Continuation of Localized Coherent Structures in Nonlocal Neural Field Equations

Rankin, James; Avitabile, Daniele; Baladron, Javier; Faye, Grégory; Lloyd, David J. B.

doi:10.1137/130918721

Cited by 55 publications

(58 citation statements)

References 60 publications

(78 reference statements)

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“…Note that generically, our conditions on the Fourier transform of connectivity kernel K imply that, in real space, the kernel is locally excitatory while it presents lateral modulations of inhibition and excitation. This specific form of kernels have already been used in the literature [25,28] to analyze stationary multi-bump solutions of equation (5.5) and is also in agreement with experimentally recorded cortical connections in cat visual areas [8]. Amplitude Equations As previously stated, at = 0, there is a bifurcation of the stationary state u = 0 where we have at the same time a Turing instability and a pitchfork bifurcation for the kinetics (5.10).…”

Section: Hypothesis (H)supporting

confidence: 78%

Linear spreading speeds from nonlinear resonant interaction

2017

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We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that facilitates the resonant coupling. We prove that these resonant speeds give the correct invasion speed in a simple example, we show that fronts with speeds slower than the resonant speed are unstable, and corroborate our speed criterion numerically in a variety of model equations, including a nonlocal scalar neural field model.

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Section: Hypothesis (H)supporting

confidence: 78%

Linear spreading speeds from nonlinear resonant interaction

2017

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“…These lines meet at two codimension-2 points (where the Turing bifurcation line changes color) that agree with the results of the weakly nonlinear analysis. Moreover, we computed numerically a bifurcation diagram of the NFM, using the spectral method developed in Reference [36] and available with Reference [37]. The results, presented in Figure (4,c) confirm that the unstable BS bifurcates subcritically for the SHS.…”

Section: B Spatially Homogeneous States (Shs) and Their Stability Smentioning

confidence: 63%

Synchrony-induced modes of oscillation of a neural field model

et al. 2017

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We investigate the modes of oscillation of heterogeneous ring networks of quadratic integrate-and-fire (QIF) neurons with nonlocal, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogously to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially homogeneous state and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states and are maintained away from onset.

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“…Localised square patterns have also been observed in the same equation with an additional nonlinear gradient term [28]. Planar neuronal models also exhibit similar exotic solutions [29]. Patchwork quilt state, i.e., regular triangles, [30] is foreseen in bistable systems with the symmetry u → −u.…”

Section: Introductionmentioning

confidence: 75%

Snakes in square, honeycomb and triangular lattices

Kusdiantara

Susanto

2019

Nonlinearity

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We present a study of time-independent solutions of the two-dimensional discrete Allen-Cahn equation with cubic and quintic nonlinearity. Three different types of lattices are considered, i.e., square, honeycomb, and triangular lattices. The equation admits uniform and localised states. We can obtain localised solutions by combining two different states of uniform solutions, which can develop a snaking structure in the bifurcation diagrams. We find that the complexity and width of the snaking diagrams depend on the number of "patch interfaces" admitted by the lattice systems. We introduce an active-cell approximation to analyse the saddle-node bifurcation and stabilities of the corresponding solutions along the snaking curves. Numerical simulations show that the active-cell approximation gives good agreement for all of the lattice types when the coupling is weak. We also consider planar fronts that support our hypothesis on the relation between the complexity of a bifurcation diagram and the number of interface of its corresponding solutions.

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Continuation of Localized Coherent Structures in Nonlocal Neural Field Equations

Cited by 55 publications

References 60 publications

Linear spreading speeds from nonlinear resonant interaction

Linear spreading speeds from nonlinear resonant interaction

Synchrony-induced modes of oscillation of a neural field model

Snakes in square, honeycomb and triangular lattices

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