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Patterns generation and transition matrices in multi-dimensional lattice models

2005; American Institute of Mathematical Sciences; Volume: 13; Issue: 3 Linguagem: Inglês

10.3934/dcds.2005.13.637

1553-5231

Jung-Chao Ban, Song-Sun Lin,

Neural dynamics and brain function

In this paper we develop a general approachfor investigating pattern generation problems in multi-dimensionallattice models. Let $\mathcal S$ be a set of $p$ symbols orcolors, $\mathbf Z_N$ a fixed finite rectangular sublattice of$\mathbf Z^d$, $d\geq 1$ and $N$ a $d$-tuple of positiveintegers. Functions $U:\mathbf Z^d\rightarrow \mathcal S$ and$U_N:\mathbf Z_N\rightarrow \mathcal S$ are called a globalpattern and a local pattern on $\mathbf Z_N$, respectively. Weintroduce an ordering matrix $\mathbf X_N$ for $\Sigma_N$, theset of all local patterns on $\mathbf Z_N$. For a larger finitelattice , , we derive arecursion formula to obtain the ordering matrix of from$\mathbf X_N$. For a given basic admissible local patterns set$\mathcal B\subset \Sigma_N$, the transition matrix$\mathbf T_N(\mathcal B)$ is defined. For each , denoted by the set of all localpatterns which can be generated from $\mathcal B$, the cardinalnumber of is the sum of entriesof the transition matrix which can be obtained from $\mathbf T_N(\mathcal B)$recursively. The spatial entropy $h(\mathcal B)$ can be obtainedby computing the maximum eigenvalues of a sequence of transitionmatrices $\mathbf T_n(\mathcal B)$. The results can be appliedto study the set of global stationary solutions in various LatticeDynamical Systems and Cellular Neural Networks.