Fig. 4. Schematic summary of DCM for fMRI. The dynamics in a system of interacting neuronal populations (left panel), which are not directly observable by fMRI, are modeled using a bilinear state equation (right panel). Integrating the state equation gives predicted neural dynamics (x) which are transformed into predicted BOLD responses (y) by means of a haemodynamic forward model (A). Neural and haemodynamic parameters are adjusted jointly such that the differences between predicted and measured BOLD series are minimized. The neural dynamics are determined by experimental manipulations that enter the model in the form of external inputs. Driving inputs (ui; e.g. sensory stimuli) elicit local responses which are propagated through the system according to the intrinsic connections. The strengths of these connections can be changed by modulatory inputs (u2; e.g. changes in task, attention, or due to learning). Note that in this figure the structure of the system and the scaling of the inputs have been chosen arbitrarily
7.2 DCM for Event-Related Potentials (ERPs)
ERPs as measured with EEG or MEG have been used for decades to study electrophysiological correlates of cognitive operations. Nevertheless, the neu-robiological mechanisms that underlie their generation are still largely unknown. DCM for ERPs was developed as a biologically plausible model to understand how event-related responses result from the dynamics in coupled neural ensembles (David et al. 2006).
DCM for ERPs rests on a neural mass model, developed by David & Friston (2003) as an extension of the model by Jansen & Rit (1995), which uses established connectivity rules in hierarchical sensory systems (Felleman & Van Essen 1992) to assemble a network of coupled cortical sources. These rules characterize connections with respect to their laminar patterns of origin and termination and distinguish between (i) forward (or bottom-up) connections originating in agranular layers and terminating in layer 4, (ii) backward (or top-down) connections originating and terminating in agranular layers, and (iii) lateral connections originating in agranular layers and targeting all layers. These long-range (extrinsic or inter-areal) cortico-cortical connections are excitatory, using glutamate as neurotransmitter, and arise from pyramidal cells.
Each region or source is modeled as a microcircuit following the model by David & Friston (2003). Three neuronal subpopulations are combined in this circuit and assigned to granular and supra-/infragranular layers. A population of excitatory pyramidal (output) cells receives inputs from inhibitory and excitatory populations of interneurons via intrinsic (intra-areal) connections. Within this model, excitatory interneurons can be regarded as spiny stellate cells found predominantly in layer 4 and in receipt of forward connections. Excitatory pyramidal cells and inhibitory interneurons are considered to occupy infra- and supragranular layers and receive backward and lateral inputs (see Fig. 5).
The neural state equations are summarized in Fig. 5. To perturb the system and model event-related responses, the network receives inputs via input connections. These connections are exactly the same as forward connections and deliver input u to the spiny stellate cells in layer 4. Input u represents afferent activity relayed by subcortical structures and are modelled as two parameterized components, a gamma density function (representing an event-related burst of input that is delayed and dispersed by subcortical synapses
Fig. 5. Schematic of the neural model in DCM for ERPs. This schema shows the state equations describing the dynamics of a microcircuit representing an individual region (source). Each region contains three subpopulations (pyramidal, spiny stellate and inhibitory interneurons) that are linked by intrinsic connections and have been assigned to supragranular, granular and infragranular cortical layers. Different regions are coupled through extrinsic (long-range) excitatory connections that follow the laminar patterns of forward, backward and lateral connections, respectively
Fig. 5. Schematic of the neural model in DCM for ERPs. This schema shows the state equations describing the dynamics of a microcircuit representing an individual region (source). Each region contains three subpopulations (pyramidal, spiny stellate and inhibitory interneurons) that are linked by intrinsic connections and have been assigned to supragranular, granular and infragranular cortical layers. Different regions are coupled through extrinsic (long-range) excitatory connections that follow the laminar patterns of forward, backward and lateral connections, respectively and axonal conduction) and a discrete cosine set (representing fluctuations in input over peristimulus time). The influence of this input on each source is controlled by a parameter vector C (see David et al. 2006 for details). Overall, the DCM is specified in terms of the state equations shown in Fig. 5 and a linear output equation d = f (x,u,^) y = Lx0 + e (15)
where x0 represents the transmembrane potential of pyramidal cells and L is a lead field matrix coupling electrical sources to the EEG channels (Kiebel et al. 2006). In comparison to DCM for fMRI, the forward model is a simple linearity as opposed to the nonlinear haemodynamic model in DCM for fMRI. In contrast, the state equations of DCM for ERPs are much more complex and realistic (cf. Fig. 5). As an example, the state equation for the inhibitory subpopulation is dx7 ~dt
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