Neural field dynamics under variation of local and global connectivity and finite transmission speed
Introduction
Large-scale neural network models are thought to be involved in the implementation of the cognitive function of the brain [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. In particular, peripheral areas are more functionally differentiated, whereas the ensuing cognitive integration appears to require more global network activations. The properties of the global network dynamics will then naturally depend on the large-scale, i.e. global, connectivity of the network components, as well as their local connectivity and dynamics [12], [13], [9], [14], [15]. So far, theoretical efforts have focussed almost exclusively on the study of networks with discretely connected nodes and complicated connectivity, but simple local dynamics, in which all neural activity is lumped into a single neural mass ([16], [17], [18], [19], [20], [21], [22]; see [23], [9] for reviews). As an additional complication, large-scale brain networks will be affected by time delays via signal transmission along the connecting pathways. The time delays may reach up to 200 ms [24] for the human brain, which is on a similar time scale as human brain function and hence is not negligible. Complementary efforts focussed on spatially continuous neural fields, which describe the temporal change of neural activity on a local spatial scale, typically within a brain area ([25], [26], [27], [28], [29], [30]; see [23], [9], [31] for reviews). These approaches use a translationally invariant, so-called homogeneous, connectivity and also take time delays into consideration, which, however, given the small spatial extent of a brain area, play a lesser role. To understand the neural basis of cognition, theoretical and analytical means must be developed which are specifically targeted to the properties of large-scale network dynamics integrating local and global contributions. Existing attempts so far include neural field theories which approximate the large-scale components of the connectivity matrix as translationally invariant and exponentially decaying over space, but on a larger spatial scale than is biologically realistic [32], [33], [34], [35], [36], [37]. These approaches have been successful in capturing various phenomena of large-scale brain dynamics including characteristic electroencephalographic power spectra [32], [36], epilepsy [38] and magnetoencephalographic activity during sensorimotor coordination [34], [35]. No systematic investigation, however, has been performed so far testing the validity of the translationally invariant approximation, and this remains an exciting challenge in theoretical neuroscience. Only very recently have computational and theoretical studies been performed exploring the effects of a large-scale connecting pathways upon the local dynamics of coupled brain areas, in particular neural fields, and vice versa; that is, the contributions of local architectures to the entire network dynamics. The first systematic but mostly computational study was performed by Jirsa and Kelso [39], who introduced the notion of a two-point connection embedded in a neural field and studied the spatiotemporal bifurcations as the length of the two-point connection was varied. Later research demonstrated analytically how the stability of the equilibrium point of the neural field depends on the length and coupling strength of the two-point connection [40], [41]. The implementation of multiple two-point connections obeying power-law and small-world distributions in neural fields (though with no time delays) has been shown computationally to result in a rich spatiotemporal dynamics [42], [43]. Other approaches mimicking heterogeneity in neural fields other than connectivity include non-uniform parameter distributions [44]. The equilibrium point solution of large-scale brain networks has received particular attention when biologically realistic anatomical connectivity matrices based on white matter fibre tracts became available to theoretical neuroscientists. Sporns and colleagues [45] studied the synchronization characteristics of the macaque connectivity matrix using chaotic oscillators and no time delays in the context of the brain’s resting state. Jirsa and colleagues [46], [47], [48] demonstrated that the equilibrium point solution of a network with biologically realistic primate connectivity and time delays is of key relevance for the understanding of the emergence of the brain’s spatiotemporal resting state dynamics. Specifically, when noise drives the neural network in the vicinity of the phase space around the stable equilibrium point, then the resulting network dynamics resembles the spatiotemporal brain dynamics observed experimentally in electroencephalography (EEG), magnetoencephalograpy (MEG) and functional magnetic resonance imaging (fMRI) [46]. For these reasons it is imperative to understand the anatomical and physiological determinants that determine the stability of the network’s equilibrium point and its dynamic vicinity in phase space.
To do so, here we take the following approach: We identify a convenient mathematical representation for the network dynamics with a locally invariant (homogeneous) and globally variant (heterogeneous) architecture via a one-dimensional spatiotemporally continuous integral–differential field equation with space-dependent delay. In particular, we prove that the subsequent discussions apply to arbitrary fixed point states of the entire network dynamics. Then we discuss the specific case of diffusive coupling, which expresses the locally invariant short-range (homogeneous) connectivity, and perform a complete analysis thereof. Diffusion is biologically unrealistic, but it nicely illustrates our approach, because it reduces the discussion to the analysis of the well-known Ginzburg–Landau equation with time delay. In particular, we generalize the analysis to include distributed transmission speeds. Next, we consider various biologically more realistic local architectures and expand our stability analysis to these cases. Specifically, we discuss the developmentally important effect of non-identical transmission speeds for local and global connecting pathways.
Section snippets
Mathematical framework for the neural field dynamics with local and global connectivity
Let be the neural field capturing the population activity at time point and position . The dynamics of the neural field can then be described by the following integro–differential equation: where represents the intrinsic time scale and is a scaling constant. The dot indicates the first time derivative. The spatial domain of the neural field is denoted by , where and is the
Neural fields with local diffusion and global two-point heterogeneous connections
This example illustrates the mutual effects of local homogeneous and global heterogeneous connections. Consider a one-dimensional spatially continuous diffusive field whose dynamics is given by the RGLE. Additionally, a heterogeneous connection is embedded between two points of the field where the activity is transmitted from one point to another, as shown in Fig. 1. The system can then be described as follows: The
Dispersion and distributed transmission velocities
The neural field captures the neural population activity at a given location . The axonal transmission speed depends mainly on the degree of myelination of the axons, which is diverse along the axis of the axon and will include a degree of variation. In a neural population model, the total axonal input into the population is the sum over all axonal inputs and will hence have a distributed propagation time. As shown in [21], distributed propagation times are equivalent to the distributed
Neural fields with locally general and globally heterogeneous two-point heterogeneous connectivity
In the previous sections we discussed the biologically unrealistic case that the effects of the local connectivity function are captured by a diffusion process. We now wish to generalize our discussion to various locally more realistic connectivity kernels. Additional consideration will be given to the fact that transmission speeds through the heterogeneous fiber system are generally by a factor 10 greater than the speeds in the local unmyelinated architecture. A suitable form of the
Summary and discussion
In this article we have developed a toy problem which addresses a key issue in biologically realistic large-scale networks, that is the interplay between local and global architectures leading to a large-scale network dynamics. We studied the contributions of the global connectivity by way of a two-point connection with a finite transmission speed. Our specific objective was to identify the conditions under which a fixed point solution loses its stability. We summarize the results of our
Acknowledgements
We would like to thank Felix Almonte and Arpan Banerjee for many valuable discussions and computational advice. This research was funded by the grants Brain NRG JSM22002082 and ATIP (CNRS).
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