershoot. To a good approximation it behaves as a linear system (Boynton et al., 1996), which is what allows it to be modelled with a compact function.
  • The double-gamma model (Friston et al., 1998) fits the HRF’s shape, including the undershoot, but does not explain it mechanistically.
  • The balloon (Buxton, Wong & Frank, 1998) and Windkessel (Mandeville et al., 1999) models explain the undershoot mechanistically, as slow venous compartment deflation lagging behind blood flow’s return to baseline.
  • The Windkessel model has been extended to separate vascular compartments (Huppert et al., 2007) and to model the neural-to-arteriolar input itself (Mesquita, Huppert & Boas, 2009).
  • Every model here is a simplification. The response is not perfectly linear, and its shape varies by region, task, age, and health. Knowing a model’s assumptions is part of knowing what it can and cannot tell you.

📚 Further Reading & Key References

  1. Boynton, G.M., Engel, S.A., Glover, G.H. & Heeger, D.J. (1996). “Linear systems analysis of functional magnetic resonance imaging in human V1.” Journal of Neuroscience, 16(13), 4207–4221. [doi] — The linear-systems foundation and single-gamma model, Section 1.
  2. Friston, K.J., Josephs, O., Rees, G. & Turner, R. (1998). “Nonlinear event-related responses in fMRI.” Magnetic Resonance in Medicine, 39(1), 41–52. — The double-gamma model and the nonlinearity/refractoriness finding, Sections 1 and 3.
  3. Buxton, R.B., Wong, E.C. & Frank, L.R. (1998). “Dynamics of blood flow and oxygenation changes during brain activation: the balloon model.” Magnetic Resonance in Medicine, 39(6), 855–864. — The balloon model, Section 2, first referenced in Lesson 1.
  4. Mandeville, J.B., Marota, J.J.A., Ayata, C., et al. (1999). “Evidence of a cerebrovascular post-arteriole Windkessel with delayed compliance.” Journal of Cerebral Blood Flow & Metabolism, 19(6), 679–689. — The Windkessel model, Section 2, first referenced in Lesson 1.
  5. Huppert, T.J., Allen, M.S., Benav, H., Jones, P.B. & Boas, D.A. (2007). “A multicompartment vascular model for inferring baseline and functional changes in cerebral oxygen metabolism and arterial dilation.” Journal of Cerebral Blood Flow & Metabolism, 27(6), 1262–1279. — The three-compartment extension, Section 2.
  6. Mesquita, R.C., Huppert, T.J. & Boas, D.A. (2009). “Exploring neuro-vascular and neuro-metabolic coupling in rat somatosensory cortex.” Physics in Medicine and Biology, 54(2), 175–185. — The linear transfer-function refinement, Section 2.
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