Abstract
For the Hartline-Ratliff model the exact conditions for the uniqueness of the stationary state are determined. Also sufficient conditions for dynamic stability are derived.
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Amari S (1977) Dynamics of pattern formation in lateralinhibition type neural fields. Biol Cybern 27:77–87
Bauer FL, Deutsch E, Stoer J (1969) Abschätzungen für die Eigenwerte positiver Operatoren. Lin Alg Appl 2:275–301
Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Academic Press, New York
Coleman BD, Maxwell JA, Renninger GH (1980) On oscillatory responses of the Limulus retina. Biol Cybern 37:125–129
Hadeler KP (1972) Bemerkugen zu einer Arbeit von W. Wetterling über positive Operatoren. Num Math 19:260–265
Hadeler KP (1974) On the theory of lateral inhibition. Kybernetik 14:161–165
Hadeler KP, Tomiuk J (1977) Periodic solutions of difference-differential equations. Arch Ration Mech Anal 65:87–95
Hartline HK, Ratliff F (1958) Spatial summation of inhibitory influences in the eye ofLimulus, and the mutual interaction of receptor units. J Gen Phys 41:1049–1066
Hartline HK, Ratliff F (1974) Studies on excitation and inhibition in the retina. Chapman and Hall, London (in particular p. 315ff.)
Hirsch MW (1982) Systems of differential equations which are competitive or cooperative. I. Limit sets. SIAM J Math Anal 13:167–179
Hirsch MW (1985) Systems of differential equations which are competitive or cooperative. II. Convergence everywhere. SIAM J Math Anal 16:423–439
Kirschfeld K, Reichardt W (1964) Die Verarbeitung stationärer optischer Nachrichten im Komplexauge vonLimulus. Kybernetik 2:43–61
Kuhn D (1986) Über stückweise lineare Abbildungen. Diplomarbeit, Tübingen
Kuhn D, Löwen R (1987) Piecewise affine bijections of ℝn, and the equationSx +−Tx -=y. Lin Alg Appl (in press)
Matsuoka K (1985) Sustained oscillations by mutually inhibiting neurons with adaptation. Biol Cybern 52:367–376
Meinhardt H (1982) Models of biological pattern formation. Academic Press, New York
Morishita I, Yajima A (1972) Analysis and simulation of networks of mutually inhibiting neurons. Kybernetik 11:154–165
Reichardt W, McGinitie G (1962) Zur Theorie der lateralen Inhibition. Kybernetik 1:155–165
Samelson H, Thrall RM, Wesler O (1958) A partition theorem of euclideann-spaces. Proc Am Math Soc 9:805–807
Tokura T, Morishita I (1977) Analysis and simulation of doublelayer neural networks with mutually inhibiting interconnections. Biol Cybern 25:83–92
Varjú D (1962) Vergleich zweier Modelle für die laterale Inhibition. Kybernetik 1:100–208
Varjú D (1965) On the theory of lateral inhibition, Cybernetics of neural processes. In: Cainanello EP (ed): Proceedings. Istituto di Fisica teoria, U. di Napoli
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Hadeler, K.P., Kuhn, D. Stationary states of the Hartline-Ratliff model. Biol. Cybernetics 56, 411–417 (1987). https://doi.org/10.1007/BF00319520
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DOI: https://doi.org/10.1007/BF00319520