This model is composed of two cascaded linear systems. The first system is characterized by the neuronal impulse response function (NRF). The NRF varies across locations and represents the power of regional synaptic currents r⇀(r,t), with a very short duration Ts up to several hundred milliseconds. The task-induced or stimulus-evoked electrophysiological response, in terms of the biophysical energy, is modeled as the convolution of the tasks or stimuli with the NRF. As the second system, the neurovascular coupling is simply modeled as a location-independent and time-invariant linear system, characterized by the hemodynamic impulse response function (HRF), denoted as h(t). Therefore, the BOLD fMRI signal f (r,t), in response to a block of sustained stimuli, can be written as Eq. (2). (2)f(r,t)=∑n=1Nδ(t−nTISI)⊗s2(r,t)⊗h(t)+fn(r,t) where δ (t) is a delta function, TISI is the inter-stimulus-interval (ISI), fn(r,t) is the noise, and ⊗ stands for convolution. Eq. (2) is valid when the ISI is longer than the neuronal refractory period and the linear neurovascular coupling holds true.
