Canonical Microcircuit Model for Effective Connectivity

A neural mass model based on a canonical microcircuit (Fig. 4A; cf. Pinotsis et al. 2012 (link)) was used for a subsequent DCM analysis, where the observed effects of experimental manipulations on ERFs are modeled as contextual changes in effective connectivity in a network comprised of several neural sources. In canonical microcircuit DCMs, the activity at each source is modeled using ordinary differential equations that describe changes in postsynaptic voltage and current in 4 neuronal populations. The 4 neural populations (spiny stellate cells in Layer 4, superficial and deep pyramidal cells in Layers 2/3 and 5/6, respectively, and inhibitory interneurons) are equipped with distinct profiles of ascending and descending connectivity both intrinsically (coupling neural populations within a source) and extrinsically (linking different sources). Specifically, spiny stellate cells in Layer 4 and deep pyramidal cells are thought to receive ascending (bottom-up) input, whereas superficial pyramidal cells and inhibitory interneurons receive descending (top-down) input. Crucially, there is a laminar asymmetry in terms of the output of each source—superficial pyramidal cells propagate signals to hierarchically higher areas (bottom-up or ascending), whereas deep pyramidal cells propagate signals to hierarchically lower areas (top-down or descending). Within sources, neural populations are interconnected with excitatory and inhibitory connections. Mathematically, the dynamics at each source are described by a set of coupled differential equations:
$\begin{array}{ll}& {\dot{V}}_{\mathrm{SS}}=I_{\mathrm{SS}}\\ & {\dot{I}}_{\mathrm{SS}}={\kappa }_{\mathrm{ss}}(A^{\mathrm{F}}\sigma (V_{\mathrm{SP}})-{\gamma }_{\mathrm{SS}\to \mathrm{S}{\mathrm{S}}^{\sigma }}(V_{\mathrm{SS}})-{\gamma }_{\mathrm{SP}\to \mathrm{S}{\mathrm{S}}^{\sigma }}(V_{\mathrm{SP}})\\ & -{\gamma }_{\mathrm{II}\to \mathrm{S}{\mathrm{S}}^{\sigma }}(V_{\mathrm{II}})Cu)-2{\kappa }_{\mathrm{SS}}{V}_{\mathrm{SS}}-{\kappa }_{\mathrm{SS}}^2{I}_{\mathrm{SS}}\end{array}$
$\begin{array}{ll}& {\dot{V}}_{\mathrm{II}}=I_{\mathrm{II}}\\ & {\dot{I}}_{\mathrm{II}}={\kappa }_{\mathrm{II}}(-A^{\mathrm{B}}\sigma (V_{\mathrm{DP}})+{\gamma }_{\mathrm{SS}\to \mathrm{I}{\mathrm{I}}^{\sigma }}(V_{\mathrm{SS}})+{\gamma }_{\mathrm{DP}\to \mathrm{I}{\mathrm{I}}^{\sigma }}(V_{\mathrm{DP}})\\ & -{\gamma }_{\mathrm{II}\to \mathrm{I}{\mathrm{I}}^{\sigma }}(V_{\mathrm{II}}))-2{\kappa }_{\mathrm{II}}{V}_{\mathrm{II}}-{\kappa }_{\mathrm{II}}^2{I}_{\mathrm{II}}\end{array}$
$\begin{array}{ll}& {\dot{V}}_{\mathrm{SP}}=I_{\mathrm{SP}}\\ & {\dot{I}}_{\mathrm{SP}}={\kappa }_{\mathrm{SP}}(-A^{\mathrm{B}}\sigma (V_{\mathrm{DP}})+{\gamma }_{\mathrm{SS}\to \mathrm{S}{\mathrm{P}}^{\sigma }}(V_{\mathrm{SS}})-{\gamma }_{\mathrm{SP}\to \mathrm{S}{\mathrm{P}}^{\sigma }}(V_{\mathrm{SP}}))\\ & -2{\kappa }_{\mathrm{SP}}{V}_{\mathrm{SP}}-{\kappa }_{\mathrm{SP}}^2{I}_{\mathrm{SP}}\end{array}$
$\begin{array}{ll}& {\dot{V}}_{\mathrm{DP}}=I_{\mathrm{DP}}\\ & {\dot{I}}_{\mathrm{DP}}={\kappa }_{\mathrm{DP}}(A^{\mathrm{F}}\sigma (V_{\mathrm{SP}})-{\gamma }_{\mathrm{DP}\to \mathrm{D}{\mathrm{P}}^{\sigma }}(V_{\mathrm{DP}})-{\gamma }_{\mathrm{II}\to \mathrm{D}{\mathrm{P}}^{\sigma }}(V_{\mathrm{II}}))\\ & -2{\kappa }_{\mathrm{DP}}{V}_{\mathrm{DP}}-{\kappa }_{\mathrm{DP}}^2{I}_{\mathrm{DP}}\end{array}$
Here, the 4 neuronal populations are indicated by subscripts SS (spiny stellate cells), II (inhibitory interneurons), SP (superficial pyramidal cells), and DP (deep pyramidal cells). V_m and I_m denote the voltage and current of population m, with synaptic rate constant ${\kappa }_{m}s$ . C is a sigmoid operator transforming the postsynaptic potential into firing rate, A^F and A^B represent the extrinsic (between regions) forward and backward connections, and γ_m→n encode the intrinsic (within-region) connection from population m to n. Finally, the changes in current of spiny stellate cells at the lowest level of the hierarchy also depend on thalamic input u scaled by its weight C. This canonical microcircuit model has been used in several previous DCM studies of synaptic gain (e.g., Boly et al. 2012; Brown and Friston 2013 (link)).
