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Neural Masses and Fields in Dynamic Causal Modeling

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Neural Masses and Fields in Dynamic Causal Modeling

Rosalyn Moran et al. Front Comput Neurosci.

Abstract

Dynamic causal modeling (DCM) provides a framework for the analysis of effective connectivity among neuronal subpopulations that subtend invasive (electrocorticograms and local field potentials) and non-invasive (electroencephalography and magnetoencephalography) electrophysiological responses. This paper reviews the suite of neuronal population models including neural masses, fields and conductance-based models that are used in DCM. These models are expressed in terms of sets of differential equations that allow one to model the synaptic underpinnings of connectivity. We describe early developments using neural mass models, where convolution-based dynamics are used to generate responses in laminar-specific populations of excitatory and inhibitory cells. We show that these models, though resting on only two simple transforms, can recapitulate the characteristics of both evoked and spectral responses observed empirically. Using an identical neuronal architecture, we show that a set of conductance based models-that consider the dynamics of specific ion-channels-present a richer space of responses; owing to non-linear interactions between conductances and membrane potentials. We propose that conductance-based models may be more appropriate when spectra present with multiple resonances. Finally, we outline a third class of models, where each neuronal subpopulation is treated as a field; in other words, as a manifold on the cortical surface. By explicitly accounting for the spatial propagation of cortical activity through partial differential equations (PDEs), we show that the topology of connectivity-through local lateral interactions among cortical layers-may be inferred, even in the absence of spatially resolved data. We also show that these models allow for a detailed analysis of structure-function relationships in the cortex. Our review highlights the relationship among these models and how the hypothesis asked of empirical data suggests an appropriate model class.

Keywords: dynamic causal modeling; electroencephalography; local field potential (LFP); magnetoencephalography (MEG); neural mass models.

Figures

Figure 1
Figure 1
Convolution-based neural mass models: “ERP” and “LFP”. Neural mass model used to represent a cortical source. Three cell subpopulations contribute to the ongoing dynamics. These include spiny stellate cells in granular layer IV, pyramidal cells and inhibitory interneurons in extra granular layers (II and III; V and VI). Intrinsic connections, γ1,2,3,4,5 link subpopulations in each source. Neuronal states include currents, i, and membrane potentials v. Extrinsic connections enter at specific cortical layers (see main text). Right: The functions controlling ongoing dynamics and their parameterization are summarized by synaptic kernels, which are used to convolve presynaptic (firing) input [a sigmoidal function of presynaptic membrane depolarization σ(v)] to produce postsynaptic depolarization (v), dependent on membrane time constants (1/κe/i) and average post-synaptic depolarizations (He/i) at excitatory (e) and inhibitory (i) synapses.
Figure 2
Figure 2
Conductance-based neural mass models: “NMM” and “MFM”. This figure shows Morris–Lecar-type differential equations describing the time evolution of a single cell current (capacitance × change of membrane potential: CV˙) and conductance (g) at inhibitory interneurons (extra granular layers), spiny stellate cells (granular layers) and pyramidal cells (extra granular layers). In this model, all cell types possess AMPA receptors, GABAA; with ion-channel time constants (1/κe/i). Layers are connected with strengths parameterized by γ VL, VE, and VI are reversal potentials for leak potassium channels, sodium, and chloride channels, respectively, at VT is the threshold potential. NMDA receptors at pyramidal cells and inhibitory interneurons can be added using a conductance equation of similar form, weighted by a voltage dependent switch (Moran et al., 2011a,b). For a full population Fokker-Planck characterization see Marreiros et al. (2008).
Figure 3
Figure 3
Canonical microcircuit neural field model: “CMC”. This figure shows the evolution equations that specify a canonical microcircuit (CMC) neural mass model of a single source. This model contains four populations occupying different cortical layers: the pyramidal cell population of the Jansen and Rit model is effectively split into two subpopulations allowing a separation of the neuronal populations that elaborate forward and backward connections in cortical hierarchies. As with the ERP and LFP models, second-order differential equations (shown earlier in Figure 1 decomposed into two first order ODEs), mediate a linear convolution of presynaptic activity [a sigmoidal function of presynaptic membrane depolarization σ(v)] to produce postsynaptic depolarization (v), dependent on membrane time constants (1/κe/i) and average post-synaptic depolarizations (He/i) at excitatory (e) and inhibitory (i) synapses. This depolarization gives rise to firing rates within each sub-population that provide inputs to other populations. Replacing connectivity parameters d, with a connectivity matrix over space and time D(x,t) enables one to generalize the neural mass model to a neural field model. This effectively converts the ordinary differential equations in this figure into partial differential equations or neural field equations.

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