Systematic approximations of neural fields through networks of neural masses in the virtual brain

Highlights

• Guidelines for building full brain network models are provided.

• The spatial sampling problem of neural fields is analyzed for arbitrary dimensions.

• Key result: The Gaussian connectivity function is easier to sample than the Laplacian.

Abstract

Full brain network models comprise a large-scale connectivity (the connectome) and neural mass models as the network's nodes. Neural mass models absorb implicitly a variety of properties in their constant parameters to achieve a reduction in complexity. In situations, where the local network connectivity undergoes major changes, such as in development or epilepsy, it becomes crucial to model local connectivity explicitly. This leads naturally to a description of neural fields on folded cortical sheets with local and global connectivities. The numerical approximation of neural fields in biologically realistic situations as addressed in Virtual Brain simulations (see http://thevirtualbrain.org/app/ (version 1.0)) is challenging and requires a thorough evaluation if the Virtual Brain approach is to be adapted for systematic studies of disease and disorders. Here we analyze the sampling problem of neural fields for arbitrary dimensions and provide explicit results for one, two and three dimensions relevant to realistically folded cortical surfaces. We characterize (i) the error due to sampling of spatial distribution functions; (ii) useful sampling parameter ranges in the context of encephalographic (EEG, MEG, ECoG and functional MRI) signals; (iii) guidelines for choosing the right spatial distribution function for given anatomical and geometrical constraints.

Introduction

With the availability of full brain structural connectivity information, the so-called connectome (Sporns, 2011), a new type of network models emerged that needed to address novel challenges characteristic for this macroscopic level of description. These challenges included the handling of a complex connectivity matrix in three-dimensional physical space, the inclusion of many time delays as a function of fiber lengths, as well as the correct choice of the mathematical model for a network node. Depending on the structural parcellation (Kötter and Wanke, 2005), a network node typically comprised a brain region of the size of multiple square centimeters (Hagmann et al., 2008). The full brain modeling approach proved successful in explaining the mechanisms underlying the emergence of network patterns and their coherent intermittent dynamics for resting state conditions (Deco and Jirsa, 2012, Deco et al., 2009, Deco et al., 2011, Ghosh et al., 2008). Crucial elements of the resting state dynamics include stochastics and multistability (Freyer et al., 2009, Freyer et al., 2011, Freyer et al., 2012). The resting state patterns and its dynamics are robust and reproducible within healthy populations (Damoiseaux et al., 2006, He et al., 2007, Mantini et al., 2007, Raichle et al., 2001), but differ quite significantly across diseases, such as schizophrenia, autism, epilepsy and others (Buckner et al., 2008, Cherkassky et al., 2006, Corbetta et al., 2005, Kennedy et al., 2006, Symond et al., 2005, Uhlhaas and Singer, 2006, Uhlhaas and Singer, 2010), as well as the aging brain (Beason-Held et al., 2009, Damoiseaux et al., 2008, Fransson et al., 2007, Koch et al., 2010, Supekar et al., 2010). For this reason the resting state dynamics finds enormous interest as a potential biomarker for disease or disorder. Of particular interest in these network manipulations is the ratio of local versus global connectivity. Local connectivity represents intracortical connections, whereas the global connectivity is the connectome comprising the white matter fibers between cortical and subcortical areas. Often local connectivity is absorbed in the parameters of the network node model. Manipulations of local connectivity (such as pruning of fibers during development or sprouting in epileptic tissue), however, are a key to modeling studies in a number of situations and require a representation of the full brain network, in which the brain region as a network node acquires more complexity and encompasses the notion of local connectivity. A natural extension of a network of coupled brain regions towards a spatially continuous sheet is provided by the neural field theory (Coombes, 2010, Deco et al., 2008, Jirsa, 2004). The aim of this work is to develop accurate numerical approximations of neural field models on folded cortical sheets with local and global connectivities, the latter typically obtained from tractographic data.

The numerical approximation of a neural field is realized via a network of neural masses. Neural field models cover the continuous description of interacting neural ensembles. A neural ensemble refers to a local set of commonly interacting neurons (Freeman, 1992). An ensemble of neurons of a certain class (e.g., due to receptor, location and/or morphological classifications) can be described in terms of mean firing rate and mean postsynaptic potential as a so-called neural mass. Hence, a neural mass model is a lumped representation, especially neglecting the spatial extend of a neural ensemble (Spiegler et al., 2010). Note that a single neural ensemble (or even a cortical area under certain functional circumstances Spiegler et al., 2011) can be described as a set of interacting neural masses, for instance, a neural mass of pyramidal cells, a neural mass of glutamatergic spiny stellate cells and a neural mass of GABAergic neurons (Spiegler et al., 2010). Usually nonlinear differential equations are used to mimic the complex behavior of neural ensembles. These equations are difficult to solve analytically and therefore such models are, nowadays, translated into a digital scheme for integration. This approach brings a discretization problem in its wake. That means that a neural field model is translated into a network of spatially separated sets of neural masses. For this purpose, a crucial step is the sampling of the spatial distribution function of the connections.

The sampling task can be addressed from three points of view with the aim: (i) to construct a model based on biophysical or theoretical considerations, (ii) to describe spatial patterns in specific empirical data, or (iii) to assess dynamics of an implemented neural field model. All three approaches are subject to restrictions of the considered underlying techniques. The resolution of human brain measurements, for instance, electrocorticography (ECoG) and functional magnetic resonance imaging (functional MRI) is in the order of mm (e.g., Blakely et al., 2008, Freeman, 2000, Yoo et al., 2004, Zhang et al., 2008). In the case of electro- and magneto-encephalography (EEG and MEG) the spatial resolution is in the order of cm (e.g., Freeman et al., 2003, Hämäläinen et al., 1993, Malmivuo and Suihko, 2004, Srinivasan et al., 1999). Moreover, implementations of a model in a digital regime underlie, for instance, a finite representation of numbers and its precision, and a finite number of network nodes.

This work deals with the spatial sampling of two specific but widely used choices of the homogeneous connectivity distribution function, namely, the sum of Gaussian (e.g., Amari, 1977, Atay and Hutt, 2005, Markounikau et al., 2010) and the sum of Laplace distributions (e.g., Jirsa and Haken, 1996). Because the main focus is on cortical dynamics, the spatial distribution function is to a first approximation assumed to be independent of the location on the cortical surface.

The main three emphases of this paper are: (i) what is the error that appears due to sampling a spatial distribution function; (ii) what are the usable parameters for a spatial distribution function considering the resolution of such measurements as EEG, MEG, ECoG and functional MRI; and (iii) how can the parameters of a spatial distribution function be specified giving a specific discrete approximation of a cortical geometry.

This work extends previous studies of neural fields, such as Bojak et al. (2010) and Freestone et al. (2011) in three points: (i) the sampling procedure can be assessed using two measures (ii) that are applicable in any dimensional physical space (iii) for two widely used spatial distribution functions.