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Journal of Computational Neuroscience

01.04.2009

The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics

verfasst von: John R. Cressman Jr., Ghanim Ullah, Jokubas Ziburkus, Steven J. Schiff, Ernest Barreto

Erschienen in: Journal of Computational Neuroscience | Ausgabe 2/2009

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Abstract

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. In this first paper, we construct a mathematical model consisting of a single conductance-based neuron together with intra- and extracellular ion concentration dynamics. We formulate a reduction of this model that permits a detailed bifurcation analysis, and show that the reduced model is a reasonable approximation of the full model. We find that competition between intrinsic neuronal currents, sodium-potassium pumps, glia, and diffusion can produce very slow and large-amplitude oscillations in ion concentrations similar to what is seen physiologically in seizures. Using the reduced model, we identify the dynamical mechanisms that give rise to these phenomena. These models reveal several experimentally testable predictions. Our work emphasizes the critical role of ion concentration homeostasis in the proper functioning of neurons, and points to important fundamental processes that may underlie pathological states such as epilepsy.

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Depending on the stability of the periodic orbit involved, Hopf bifurcations are classified as sub- or supercritical.

http://senselab.med.yale.edu/modeldb/

The stable and unstable periodic orbits involved in this scenario appear via a saddle-node bifurcation at a slightly smaller parameter value that is extremely close to that of the Hopf bifurcation. Thus, the sequence of bifurcations is not immediately apparent in Fig. 2. The abruptness of these transitions, and the difficulty in resolving them numerically, is due to the “canard” mechanism (Dumortier and Roussarie 1996; Wechselberger (2007)).

A canard similar to that described previously occurs here, so that the Hopf and the saddle-node bifurcations on the left sides of Figs. 3a and b occur in extremely narrow intervals of the parameter.

In Fig. 3a, the equilibrium curve does not extend all the way to zero because of the constant chloride leak current.

Note that oscillations may persist slightly outside of the RO, where a stable periodic orbit coexists with the stable equilibrium solution; see, for example, the right side of Fig. 3a.

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Titel: The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics
verfasst von: John R. Cressman Jr.
Ghanim Ullah
Jokubas Ziburkus
Steven J. Schiff
Ernest Barreto
Publikationsdatum: 01.04.2009
Verlag: Springer US
Erschienen in: Journal of Computational Neuroscience / Ausgabe 2/2009
Print ISSN: 0929-5313
Elektronische ISSN: 1573-6873
DOI: https://doi.org/10.1007/s10827-008-0132-4

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