Table 1.

This table presents the expressions that relate unnormalised and normalised measures of second-order statistical dependencies among data to the underlying process generating those data. Table 1a specifies the generative (state-space) model in terms of stochastic differential equations of motion and a static nonlinear observer function. The random fluctuations that perturb the motion of hidden (neuronal) states and the observation noise are characterised in terms of their second-order statistics; namely their covariance or spectral density. These state-space models can be formulated in terms of a convolution of the fluctuations, where the (first order Volterra) convolution kernels are a function of the model parameters. Table 1b shows how these kernels can generate any characterisation of the ensuing dependencies among the data – as cross-covariance functions of lag or time, cross spectral density functions of frequency and autoregressive formulations – in terms of autoregression coefficients and associated directed transfer functions. The expressions have been simplified and organised to illustrate the formal symmetry among the relationships. The key point to take from these expressions is that any characterisation can be derived analytically from any other using Fourier transforms F[·], expectations E[·] convolution operators * and Kronecker tensor products ⊗. Variables with a ~ denote matrices whose columns contain lagged functions of time and † denotes the conjugate transpose. Table 1c provides standardised versions of the second order statistics in Table 1b. These include the cross-correlation function, coherence, Geweke Granger causality and the normalised (Kaminski) directed transfer functions. These results mean that we can generate the expected Granger causality from the parameters of any generative model in exactly the same way that any other data feature can be generated. Note that in going from a parameterised generative model to the second-order statistics, there is no return. In other words, although second-order statistics can be generated given the model parameters, model parameters cannot be derived from second-order statistics. This is the (inverse) problem solved by DCM for complex cross spectra — that requires a generative model.

a: state-space model
State space model	Random fluctuations	Convolution kernels
$\begin{array}{c}\dot{\mathit{x}}\left(\mathit{t}\right)=\mathit{f}\left(\mathit{x},\mathit{\theta }\right)+\mathit{v}\left(\mathit{t}\right)\\ \mathit{y}\left(\mathit{t}\right)=\mathit{g}\left(\mathit{x},\mathit{\theta }\right)+\mathit{w}\left(\mathit{t}\right)\end{array}$	$\begin{array}{c}\mathit{E}[\mathit{v}\left(\mathit{t}\right)\cdot \mathit{v}(\mathit{t}-\mathit{\tau }\left){}^{\mathit{T}}\right]={\mathit{\Sigma }}_{\mathit{v}}(\mathit{\tau },\mathit{\theta })\\ \mathit{E}[\mathit{w}\left(\mathit{t}\right)\cdot \mathit{w}(\mathit{t}-\mathit{\tau }\left){}^{\mathit{T}}\right]={\mathit{\Sigma }}_{\mathit{w}}(\mathit{\tau },\mathit{\theta })\end{array}$	$\begin{array}{c}\mathit{y}\left(\mathit{t}\right)=\mathit{k}\left(\mathit{\tau }\right)\ast \mathit{v}\left(\mathit{t}\right)+\mathit{w}\left(\mathit{t}\right)\\ \mathit{k}\left(\mathit{\tau }\right)={\nabla }_{\mathit{x}}\mathit{g}({\mathit{x}}_0,\mathit{\theta })\cdot exp\left(\mathit{\tau }\cdot {\nabla }_{\mathit{x}}\mathit{f}\left({\mathit{x}}_0,\mathit{\theta }\right)\right)\end{array}$

