Bifurcation structure of a discrete neuronal equation (Q1825037)

The general equation \(x_{n+1}=\mathbf{1}(\theta -\sum_{0\leq i\leq n}a(i)x_{n-i})\) (where \(\mathbf{1}\) is the characteristic function of [0,\(\infty [\) in \({\mathbb{R}}\), \(\theta\in {\mathbb{R}}\), and a(i)\(\in {\mathbb{R}})\) was first proposed by \textit{E. R. Caianello} and \textit{A. de Luca} [Kybernetik 3, 33-40 (1966)] and, with \textit{L. Ricciardi} [Kybernetik 4 (1967)], to simulate the neural response taking into account of the refractary transition period. The particular case given by \(a(i)=b^ i\), \(0<b\leq 1/2\), was previously studied by \textit{T. Kitagawa} [Math. Biosci. 18, 191-244 (1973; Zbl 0272.92002)]. The authors investigate the bifurcation diagram arising from iterations of the k-steps memory transition A(k,\(\theta\),b) given by \[ x_{n+1}=\mathbf{1}(\theta - \sum_{0\leq i<k}b^ ix_{n-i}), \] which can be reduced to A(k,\(\lambda\),\()\) for a suitable \(\lambda\). The complete bifurcation structure is described. Put \(F_{\lambda}(x_ 0,...,x_{k- 1})=\mathbf{1} (\lambda -\sum_{0\leq i<k}x_ i/2^ i)\) and call a 0-1 string \(C=C_{T-1}..C_ 0\) (of length \(| C|:=T)\) admissible if there exist k and \(\lambda\) such that C is a cycle for A(k,\(\lambda\),\()\), i.e., \[ C_{n+1}=F_{\lambda}(C_ n...C_{n-k+1}),\quad n=0,...,T-1 \] (using indices modulo T). Admissible cycle are characterized (Theorem 1) by a max-min condition involving lexicographic order on the set of cyclic permutations of C. Let \(\tau_{r,0}\) be the substitution \(10^ r\to 0\), \(10^{r+1}\to 1\) and put \(\tau_{r,1}=1-\tau_{r,0}\). If C is a cycle for A(k,\(\lambda\),\()\), then \(| C| \leq k+1\) (Corollary 2) and there exist \((r_ 1,s_ 1),...,(r_ n,s_ n)\) such that \(\tau_{r_ n,s_ n}\circ...\circ \tau_{r_ 1,s_ 1}(C)=0\) (Corollary 3). More results are given, one of them concerns the rotation number \(\rho\) (C) of C defined as the ratio of the number of 1's over the length of C. Theorem 3: The map \(\rho\) : \(R\to {\mathbb{Q}}\) is monotonic, increasing and \(\rho\) (\({\mathbb{R}})\) is the set of irreducible fractions with denominators \(\leq k+1\). This gives for the automaton A(k,\(\theta\),b) a unique attractor which is periodic. The last part relates the unbounded memory case in connection with previous results of \textit{J. Keener} [Trans. Am. Math. Soc. 261, 589-604 (1980; Zbl 0458.58016)], \textit{J. Nagumo} and \textit{S. Sato} [Kybernetik 10, 155-166 (1972; Zbl 0235.92001)], \textit{M. Yamaguti} and \textit{M. Hata} [Lect. Notes in Biomath. 45, 171-177 (1982; Zbl 0495.92004)], \textit{S. Yoschizawa} [ibid., 155-170 (1982; Zbl 0487.92006)]. Here the iteration has a unique global attractor which is (i) a cycle or (ii) a Cantor set. For a given b, case (ii) corresponds to parameters \(\lambda\) belonging to a Cantor set of Lebesgue measure zero (Theorem 4). Rotation number \(\rho_ b(\lambda)\) can be defined. It is rational in case (i) and irrational otherwise (Theorem 5). The map \(\rho_ b(\cdot)\) is a devil staircase (i.e., distribution function of a singular measure).