Source locations were based on a multiple sparse priors source reconstruction (Friston et al. 2008 (link)) of the main effect of expectation on ERF topography at 170–230 ms post-stimulus (see Results for more details). The DCM architecture (i.e., the weighted adjacency matrix of extrinsic connections among sources) was optimized using fixed-effects Bayesian model selection following a heuristic model search: First, the basic architecture was identified using responses to “unattended standards.” Changes in extrinsic connectivity were then selected under this basic architecture using responses in all conditions. Finally, expectation and attention-dependent changes in intrinsic connectivity were identified. In all 3 steps, models were inverted using a 1- to 300-ms peristimulus time window, which included both main effects of attention and expectation and their interaction. The thalamic input to A1 was modeled as a Gaussian function with a prior latency of 20 ms post-stimulus. The DCMs were completed with a spatial forward model (mapping from source dipoles to observed MEG topography) based on a single MEG shell (Nolte 2003 (link)).
The first step considered 9 competing model structures, differing in the number of sources and in the pattern of extrinsic connections (Fig. 4B). The 9 models were inverted per participant to model the “unattended standard” ERFs. These responses were considered the baseline for subsequent modulation by attention and expectation. The selected model structure was then optimized with respect to condition-specific changes in extrinsic connectivity. Sixteen competing models, each allowing for a different subset of connections (forward, backward, both, or no connections) to be modulated by either of the experimental factors (attention and/or expectation), were fitted to each participant's ERF data and compared using fixed-effects Bayesian model selection based on the free-energy approximation to their log-evidence (Friston et al. 2007 (link)). This approach implements the a priori assumption that each participant's data were generated under the same (unknown) model—and ensures that models are compared based on a tradeoff between their accuracy and complexity (Stephan et al. 2009 (link)). Finally, the model with an optimized modulation of extrinsic connections was used to compare alternative models of intrinsic modulation by attention and expectation.
The canonical microcircuit neural mass model has been considered in terms of the message passing implicit in predictive coding (Bastos et al. 2012 (link)). Crucially, the precision of prediction errors pertaining to hidden causes (that link levels of hierarchical models) and states (that link dynamics over time within one level) have been associated with the gain of superficial pyramidal cells and inhibitory interneurons, respectively (Feldman and Friston 2010 (link); Friston 2010 (link)). Given the literature explaining both attention and sensory learning in terms of precision of prediction errors and the underlying synaptic gain (Brown and Friston 2013 (link); Moran et al. 2013 (link)), the alternative models of intrinsic modulation by attention and/or expectation allowed for activity-dependent gain modulation of either superficial pyramidal cells or inhibitory interneurons at different levels of the processing hierarchy, resulting in 7 models per experimental factor. As mentioned above, the models were compared based on their free-energy approximation to log model evidence using a fixed-effects Bayesian model selection. The winning model was used to infer the posterior connectivity and gain parameters after Bayesian parameter averaging (Garrido, Kilner, Kiebel, Friston et al. 2007 (link)).