Chimera patterns in two-dimensional networks of coupled neurons

Chimera patterns in two-dimensional networks of coupled neurons

We discuss synchronization patterns in networks of FitzHugh-Nagumo and Leaky Integrate-and-Fire oscillators coupled in a two-dimensional toroidal geometry. Common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observe in the two models follow similar rules. For example the diameter of the rings grows linearly with the coupling radius.

One. INTRODUCTION

Chimeras are hybrid states that emerge spontaneously, combining both coherent and incoherent parts. First found in identical and symmetrically coupled phase oscillators, chimera states have been the focus of extensive research for over a decade now. Both theoretical and experimental works have shown that this counter-intuitive collective phenomenon may arise in numerous systems including mechanical, chemical, electro-chemical, electro-optical, electronic, and superconducting coupled oscillators.

The phenomenon of chimera states has also been addressed in networks of biological neural oscillators. In particular, Hindmarsh-Rose neural oscillators have been studied in networks with nonlocal and nearest-neighbor coupling, as well as in modular networks consisting of communities. Potential relevance of chimera states in this context include bump states and the phenomenon of unihemispheric sleep observed in birds and dolphins, which sleep with one eye open, meaning that half of the brain is synchronous with the other half being asynchronous. Furthermore, it has been recently hypothesized that chimera states are the route of onset or termination of epileptic seizures.

Chimera states have also been reported in the one-dimensional FitzHugh-Nagumo model with nonlocal connectivity. Patches of synchrony were observed within the incoherent domains giving rise to multichimera states, when the coupling constant increased above the weak limit. The multiplicity of the state, that is, the number of (in)coherent regions, depended on the coupling strength and range. These multichimera states were shown to be robust when small inhomogeneities in the coupling topology (with identical elements) or inhomogeneous elements with regular nonlocal coupling were introduced. For a constant coupling strength and given number of links, hierarchical connectivity was shown to induce nested multichimera patterns. Coexistence of coherent and incoherent domains were also observed in systems of excitable FitzHugh-Nagumo elements under the influence of noise. As Semenova et al. stress, the noise has often a constructive role: It shifts the dynamics of identical excitable elements into the oscillatory domain, giving then rise to chimera states. Moreover, it is possible to control the position of coherent and incoherent domains of a multichimera state by introducing a block of excitable FitzHugh-Nagumo elements in appropriate positions, or to generate a chimera directly from the synchronous state. This observation offers promising ideas in terms of achieving desired states by a local modification of the system parameters. For instance, it may be possible in targeted medication to modify incoherent neuron dynamics by changing only the potential of a few neurons locally, leaving the coupling topology of the rest of the network untouched. In addition, Omelchenko et al. proposed an alternative control scheme that extends the lifetime of chimeras, which are known to be chaotic transients, and at the same time reduces the erratic drift present in small networks. Controlling the position can also be achieved by introducing an asymmetric coupling strength.

Experimentally, the FitzHugh-Nagumo model has been studied by Essaki Arumugam and Spano, in connection to synchronization phenomena associated with neurological disorders such as epilepsy. In their study, they implemented nine FitzHugh-Nagumo neurons linked in a ring topology, via discrete electronics. Introducing nonlocal coupling, a chimera state appeared, while for local connectivity either fully synchronous or asynchronous states were observed. Their results indicate that epilepsy can be understood as a topological disease, strongly related to the connectivity of the underlying network of neurons.

ders such as epilepsy [32]. In their study, they imple- mented nine FHN neurons linked in a ring topology, via discrete electronics. Introducing nonlocal coupling, a chimera state appeared, while for local connectivity ei- ther fully synchronous or asynchronous states were ob- served. Their results indicate that epilepsy can be un- derstood as a topological disease, strongly related to the connectivity of the underlying network of neurons.

Chimera states were recently reported in the one-dimensional Leaky Integrate-and-Fire model, a system describing the spiking behavior of neuron cells. It was shown that identical Leaky Integrate-and-Fire oscillators nonlocally linked in a one-dimensional ring geometry give rise to multichimera states whose multiplicity depends both on the coupling strength and the refractory period of the neuron cells. In analogy with the FitzHugh-Nagumo model, the introduction of a hierarchical topology in the coupling induced nested chimera states and also transitions between multichimera states with different multiplicities. Using a different geometrical setup of two populations of identical Leaky Integrate-and-Fire oscillators coupled via excitatory coupling, Olmi et al. studied the onset of chimeras as well as states characterized by a different degree of synchronization in the two populations. Irregular synchronization phenomena have been reported for Leaky Integrate-and-Fire elements even in the case of all-to-all connectivity.

Chimera states have mainly been investigated in one-dimensional systems. Recently, works involving two and three-dimensional oscillator arrays have revealed new types of chimera states, depending on the coupling function. These studies have mainly focused on phase oscillators. In this paper, we go beyond the simple Kuramoto model and consider a two-dimensional network configuration using two different models related to neuronal spiking activity. This topology is motivated from medical experiments where thin brain slices are cultured in Petri dishes and various electrical and chemical properties are recorded. The two-dimensional nonlocal connectivity studied here can be considered as an approximation of the acute brain slices, whose connectivity is certainly more complex.

Our investigation follows a parallel presentation of common patterns in the two-dimensional FitzHugh-Nagumo and Leaky Integrate-and-Fire systems highlighting the main features that are responsible for the formation of each pattern in the where we fix the coupling strength at sigma equals zero point one in the two systems. In Section Two, the two models are briefly recapitulated. In Section Three, main attributes of spot and ring chimeras are presented for the two models, and their common and different properties are discussed. Similarly, in Sections Four and Five stripe and grid chimeras are discussed, respectively. Additional miscellaneous chimeras as well as other patterns are summarized in the appendices. Finally, the main results and open problems are discussed in a brief concluding section.