Turing instability and pattern formation in a two-population neuronal network model

Wyller, John; Blomquist, Patrick; Einevoll, Gaute T.

doi:10.1016/j.physd.2006.10.004

Cited by 35 publications

(60 citation statements)

References 30 publications

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“…Astonishing examples are found in morphogenesis, the branch of embryology devoted to investigating the development of patterns and forms in biology 11 . Moreover, spatial patterns of cortical activation observed with electroencephalography, magnetoencephalography and functional magnetic resonance imaging have been argued to originate from the spontaneous self-organization of interacting populations of excitatory and inhibitory neurons 12 . In this case, travelling waves can also develop following a symmetry breaking instability of a homogeneous fixed point 13 .…”

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confidence: 99%

The theory of pattern formation on directed networks

et al. 2014

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Dynamical processes on networks have generated widespread interest in recent years. The theory of pattern formation in reaction-diffusion systems defined on symmetric networks has often been investigated, due to its applications in a wide range of disciplines. Here we extend the theory to the case of directed networks, which are found in a number of different fields, such as neuroscience, computer networks and traffic systems. Owing to the structure of the network Laplacian, the dispersion relation has both real and imaginary parts, at variance with the case for a symmetric, undirected network. The homogeneous fixed point can become unstable due to the topology of the network, resulting in a new class of instabilities, which cannot be induced on undirected graphs. Results from a linear stability analysis allow the instability region to be analytically traced. Numerical simulations show travelling waves, or quasi-stationary patterns, depending on the characteristics of the underlying graph.

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confidence: 99%

The theory of pattern formation on directed networks

et al. 2014

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“…2, 3 for the model with equal synaptic weights demonstrate that the choice of temporal coupling kernel can have a large effect on the stability properties of the (common) equilibrium point. A general observation is that instability is promoted by large inhibitory time constants τ , a feature also observed for Turing-type instabilities in analogous rate models for a one-dimensional spatial continuum of neurons (Wyller et al 2007). …”

Section: Discussionmentioning

confidence: 67%

“…In the context of models in one or more spatial dimensions the present instability in the zero-dimensional spatial model corresponds to an infinite-wavelength instability, contrasting with the finite-wavelength instabilities generating the periodic structures, see, e.g., Wyller et al (2007). A natural extension of this work would thus be to see how the finite-wavelength instabilities in models in one spatial dimension are affected by other choices of temporal coupling kernels.…”

Section: Discussionmentioning

confidence: 96%

“…With a one-or two-dimensional spatial continuum a richer variety of instabilities from the (homogeneous) background states is possible, for example stationary periodic patterns and spatiotemporally oscillations, see, e.g., Ermentrout and Cowan (1979a,b), Ermentrout and Cowan (1980a,b), Hutt et al (2003), Curtu and Ermentrout (2004), Hutt and Atay (2005), Hutt (2005, 2006), Wyller et al (2007) and reviews by Ermentrout (1998), Bressloff (2005, and Coombes (2005). In the context of models in one or more spatial dimensions the present instability in the zero-dimensional spatial model corresponds to an infinite-wavelength instability, contrasting with the finite-wavelength instabilities generating the periodic structures, see, e.g., Wyller et al (2007).…”

Section: Discussionmentioning

confidence: 99%

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Neural network firing-rate models on integral form

2007

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Firing-rate models describing neural-network activity can be formulated in terms of differential equations for the synaptic drive from neurons. Such models are typically derived from more general models based on Volterra integral equations assuming exponentially decaying temporal coupling kernels describing the coupling of pre- and postsynaptic activities. Here we study models with other choices of temporal coupling kernels. In particular, we investigate the stability properties of constant solutions of two-population Volterra models by studying the equilibrium solutions of the corresponding autonomous dynamical systems, derived using the linear chain trick, by means of the Routh-Hurwitz criterion. In the four investigated synaptic-drive models with identical equilibrium points we find that the choice of temporal coupling kernels significantly affects the equilibrium-point stability properties. A model with an alpha-function replacing the standard exponentially decaying function in the inhibitory coupling kernel is in most of our examples found to be most prone to instability, while the opposite situation with an alpha-function describing the excitatory kernel is found to be least prone to instability. The standard model with exponentially decaying coupling kernels is typically found to be an intermediate case. We further find that stability is promoted by increasing the weight of self-inhibition or shortening the time constant of the inhibition.

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“…In particular, the unknown complex amplitude W enters the definition of the right-hand side B 3 . By imposing the solvability condition (U *…”

Section: B Multiscale Analysis: a Topology Dependent Stuart-landau Ementioning

confidence: 99%

Multiple-scale theory of topology-driven patterns on directed networks

Contemori¹,

Patti²,

Fanelli³

et al. 2016

Phys. Rev. E

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Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from computer networks to traffic systems. Due to the peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points can turn unstable, on a directed support, because of the topology of the network, a phenomenon which cannot be induced on undirected graphs. A linear analysis can be performed to single out the conditions that underly the instability. The complete characterization of the patterns, which are eventually attained beyond the linear regime of exponential growth, calls instead for a full nonlinear treatment. By performing a multiple time scale perturbative calculation, we here derive an effective equation for the nonlinear evolution of the amplitude of the most unstable mode, close to the threshold of criticality. This is a Stuart-Landau equation the complex coefficients of which appear to depend on the topological features of the embedding directed graph. The theory proves adequate versus simulations, as confirmed by operating with a paradigmatic reaction-diffusion model.

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Turing instability and pattern formation in a two-population neuronal network model

Cited by 35 publications

References 30 publications

The theory of pattern formation on directed networks

The theory of pattern formation on directed networks

Neural network firing-rate models on integral form

Multiple-scale theory of topology-driven patterns on directed networks

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