Table 4.

Glossary of variables and expressions.

Variable	Description
$u_m$	The m-th of M experimental inputs as a function of time
$v_i^{(j)}$	The i-th (neuronal) state in region j; e.g., mean depolarisation of a neuronal population
$\sigma (v_i^{(j)})$	The neuronal firing rate – a sigmoid squashing function of depolarisation
$q_i^{(j)},p_i^{(j)},r_i^{(j)}$	(intrinsic, extrinsic and combined) presynaptic input to the i-th population of region j; eliciting postsynaptic and neurovascular responses; e.g., by depolarising astrocytes
$s$	Neurovascular signal; e.g., intracellular astrocyte calcium levels
$h_1,h_2,h_3,h_4$	Haemodynamic states: h1 - vasodilatory signal (e.g., NO), h2 - blood flow, h3 - blood h4 - volume deoxyhaemoglobin content
$L_i$	Lead field vector mapping from (neuronal) states to measured (electrophysiological) responses
$g_v(\omega ),g_o(\omega ),g_y(\omega )$	Spectral density of (neuronal) state fluctuations, observation error and ensuing measurement, respectively
${\partial }_{x}f$	System Jacobian or derivative of system flow with respect to (neuronal) states
$k(t)=FT[K(\omega )]$	First order kernel mapping from inputs to responses; c.f., an impulse response function of time. This is the Fourier transform the transfer function
$K(\omega )=FT[k(t)]$	Transfer function of frequency modulating the power of endogenous neuronal fluctuations to produce a (cross spectral density) response. This is the Fourier transform of the kernel
$\lambda$	Eigenvalues of the transfer function
$\mu ,{\mu }^{-}$	Eigenvectors of transfer function and their generalised inverse