Neurovascular Coupling & BOLD Signal (2D)

Neural Stimulus

Amplitude A 1.00

Duration (s) 4

Vascular Parameters

Transit time τ (s) 2.0

Grubb exponent α 0.38

Live Readout

CBF (f_in)

1.000

Blood vol (v)

1.000

dHb content (q)

1.000

BOLD ΔS/S₀

0.0000

Elapsed time

0.0 s

Peak BOLD

0.0000

How it works

The Buxton balloon model treats the venous vasculature as an expandable balloon. Neural firing drives an inflow pulse f_in, which the coupled ODEs below convert into blood volume v and deoxyhemoglobin content q, from which the fMRI BOLD signal is derived:

f_in(t) = 1 + A·(1 − e^(−t/τ_on))        during stimulus
dv/dt   = (f_in − v^(1/α)) / τ
dq/dt   = (f_in·E(f_in)/E0 − v^(1/α)·(q/v)) / τ
BOLD    = V0·[k1(1−q) + k2(1−q/v) + k3(1−v)]

Integrated with explicit Euler at a fixed 20 ms sub-step. The chart plots four synchronized time-courses over a rolling 30 s window so you can watch the several-second lag between the stimulus band (shaded) and the characteristic hemodynamic response peak, plus the post-stimulus undershoot.

τ — transit time; larger values smear and lower the BOLD peak.
α — Grubb exponent linking outflow to blood volume; smaller α speeds volume recovery.
Undershoot — q drains more slowly than v returns to baseline, pulling BOLD briefly negative.

f_in (CBF) volume v dHb q BOLD ×10