b: second-order dependencies
	Cross covariance Σ(t)	Cross spectral density g(ω)	Autoregression coefficients a	Directed transfer functions S(ω)
Cross covariance Σ(t)	Σ(t) = k(τ) ∗ Σv ∗ k(τ) + Σw	Σ(t) = F− 1[g(ω)]	$\mathit{C}={\left(\mathit{I}-\tilde{\mathit{a}}\right)}^{-1}\left({\displaystyle {\sum }_{\mathit{z}}\otimes \mathit{I}}\right){\left(\mathit{I}-\tilde{\mathit{a}}\right)}^{-\mathit{T}}$	Σ(t) ∝ F− 1[S(ω) ⋅ Σz ⋅ S(ω)†]
Cross spectral density g(ω)	g(ω) = F[Σ(τ)]	$\begin{array}{l}\mathit{g}\left(\mathit{\omega }\right)=\mathit{K}\left(\mathit{\omega }\right)\cdot {\mathit{g}}_{\mathit{v}}\cdot \mathit{K}{\left(\mathit{\omega }\right)}^{†}+{\mathit{g}}_{\mathit{w}}\\ \mathit{K}\left(\mathit{\omega }\right)=\mathbf{F}\left[\mathit{k}\left(\mathit{\tau }\right)\right]\end{array}$	$\begin{array}{l}\mathit{g}\left(\mathit{\omega }\right)=\mathit{S}\left(\mathit{\omega }\right)\cdot {\mathit{\Sigma }}_{\mathit{z}}\cdot \mathit{S}{\left(\mathit{\omega }\right)}^{†}\\ \mathit{S}\left(\mathit{\omega }\right)={\left(\mathit{I}-\mathbf{F}\left[\mathit{a}\right]\right)}^{-1}\end{array}$	$\begin{array}{c}\mathit{g}\left(\mathit{\omega }\right)\propto \mathit{S}\left(\mathit{\omega }\right)\cdot {\mathit{\Sigma }}_{\mathit{z}}\cdot \mathit{S}{\left(\mathit{\omega }\right)}^{†}\\ =\mathit{\Psi }\left(\mathit{\omega }\right)\cdot \mathit{\Psi }{\left(\mathit{\omega }\right)}^{†}\end{array}$
Autoregression coefficients a	$\mathit{a}={\mathit{C}}^{-1}\tilde{\mathit{\Sigma }}$	$\begin{array}{c}\mathit{a}={\mathit{C}}^{-1}\tilde{\mathit{\Sigma }}\\ \mathit{\Sigma }\left(\mathit{\tau }\right)={\mathbf{F}}^{-1}\left[\mathit{g}\left(\mathit{\omega }\right)\right]\end{array}$	$\begin{array}{c}\mathit{y}=\tilde{\mathit{y}}\cdot \mathit{a}+\mathit{z}\Rightarrow \\ \mathit{a}=\mathbf{E}{\left[{\tilde{\mathit{y}}}^{\mathit{T}}\tilde{\mathit{y}}\right]}^{-1}\mathbf{E}\left[{\tilde{\mathit{y}}}^{\mathit{T}}\mathit{y}\right]\\ \\ {\mathit{\Sigma }}_{\mathit{z}}=\mathbf{E}\left[{\mathit{z}}^{\mathit{T}}\mathit{z}\right]\propto \mathit{\psi }\left(0\right)\cdot \mathit{\psi }{\left(0\right)}^{†}\\ =\mathit{\Sigma }\left(0\right)-{\tilde{\mathit{\Sigma }}}^{\mathit{T}}{\mathit{C}}^{-1}\tilde{\mathit{\Sigma }}\end{array}$	$\begin{array}{c}\mathit{a}={\mathbf{F}}^{-1}\left[\mathit{A}\left(\mathit{\omega }\right)\right]\\ \mathit{A}\left(\mathit{\omega }\right)=\mathit{I}-\mathit{S}{\left(\mathit{\omega }\right)}^{-1}\\ \\ \mathit{S}\left(\mathit{\omega }\right)=\mathit{\Psi }\left(\mathit{\omega }\right)\cdot \mathit{\psi }{\left(0\right)}^{-1}\\ \mathit{\psi }={\mathbf{F}}^{-1}\left[\mathit{\Psi }\left(\mathit{\omega }\right)\right]\end{array}$
Directed transfer functions S(ω)	$\mathit{S}\left(\mathit{\omega }\right)={\left(\mathit{I}-\mathbf{F}\left[{\mathit{C}}^{-1}\tilde{\mathit{\Sigma }}\right]\right)}^{-1}$	$\begin{array}{c}\mathit{S}\left(\mathit{\omega }\right)={\left(\mathit{I}-\mathbf{F}\left[{\mathit{C}}^{-1}\tilde{\mathit{\Sigma }}\right]\right)}^{-1}\\ \mathit{\Sigma }\left(\mathit{\tau }\right)={\mathbf{F}}^{-1}\left[\mathit{g}\left(\mathit{\omega }\right)\right]\end{array}$	$\begin{array}{c}\mathit{S}\left(\mathit{\omega }\right)={\left(\mathit{I}-\mathit{A}\left(\mathit{\omega }\right)\right)}^{-1}\\ \mathit{A}\left(\mathit{\omega }\right)=\mathbf{F}\left[\mathit{a}\right]\end{array}$	$\begin{array}{c}\mathit{Y}\left(\mathit{\omega }\right)=\mathit{A}\left(\mathit{\omega }\right)\cdot \mathit{Y}\left(\mathit{\omega }\right)+\mathit{Z}\left(\mathit{\omega }\right)\\ =\mathit{S}\left(\mathit{\omega }\right)\cdot \mathit{Z}\left(\mathit{\omega }\right)\end{array}$

