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  • 1
    Electronic Resource
    Electronic Resource
    Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells—I. Reduced systems (1979)
    Springer
    Bulletin of mathematical biology 41 (1979), S. 665-686 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462421
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  • 2
    Electronic Resource
    Electronic Resource
    Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells. II. Complete system (1980)
    Springer
    Bulletin of mathematical biology 42 (1980), S. 221-238 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper deals with a model describing the behavior of barium-treatedApalysia neurons. The model is represented by a dynamical system, so-called “complete system”, defined in R4 and depending on a small parameter. The study of this system under zero membrane current conditions was performed with the use of the qualitative theory of singular perturbations. We show that this system has a stable periodic solution of the discontinuous type when the small parameter tends to 0+. A reduced system defined in R3, associated to the complete system was also studied: it corresponds to a constant activation of the inward current. We demonstrate that the corresponding hypothetical cell remains silent under zero current conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02464639
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  • 3
    Electronic Resource
    Electronic Resource
    Qualitative study of a dynamical system for metrazol-induced paroxysmal depolarization shifts (1984)
    Springer
    Bulletin of mathematical biology 46 (1984), S. 903-922 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The convulsant pentylenetetrazol (PTZ) was used to trigger spike bursts and paroxysmal discharges inAplysia neurons. Voltage clamp experiments showed that PTZ induced a slow voltage-dependent potassium current and a persistent inward current. These currents are incorporated into a membrane model together with modified spike-generating Hodgkin-Huxley equations. From these data a metaphoric model is constructed and represented by a slow-fast dynamical system defined inR 4. With some values of the main physiological parameters, the system might have limit cycles for the fast dynamic. A qualitative study of the system shows that it satisfactorily reproduces the various observed patterns produced by PTZ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462078
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  • 4
    Electronic Resource
    Electronic Resource
    An asymmetrical kinetic model for veratridine interactions with sodium channels in molluscan neurons (1982)
    Springer
    Bulletin of mathematical biology 44 (1982), S. 231-258 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Veratridine alkaloid induces bi-stability or saw-tooth-shaped long potential waves in molluscan neurons. Voltage clamp experiments reveal the production of a slow sodium current whose changes are described by an asymmetric kinetic diagram relating the states of the sodium channels. Methods of the qualitative theory of differential equations were used to determine the condition necessary for such a model to have either an oscillatory solution or a bi-stable behavior. The kinetic diagram was modified to account for the frequency dependence of the slow sodium current production upon repeated short depolarizations. The modified kinetic diagram suggests that open and inactivated sodium channels are turned into channels with slow kinetic parameters; the transition from open channels would be fast, irreversible and restricted to part of the open channels, whereas that from inactivated channels would be slow and fully reversible upon repolarization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02463249
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  • 5
    Electronic Resource
    Electronic Resource
    Solutions périodiques discontinues pour l'approximation singulière d'un modèle neurophysiologique dans R4 — une métaphore dans R3 avec chaos (1983)
    Springer
    Journal of mathematical biology 17 (1983), S. 67-92 
    Details
    ISSN: 1432-1416
    Keywords: Oscillateurs biologiques ; Perturbations singulières ; Bifurcations ; Catastrophes ; Chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Résumé Il s'agit de l'étude qualitative d'un problème de perturbations singulières, lié à un modèle de comportement électrique de la membrane d'une cellule nerveuse traitée par le baryum. On démontre, pour l'approximation singulière de ce modèle, l'existence de solutions périodiques géométriquement distinctes, dont on étudie les bifurcations en fonction de l'intensité du courant transmembranaire. Certaines de ces solutions traversent un point pseudo-singulier et ne sont pas individualisables lorsqu'il y a engouffrement de trajectoires lentes en ce point. Une métaphore de ce problème conduit, pour une valeur donnée du petit paramètre, à des solutions chaotiques par doublement de période.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276115
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  • 6
    Electronic Resource
    Electronic Resource
    Points singuliers non élémentaires de quelques classes de systèmes dynamiques plans (1971)
    Springer
    Annali di matematica pura ed applicata 89 (1971), S. 321-351 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Résumé A partir de la méthode projective exposée en[5], on détermine la nature du point singulier non élémentaire, supposé à l’origine, pour quelques classes de systèmes dynamiques plans, et en particulier, pour un système dont les termes de plus bas degré ont un diviseur réel commun.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02414952
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  • 7
    Electronic Resource
    Electronic Resource
    Sur les points singuliers multiples de systèmes dynamiques dansR 2 (1968)
    Springer
    Annali di matematica pura ed applicata 79 (1968), S. 35-69 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Sommaire On étudie, dans le voisinage d’un point singulier, le comportement des trajectoires d’un système dynamique à deux variables défini par deux formes homogènes de même degré.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02415178
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