Abstract
This study is related to a model describing the behavior of barium-treatedAplysia neurons generating regular burst-plateau patterns. The model is represented by an autonomous dynamical system, defined inR 4 and depending on a small parameter. This paper is restricted to the qualitative study of three “reduced systems” deduced from the “complete system”. Part of the study is performed with the use of the qualitative theory of singular perturbations. The predicted behaviors are compared with experimental results.
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Literature
Andronov, A. A., A. A. Vitt and S. E. Khaikin. 1966.Theory of Oscillators, English translation. Reading, Mass.: Addison Wesley.
Argémi, J. 1968. “Sur les Points Singuliers Multiples de Systèmes Dynamiques dansR 2.”Annali Mat. Pura Appl.,79, 35–70.
Arvanitaki, A. 1962. “Plateaux de Dépolarisation et Trains Paroxystiques de Pointes en Hyperthermie sur Certains Neurones Identifiables d'Aplysia.”C.R. Acad. Sci., Paris,255, 1523–1525.
Bendixson, I. 1901. “Sur les Courbes Définies par des Équations Différentielles.”Acta Math., Scand.,24, 1–88.
Both, R., W. Finger and R. A. Chaplain. 1976. “Model Predictions of the Ionic Mechanisms Underlying the Beating and Bursting Pacemaker Characteristics of Molluscan Neurons.”Biol. Cybernetics,23, 1–11.
Chalazonitis, N. 1977. “Introduction to Neuronal Burst Activity.”J. Physiol, Paris,73, 441–452.
Dubois-Violette, P. L. 1943. “Sur les Points Singuliers Exceptionnels des Équations Différentielles du Premier Ordre Considérés Comme Limites de Points Singuliers Simples.”C.R. Acad. Sci., Oaris,217, 567–569.
Ducreux, C. and M. Gola. 1975. “Ondes Paroxysmales Induites par le Métrazol (PTZ) sur les Neurones d'Helix p.: Modèle Fonctionnel.”Pflüger Arch.,361, 43–53.
— and—. 1977. “Ionic Mechanism of Ba2+-Induced Square-Shaped Potential Waves in Molluscan Neurons.”Brain Res.,123, 384–389.
Eckert, R. and H. D. Lux. 1976. “A Voltage-Sensitive Persistent Calcium Conductance in Neuronal Somata ofHelix.J. Physiol., Lond.,254, 129–151.
El'sgol'c, L. E. 1964.Qualitative Methods in Mathematical Analysis. Am. Math. Soc. Providence, R.I.
Fitz-Hugh, R. 1960. “Thresholds and Plateus in the Hodgkin-Huxley Nerve Equations.”J. Gen. Physiol.,43, 867–896.
—, 1961. “Impulses and Physiological States in Theoretical Models of Nerve Membrane.”Biophys. J.,1, 445–466.
Gola, M. 1974. “Neurones à Ondes-Salves des Mollusques. Variations Cycliques Lentes des Conductances Ioniques.”Pflügers Arch.,352, 17–36.
Gola, M., C. Ducreux and H. Chagneux. 1977. “Ionic Mechanism of Slow Potential Wave Production in Barium-TreatedAplysia Neurons.”J. Physiol., Paris,73, 407–440.
Leicht, R., H. Meves and H. H. Wellhoner. 1971. “The Effect of Veratridine onHelix pomatia Neurones.”Pflügers Arch.,323, 50–62.
Mathieu, P. A. and F. A. Roberge. 1971. “Characteristics of Pacemaker Oscillations inAplysia Neurons.”Can. J. Physiol. Pharmac.,49, 787–795.
Magura, I. S. 1977. “Long-Lasting Inward Current in Snail Neurons in Barium Solutions in Voltage-Clamp Conditions.”J. Membrane Biol.,35, 239–256.
Plant, R. E. 1976. “The Geometry of the Hodgkin-Huxley Model.”Computer Programs Biomedicine,6, 85–91.
— and M. Kim. 1976. “Mathematical Description of a Bursting Pacemaker Neuron by a Modification of the Hodgkin-Huxley Equations.”Biophys. J.,16, 227–244.
Pontryagin, L. S. 1957. “Asymptotic Behavior of the Solutions of Systems of Differential Equations with a Small Parameter in the Higher Derivatives.”Izv. Akad. Nauk SSSR (ser. matem.),253, 450–452. (Russian)
Tihonov, A. N. 1948. “On the Dependence of the Solutions of Differential Equations on a Small Parameter.”Math. Sb. (N.S.),22, 193–204. (Russian)
Ulbricht, W. 1969. “The Effects of Veratridine on Excitable Membranes of Nerve and Muscle.”Ergebn. Physiol., Rev. Physiol.,61, 18–71.
Wilson, W. A. and H. Wachtel. 1974. “Negative Resistance Characteristic Essential for the Maintenance of Slow Oscillations in Bursting Neurons.”Science,186, 932–934.
Zeeman, E. C. 1973. “Differential Equations for the Heartbeat and Nerve Impulse”. InDynamical Systems, Ed. M. Peixoto, pp. 683–741. New York: Academic Press.
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Argémi, J., Gola, M. & Chagneux, H. Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells—I. Reduced systems. Bltn Mathcal Biology 41, 665–686 (1979). https://doi.org/10.1007/BF02462421
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DOI: https://doi.org/10.1007/BF02462421