Stability of small-world networks of neural populations

From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity

It is clear that the topological structure of a neural network somehow determines the activity of the neurons within it. In the present work, we ask to what extent it is possible to examine the structural features of a network and learn something about its activity? Specifically, we consider how the centrality (the importance of a node in a network) of a neuron correlates with its firing rate. To investigate, we apply an array of centrality measures, including In-Degree, Closeness, Betweenness, Eigenvector, Katz, PageRank, Hyperlink-Induced Topic Search (HITS) and NeuronRank to Leaky-Integrate and Fire neural networks with different connectivity schemes. We find that Katz centrality is the best predictor of firing rate given the network structure, with almost perfect correlation in all cases studied, which include purely excitatory and excitatory–inhibitory networks, with either homogeneous connections or a small-world structure. We identify the properties of a network which will cause this correlation to hold. We argue that the reason Katz centrality correlates so highly with neuronal activity compared to other centrality measures is because it nicely captures disinhibition in neural networks. In addition, we argue that these theoretical findings are applicable to neuroscientists who apply centrality measures to functional brain networks, as well as offer a neurophysiological justification to high level cognitive models which use certain centrality measures.

Stability and instability of a neuron network with excitatory and inhibitory small-world connections

This study considers a delayed neural network with excitatory and inhibitory shortcuts. The global stability of the trivial equilibrium is investigated based on Lyapunov's direct method and the delay-dependent criteria are obtained. It is shown that both the excitatory and inhibitory shortcuts decrease the stability interval, but a time delay can be employed as a global stabilizer. In addition, we analyze the bounds of the eigenvalues of the adjacent matrix using matrix perturbation theory and then obtain the generalized sufficient conditions for local stability. The possibility of small inhibitory shortcuts is helpful for maintaining stability. The mechanisms of instability, bifurcation modes, and chaos are also investigated. Compared with methods based on mean-field theory, the proposed method can guarantee the stability of the system in most cases with random events. The proposed method is effective for cases where excitatory and inhibitory shortcuts exist simultaneously in the network.

Inference of direct and multistep effective connectivities from functional connectivity of the brain and of relationships to cortical geometry

BACKGROUND

The problem of inferring effective brain connectivity from functional connectivity is under active investigation, and connectivity via multistep paths is poorly understood.

NEW METHOD

A method is presented to calculate the direct effective connection matrix (deCM), which embodies direct connection strengths between brain regions, from functional CMs (fCMs) by minimizing the difference between an experimental fCM and one calculated via neural field theory from an ansatz deCM based on an experimental anatomical CM.

RESULTS

The best match between fCMs occurs close to a critical point, consistent with independent published stability estimates. Residual mismatch between fCMs is identified to be largely due to interhemispheric connections that are poorly estimated in an initial ansatz deCM due to experimental limitations; improved ansatzes substantially reduce the mismatch and enable interhemispheric connections to be estimated. Various levels of significant multistep connections are then imaged via the neural field theory (NFT) result that these correspond to powers of the deCM; these are shown to be predictable from geometric distances between regions.

COMPARISON WITH EXISTING METHODS

This method gives insight into direct and multistep effective connectivity from fCMs and relating to physiology and brain geometry. This contrasts with other methods, which progressively adjust connections without an overarching physiologically based framework to deal with multistep or poorly estimated connections.

CONCLUSIONS

deCMs can be usefully estimated using this method and the results enable multistep connections to be investigated systematically.

Determination of effective brain connectivity from functional connectivity using propagator-based interferometry and neural field theory with application to the corticothalamic system

It is shown how to compute both direct and total effective connection matrices (deCMs and teCMs), which embody the strengths of neural connections between regions, from correlation-based functional CMs using propagator-based interferometry, a method that stems from geophysics and acoustics, coupled with the recent identification of deCMs and teCMs with bare and dressed propagators, respectively. The approach incorporates excitatory and inhibitory connections, multiple structures and populations, and measurement effects. The propagator is found for a generalized scalar wave equation derived from neural field theory, and expressed in terms of neural activity correlations and covariances, and wave damping rates. It is then related to correlation matrices that are commonly used to express functional and effective connectivities in the brain. The results are illustrated in analytically tractable test cases.

Anatomical connectivity and the resting state activity of large cortical networks

This paper uses mathematical modelling and simulations to explore the dynamics that emerge in large scale cortical networks, with a particular focus on the topological properties of the structural connectivity and its relationship to functional connectivity. We exploit realistic anatomical connectivity matrices (from diffusion spectrum imaging) and investigate their capacity to generate various types of resting state activity. In particular, we study emergent patterns of activity for realistic connectivity configurations together with approximations formulated in terms of neural mass or field models. We find that homogenous connectivity matrices, of the sort of assumed in certain neural field models give rise to damped spatially periodic modes, while more localised modes reflect heterogeneous coupling topologies. When simulating resting state fluctuations under realistic connectivity, we find no evidence for a spectrum of spatially periodic patterns, even when grouping together cortical nodes into communities, using graph theory. We conclude that neural field models with translationally invariant connectivity may be best applied at the mesoscopic scale and that more general models of cortical networks that embed local neural fields, may provide appropriate models of macroscopic cortical dynamics over the whole brain.

Interrelating anatomical, effective, and functional brain connectivity using propagators and neural field theory

It is shown how to compute effective and functional connection matrices (eCMs and fCMs) from anatomical CMs (aCMs) and corresponding strength-of-connection matrices (sCMs) using propagator methods in which neural interactions play the role of scatterings. This analysis demonstrates how network effects dress the bare propagators (the sCMs) to yield effective propagators (the eCMs) that can be used to compute the covariances customarily used to define fCMs. The results incorporate excitatory and inhibitory connections, multiple structures and populations, asymmetries, time delays, and measurement effects. They can also be postprocessed in the same manner as experimental measurements for direct comparison with data and thereby give insights into the role of coarse-graining, thresholding, and other effects in determining the structure of CMs. The spatiotemporal results show how to generalize CMs to include time delays and how natural network modes give rise to long-range coherence at resonant frequencies. The results are demonstrated using tractable analytic cases via neural field theory of cortical and corticothalamic systems. These also demonstrate close connections between the structure of CMs and proximity to critical points of the system, highlight the importance of indirect links between brain regions and raise the possibility of imaging specific levels of indirect connectivity. Aside from the results presented explicitly here, the expression of the connections among aCMs, sCMs, eCMs, and fCMs in terms of propagators opens the way for propagator theory to be further applied to analysis of connectivity.

Dynamical reconnection and stability constraints on cortical network architecture

Stability under dynamical changes to network connectivity is invoked alongside previous criteria to constrain brain network architecture. A new hierarchical network is introduced that satisfies all these constraints, unlike more commonly studied regular, random, and small-world networks. It is shown that hierarchical networks can simultaneously have high clustering, short path lengths, and low wiring costs, while being robustly stable under large scale reconnection of substructures.