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Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs.

Hübel N, Dahlem MA - PLoS Comput. Biol. (2014)

Bottom Line: Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings.The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours).The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

View Article: PubMed Central - PubMed

Affiliation: Department of Theoretical Physics, Technische Universität Berlin, Berlin, Germany.

ABSTRACT
The classical Hodgkin-Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

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Bifurcation diagram.Fundamental bifurcation diagram in the slowest–scale dynamics, the potassium ion gain or loss through reservoirs (i.e., the bifurcation parameter). The unit of the bifurcation parameter was chosen such that it denotes the ion concentration with respect to the extracellular volume. The actual extracellular potassium concentration is the order parameter. Shown are the stable branches  and  (see Sec. Results) and the directions (arrows) of two paths of ‘pure’ flux condition: fluxes exclusively across the membrane and fluxes exclusively from (or to) reservoirs. A horizontal path is caused by a particular mixture of these fluxes that induces potassium ion concentration changes exclusively to the intracellular compartment. Ionic excitability can be understood as a cyclic process in this diagram (see text).
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pcbi-1003941-g011: Bifurcation diagram.Fundamental bifurcation diagram in the slowest–scale dynamics, the potassium ion gain or loss through reservoirs (i.e., the bifurcation parameter). The unit of the bifurcation parameter was chosen such that it denotes the ion concentration with respect to the extracellular volume. The actual extracellular potassium concentration is the order parameter. Shown are the stable branches and (see Sec. Results) and the directions (arrows) of two paths of ‘pure’ flux condition: fluxes exclusively across the membrane and fluxes exclusively from (or to) reservoirs. A horizontal path is caused by a particular mixture of these fluxes that induces potassium ion concentration changes exclusively to the intracellular compartment. Ionic excitability can be understood as a cyclic process in this diagram (see text).

Mentions: In particular SD is explained by a bistability of neuronal ion dynamics that occurs in the absence of external reservoirs. The potassium gain or loss through reservoirs provided by an extracellular bath, the vasculature or the glial cells is identified as a bifurcation parameter whose essential importance was not realized in earlier studies (see Fig. 11). Using this bifurcation parameter and the extracellular potassium concentration as the order parameter, we obtain a folded fixed point curve with the two outer stable branches corresponding to states with normal physiological function, hence named physiological branch , and to states being free-energy starved ().


Dynamics from seconds to hours in Hodgkin-Huxley model with time-dependent ion concentrations and buffer reservoirs.

Hübel N, Dahlem MA - PLoS Comput. Biol. (2014)

Bifurcation diagram.Fundamental bifurcation diagram in the slowest–scale dynamics, the potassium ion gain or loss through reservoirs (i.e., the bifurcation parameter). The unit of the bifurcation parameter was chosen such that it denotes the ion concentration with respect to the extracellular volume. The actual extracellular potassium concentration is the order parameter. Shown are the stable branches  and  (see Sec. Results) and the directions (arrows) of two paths of ‘pure’ flux condition: fluxes exclusively across the membrane and fluxes exclusively from (or to) reservoirs. A horizontal path is caused by a particular mixture of these fluxes that induces potassium ion concentration changes exclusively to the intracellular compartment. Ionic excitability can be understood as a cyclic process in this diagram (see text).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4256015&req=5

pcbi-1003941-g011: Bifurcation diagram.Fundamental bifurcation diagram in the slowest–scale dynamics, the potassium ion gain or loss through reservoirs (i.e., the bifurcation parameter). The unit of the bifurcation parameter was chosen such that it denotes the ion concentration with respect to the extracellular volume. The actual extracellular potassium concentration is the order parameter. Shown are the stable branches and (see Sec. Results) and the directions (arrows) of two paths of ‘pure’ flux condition: fluxes exclusively across the membrane and fluxes exclusively from (or to) reservoirs. A horizontal path is caused by a particular mixture of these fluxes that induces potassium ion concentration changes exclusively to the intracellular compartment. Ionic excitability can be understood as a cyclic process in this diagram (see text).
Mentions: In particular SD is explained by a bistability of neuronal ion dynamics that occurs in the absence of external reservoirs. The potassium gain or loss through reservoirs provided by an extracellular bath, the vasculature or the glial cells is identified as a bifurcation parameter whose essential importance was not realized in earlier studies (see Fig. 11). Using this bifurcation parameter and the extracellular potassium concentration as the order parameter, we obtain a folded fixed point curve with the two outer stable branches corresponding to states with normal physiological function, hence named physiological branch , and to states being free-energy starved ().

Bottom Line: Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings.The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours).The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

View Article: PubMed Central - PubMed

Affiliation: Department of Theoretical Physics, Technische Universität Berlin, Berlin, Germany.

ABSTRACT
The classical Hodgkin-Huxley (HH) model neglects the time-dependence of ion concentrations in spiking dynamics. The dynamics is therefore limited to a time scale of milliseconds, which is determined by the membrane capacitance multiplied by the resistance of the ion channels, and by the gating time constants. We study slow dynamics in an extended HH framework that includes time-dependent ion concentrations, pumps, and buffers. Fluxes across the neuronal membrane change intra- and extracellular ion concentrations, whereby the latter can also change through contact to reservoirs in the surroundings. Ion gain and loss of the system is identified as a bifurcation parameter whose essential importance was not realized in earlier studies. Our systematic study of the bifurcation structure and thus the phase space structure helps to understand activation and inhibition of a new excitability in ion homeostasis which emerges in such extended models. Also modulatory mechanisms that regulate the spiking rate can be explained by bifurcations. The dynamics on three distinct slow times scales is determined by the cell volume-to-surface-area ratio and the membrane permeability (seconds), the buffer time constants (tens of seconds), and the slower backward buffering (minutes to hours). The modulatory dynamics and the newly emerging excitable dynamics corresponds to pathological conditions observed in epileptiform burst activity, and spreading depression in migraine aura and stroke, respectively.

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Related in: MedlinePlus