c: normalised measures
Cross correlation	Coherence	Granger causality	Normalised directed transfer functions
${\mathit{\rho }}_{\mathit{ij}}\left(\mathit{\tau }\right)=\frac{{\displaystyle {\sum }_{\mathit{ij}}\left(\mathit{\tau }\right)}}{\sqrt{{\displaystyle {\sum }_{\mathit{ii}}\left(0\right)}\cdot {\displaystyle {\sum }_{\mathit{jj}}\left(0\right)}}}$	${\mathit{\gamma }}_{\mathit{ij}}\left(\mathit{\omega }\right)=\frac{{\left|{\mathit{g}}_{\mathit{ij}}\left(\mathit{\omega }\right)\right|}^2}{{\mathit{g}}_{\mathit{ii}}\left(\mathit{\omega }\right){\mathit{g}}_{\mathit{jj}}\left(\mathit{\omega }\right)}$	${\mathit{G}}_{\mathit{ij}}\left(\mathit{\omega }\right)=-ln\left(1-\left({\mathit{\Sigma }}_{\mathit{zjj}}-\frac{{\mathit{\Sigma }}_{\mathit{zij}}^2}{{\mathit{\Sigma }}_{\mathit{zii}}}\right)\frac{{\left|{\mathit{S}}_{\mathit{ij}}\left(\mathit{\omega }\right)\right|}^2}{{\mathit{g}}_{\mathit{ii}}\left(\mathit{\omega }\right)}\right)$	${\mathit{D}}_{\mathit{ij}}\left(\mathit{\omega }\right)=\frac{\left|{\mathit{S}}_{\mathit{ij}}\left(\mathit{\omega }\right)\right|}{\left|{\mathit{S}}_{\mathit{ii}}\left(\mathit{\omega }\right)\right|}$

$$\begin{array}{cccc}\hfill \mathit{C}=\mathbf{E}\left[{\tilde{\mathit{y}}}^{\mathit{T}}\tilde{\mathit{y}}\right]:{\mathit{C}}_{\mathit{ij}}=\left[\begin{array}{ccc}\hfill {\mathit{\Sigma }}_{\mathit{ij}}\left(0\right)\hfill & \hfill \cdots \hfill & \hfill {\mathit{\Sigma }}_{\mathit{ij}}\left(-\mathit{p}\right)\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\mathit{\Sigma }}_{\mathit{ij}}\left(\mathit{p}\right)\hfill & \hfill \cdots \hfill & \hfill {\mathit{\Sigma }}_{\mathit{ij}}\left(0\right)\hfill \end{array}\right]\hfill & \hfill \tilde{\mathit{\Sigma }}=\mathbf{E}\left[{\tilde{\mathit{y}}}^{\mathit{T}}\mathit{y}\right]:{\tilde{\mathit{\Sigma }}}_{\mathit{ij}}=\left[\begin{array}{c}\hfill {\mathit{\Sigma }}_{\mathit{ij}}\left(1\right)\hfill \\ \hfill ⋮\hfill \\ \hfill {\mathit{\Sigma }}_{\mathit{ij}}\left(\mathit{p}+1\right)\hfill \end{array}\right]\hfill & \hfill \tilde{\mathit{y}}=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill {\mathit{y}}_{11}\hfill & \hfill 0\hfill & \hfill \hfill & \hfill \hfill \\ \hfill {\mathit{y}}_{12}\hfill & \hfill {\mathit{y}}_{11}\hfill & \hfill 0\hfill & \hfill \hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\begin{array}{cccc}\hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill {\mathit{y}}_{21}\hfill & \hfill 0\hfill & \hfill \hfill & \hfill \hfill \\ \hfill {\mathit{y}}_{22}\hfill & \hfill {\mathit{y}}_{21}\hfill & \hfill 0\hfill & \hfill \hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\dots \right]\hfill & \hfill {\tilde{\mathit{a}}}_{\mathit{ij}}=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill {\mathit{a}}_{\mathit{ij}1}\hfill & \hfill 0\hfill & \hfill \hfill & \hfill \hfill \\ \hfill {\mathit{a}}_{\mathit{ij}2}\hfill & \hfill {\mathit{a}}_{\mathit{ij}}\hfill & \hfill 0\hfill & \hfill \hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \end{array}\right]\hfill \end{array}$$