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Chimera states in brain networks: empirical neural vs. modular fractal connectivity

Teresa Chouzouris, Iryna Omelchenko, Anna Zakharova, Jaroslav Hlinka, Premysl Jiruska, Eckehard Schöll

arXiv:1710.08219v2nlin.AO

TL;DR

The interplay of synchrony and asynchrony in brain networks matters for studying brain function and disease, including epilepsy. The paper models FitzHugh–Nagumo neurons in empirical and modular-fractal networks, finding qualitatively similar dynamical scenarios and examining seizure-like synchrony and node-removal effects.

  • Problem

    Synchrony and desynchrony are relevant to brain function and disorders, while the relation between neural-network structure and dynamical regimes motivates comparing empirical and hierarchical connectivity.

    Source: §I · pp. 1–2

  • Method

    The authors simulate coupled FitzHugh–Nagumo oscillators on empirical structural brain connectivity and a mathematically constructed modular-fractal network, then examine seizure-like dynamics and node-removal effects.

    Source: §I · p. 2, §II · p. 2

  • Results

    Both network topologies show qualitatively similar scenarios: weak coupling supports chimeras, and increasing coupling leads toward complete synchrony through solitary and frequency-synchronized states.

    Source: §III · p. 6, §VI · p. 9

  • Takeaways & Limitations

    The authors conclude that a topologically defined synchronizability index can help optimize node removal to decrease network synchronizability.

    Source: §VI · p. 9

Abstract

from arXiv · show

Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states, and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.

I. INTRODUCTION

Chimera states combine coexisting coherent and incoherent neural activity, making synchrony and desynchrony relevant to understanding brain function and disorders. The paper compares empirical and modular-fractal brain-network dynamics using FitzHugh–Nagumo oscillators, motivated by seizure research and possible applications to epilepsy surgery.

I. INTRODUCTION

  • Chimera states combine coherent and incoherent dynamics, and neural synchrony and desynchrony are relevant to both healthy brain function and disorders such as epilepsy and Parkinson’s disease.Neural synchronization patterns depend on structural connectivity, local dynamics, and coupling type and strength.Source: §I · p. 1
  • Neuronal axon connectivity has been described as hierarchical and quasi-fractal, and chimera states have been reported in networks with hierarchical connectivity.Prior work has also reported chimera-like states in several neuron models, including FitzHugh–Nagumo networks.Source: §I · p. 2
  • The study compares transitions from asynchronous behavior to synchrony through chimeras in an empirical dMRI-derived brain network and a mathematically constructed modular-fractal network.The authors use an iterative algorithm to construct a hierarchical, nested, quasi-fractal connectivity structure and FitzHugh–Nagumo oscillators to model node dynamics.Source: §I · p. 2

II. THE MODEL

The model uses N identical coupled FitzHugh–Nagumo oscillators, with activator and inhibitor variables and a rotational coupling scheme. The study fixes the timescale parameter and considers the oscillatory regime while comparing two weighted, undirected network topologies.

II. THE MODEL

  • The system consists of N identical coupled FitzHugh–Nagumo oscillators, with u and v denoting activator and inhibitor variables.The indices k = 1, ..., N are taken modulo N.Source: §II · p. 2
  • The timescale-separation parameter ε is fixed at 0.05, and the oscillatory regime is defined by |a| < 1.The threshold parameter is a and coupling strength is σ.Source: §II · p. 2
  • The network adjacency matrix is either symmetrized empirical structural connectivity from dMRI or a mathematically constructed modular-fractal topology; both are weighted and undirected.Source: §II · p. 2
  • Local interactions use a rotational coupling matrix, with coupling phase φ fixed at π/2 − 0.1 to allow information transfer between activator and inhibitor variables.Cross-couplings between these variables have been reported as essential for observing chimeras in identical nonlocally coupled FitzHugh–Nagumo systems.Source: §II · p. 2

A. Empirical structural topology

The empirical topology is a weighted, undirected structural network derived from dMRI data for one healthy subject and mapped onto 90 cortical and subcortical regions. It is used as a coarse approximation because full brain connectivity is not known and the FitzHugh–Nagumo model simplifies single-neuron dynamics.

