Characterization and dynamics of pattern formation in cellular neural networks

K. R. Crounse, L. O. Chua, P. Thiran, G. Setti

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


We study some properties of pattern formation arising in large arrays of locally coupled first-order nonlinear dynamical systems, namely Cellular Neural Networks (CNNs). We will present exact results to analyze spatial patterns for symmetric coupling and to analyze spatio-temporal patterns for anti-symmetric coupling in one-dimensional lattices, which will then be completed by approximative results based on a spatial and/or temporal frequency approach. We will discuss the validity of these approximations, which bring a lot of insight. This spectral approach becomes very convenient for the two-dimensional lattice, as exact results get more complicated to establish. In this second part, we will only consider a symmetric coupling between cells. We will show what kinds of motifs can be found in the patterns generated by 3 × 3 templates. Then, we will discuss the dynamics of pattern formation starting from initial conditions which are a small random noise added to the unstable equilibrium: this can generally be well predicted by the spatial frequency approach. We will also study whether a defect in a pure pattern can propagate or not through the whole lattice, starting from initial conditions being a localized perturbation of a stable pattern: this phenomenon is no longer correctly predicted by the spatial frequency approach. We also show that patterns such as spirals and targets can be formed by "seed" initial conditions - localized, non-random perturbations of an unstable equilibrium. Finally, the effects on the patterns formed of a bias term in the dynamics are demonstrated.
Original languageEnglish (US)
Pages (from-to)1703-1724
Number of pages22
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Issue number9
StatePublished - Jan 1 1996
Externally publishedYes

ASJC Scopus subject areas

  • Applied Mathematics
  • General
  • General Engineering
  • Modeling and Simulation


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