Chimera patterns in two-dimensional networks of coupled neurons

Abstract

We discuss synchronization patterns in networks of FitzHugh-Nagumo and Leaky Integrate-andFire 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 FitzHughNagumo 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.

Synchronization patterns in LIF neuron networks: merging nonlocal and diagonal connectivity

“…The present results show that it is a general property of the LIF model and has been previously reported in Ref. [34] for the 2D connectivity. The same effect also holds for the combined (nonlocaldiagonal) connectivity, as we will see in sec.…”

“…A recently discovered [25,26] complex synchronization pattern is the chimera state; this is a collective state characterized by the simultaneous presence of synchronous and asynchronous domains [18,19,[27][28][29][30][31][32]. Chi-mera states have been realized using a number of neuron models, such as the FitzHugh-Nagumo model [19,24,[32][33][34][35], the Hindmarsh-Rose model of bursting neurons [20-22, 29, 36-39], the LIF model [34,[40][41][42][43][44] and others. More recent studies include simulations of chime-ra arXiv:1807.11843v2 [nlin.PS] 13 Oct 2018 states in 2D lattices [34,38] and 3D spatial dimensions [34,39,[44][45][46].…”

“…Chi-mera states have been realized using a number of neuron models, such as the FitzHugh-Nagumo model [19,24,[32][33][34][35], the Hindmarsh-Rose model of bursting neurons [20-22, 29, 36-39], the LIF model [34,[40][41][42][43][44] and others. More recent studies include simulations of chime-ra arXiv:1807.11843v2 [nlin.PS] 13 Oct 2018 states in 2D lattices [34,38] and 3D spatial dimensions [34,39,[44][45][46]. Another striking synchronization example is the subthreshold oscillations: the network splits into two coexisting domains, one domain where all elements perform small-amplitude oscillations at subthreshold values and one domain where the incoherent elements develop a multileveled mean phase velocity distribution [43].…”

Chimera States With 2D Deterministic and Random Fractal Connectivity

“…They captivated scientific interest during the past 15 years due to their intriguing structural and dynamical properties and to potential applications in physics [4][5][6][7], chemistry [8][9][10], and biology [11][12][13][14][15][16]. Although original studies referred to coupled phase oscillators, later works have reported chimera states in coupled FitzHugh-Nagumo, Hindmarsh-Rose, Van der Pol, and Leaky Integrate-and-Fire (LIF) oscillator networks [17][18][19][20][21][22][23][24]. Most recent advances in the general domain of local synchronization are summarized in review articles [25][26][27][28][29].…”

“…Previous studies on 2D nonlocal connectivity with periodic, toroidal boundary conditions have demonstrated a variety of chimera patterns. Using the phase oscillator, the FitzHugh Nagumo system or the LIF neuron oscillators chimera patterns emerged in the form of coherent and incoherent single or multiple spots, rings, lines, and grids of spots [19,[30][31][32][33][34]. Some of these patterns, come as generalizations of the 1D chimera forms to 2D geometry (e.g., spots and stripes), while others are new patterns which do not have an analogy in 1D (e.g., spiral waves).…”