A. Empirical structural topology

  • The empirical network comes from dMRI data for a randomly selected healthy control who participated in the Early-Stage Schizophrenia Outcome study.The structural network is anatomical, unlike fMRI, which indirectly measures neuronal activity over time.Source: A. Empirical structural topology · p. 2
  • The cortex and subcortex are divided into 90 regions using the AAL atlas, with each region represented as a network node.Probabilistic tractography estimates streamline-based connectivity between regions.Source: A. Empirical structural topology · p. 3
  • Connectivity probabilities are symmetrized by averaging Pij and Pji, yielding a weighted, undirected 90 × 90 adjacency matrix.Each probability is based on probabilistic tracts between regions, normalized by Ns = 5000 times the number of voxels in the starting region.Source: A. Empirical structural topology · p. 3
  • The network is a coarse approximation because full human-brain connectivity is unknown and the FitzHugh–Nagumo model simplifies single-neuron dynamics.The 90 regions cover the main cortical and subcortical areas; the model is also used as a coarse-grained excitable-media model.Source: A. Empirical structural topology · p. 3

B. Topology with modular fractal connectivity

The modular-fractal topology is generated iteratively from a base adjacency matrix using Kronecker products. Its weighted structure is designed to resemble the empirical network in size and visual appearance while retaining a hierarchical modular organization.

B. Topology with modular fractal connectivity

  • An iterative algorithm generates a hierarchical, quasi-fractal connectivity matrix from an initial b × b base matrix A1.The adjacency matrix at hierarchy level n is constructed through n − 1 Kronecker products of the base matrix with itself.Source: B. Topology with modular fractal… · p. 3
  • In the binary case, the recursive substitution of nonzero entries by scaled copies of A1 and zeros by zero matrices gives a modular topology rather than a circulant ring.The construction preserves a constant row sum when A1 has a constant row sum.Source: B. Topology with modular fractal… · p. 3
  • The empirical adjacency matrix is divided into 52 equal squares of size m × m, with m = 18, and the link sums determine the elements of a 5 × 5 base matrix.The base-matrix entries are specified for the empirical data.Source: B. Topology with modular fractal… · p. 3
  • Three iterations produce the investigated modular adjacency matrix with N = 125 elements, chosen to be comparable in size and visually similar to the empirical topology.The base matrix is weighted, link weights are normalized, the networks have equal mean link weights, and self-coupling is zero.Source: B. Topology with modular fractal… · p. 3

III. THE DYNAMICAL REGIMES

Both networks show a similar progression from chimera states at weak coupling toward complete synchronization through solitary and frequency-synchronized states. The empirical network requires stronger coupling to stabilize complete synchrony than the modular fractal network.

III. THE DYNAMICAL REGIMES

  • Regime maps and stability boundaries use coupling strength σ and threshold a, alongside mean phase-velocity measures, the global Kuramoto order parameter, and Master Stability Function results.The synchronous state is stable above the orange boundary in the parameter plane.Source: §III · pp. 3–4, Figure 3 · p. 5
  • Mean phase-velocity dispersion and the global Kuramoto order parameter characterize the patterns, while the Master Stability Function identifies the synchronous state's stability region.Larger ∆ω corresponds to more pronounced frequency differences; r = 1 denotes complete synchronization, while small r denotes spatial desynchronization.Source: §III · p. 4
  • At weak coupling and approximately a ∈ (0, 0.8), both networks exhibit chimeras, whose moving incoherent domains complicate velocity-profile interpretation but are indicated by spatial and temporal coherence measures.The reported examples are stationary moving chimeras, with zero temporal coherence and constant spatial coherence.Source: §III · pp. 4–5, Figure 4 · p. 6
  • As a and σ increase, chimera states give way to frequency-synchronized states with equal oscillator mean phase velocities; modular fractal networks also exhibit breathing variants.In the breathing state, stronger and slower order-parameter oscillations accompany cyclic changes in travelling-wave speed.Source: §III · p. 5
  • At higher coupling and threshold values, solitary states coexist with frequency-synchronized states in both networks, with most oscillators synchronized and a few at different frequencies.These patterns may also be classified as weak chimera states under the cited definition.Source: §III · p. 5
  • The empirical network has two solitary-state regimes: one often contains a single oscillator that splits off, while another includes three incoherent oscillators or turbulent solitary dynamics.In the single-oscillator example, periodic strong network input produces an additional oscillation while the rest remain frequency synchronized.Source: §III · pp. 5–6
  • The modular fractal network also shows solitary states near the transition from chimeras to complete synchronization, including coexistence with frequency-synchronized states and examples with numerous solitary oscillators.Source: §III · p. 6
  • Both networks progress from weak-coupling chimeras toward complete synchronization through solitary and frequency-synchronized states, but the empirical network needs stronger coupling for complete synchrony.The modular fractal network stabilizes complete synchrony at smaller coupling strengths.Source: §III · pp. 4, 6

IV. SIMULATION OF EPILEPTIC SEIZURES

In the empirical network, chimera dynamics produce spontaneous or driven high-coherence events described as qualitatively similar to epileptic seizures. Changing coupling can alter event probability and switch states, with a delayed response after coupling changes.

IV. SIMULATION OF EPILEPTIC SEIZURES

  • The empirical network produces spontaneous high-coherence events during chimera dynamics, qualitatively resembling epileptic seizure behavior.The authors report that driven events can also occur.Source: §IV · p. 6
  • Stronger coupling within the chimera regime increases the probability of high-coherence events.Source: §IV · p. 6
  • Changing coupling can switch between chimera and solitary or frequency-synchronized states, while high synchronization may persist briefly after coupling returns to its initial value.Source: §IV · pp. 6–7

V. NODE REMOVAL AND SYNCHRONIZABILITY

The study uses a Laplacian-eigenvalue index to assess how node removal changes network synchronizability. In the empirical network, removal strategies differ in effectiveness, while the tested measures are mostly insensitive in the modular-fractal network.

V. NODE REMOVAL AND SYNCHRONIZABILITY

  • The index is the ratio of the smallest transversal Laplacian eigenvalue after removal to its unperturbed value; values below one indicate reduced synchronizability.Source: §V · p. 8
  • Removing nodes with the lowest incremental synchronizability index most effectively decreases the network’s ability to synchronize.The index tracks the change in the smallest transversal Laplacian eigenvalue after each additional node removal.Source: §V · p. 8
  • In the empirical topology, degree- or closeness-based removal outperforms random removal, while lowest-index removal gives the best results.Source: §V · p. 8
  • In the modular-fractal topology, the tested measures are not sensitive except for the highest-eigenvalue shift, consistent with its narrower degree distribution and less-distinct node properties.Source: §V · p. 8
  • Removing nodes with minimum incremental index lowers the order parameter more successfully than removing random nodes.Source: §V · p. 9

VI. CONCLUSION

The FitzHugh–Nagumo networks show qualitatively similar partially synchronized dynamics despite different connectivity structures. The paper maps dynamical regimes and considers seizure-like synchrony and node removal as possible routes to studying synchronization control.

VI. CONCLUSION

  • Empirical and modular-fractal networks show similar scenarios of complex partially synchronized states under FitzHugh–Nagumo node dynamics.The modular-fractal topology is generated by a hierarchical iterative algorithm.Source: §VI · p. 9
  • Chimera states occur at weak coupling, with solitary and frequency-synchronized states appearing between them and complete synchrony as coupling increases.Source: §VI · p. 9
  • The authors simulate seizure-like pathological synchrony initiated or terminated via chimera states and discuss suppressing synchrony through node removal.Source: §VI · p. 9
  • A topologically defined synchronizability index can help optimize node selection for reducing synchronizability, with possible relevance to epilepsy surgery.Source: §VI · p. 9

APPENDIX: DEGREE DISTRIBUTIONS OF THE CONSIDERED TOPOLOGIES

The appendix presents degree distributions for the empirical and modular-fractal network topologies. Because both are undirected, their in-degree equals their out-degree.

APPENDIX: DEGREE DISTRIBUTIONS OF THE CONSIDERED TOPOLOGIES

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