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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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Related in: MedlinePlus

Bifurcation diagram.Different representations of the bifurcation diagram of Fig. 8. Panel (a) shows the extracellular sodium concentration and includes an inset around TR4 and PD. Panel (b) presents the potassium gain/loss.
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pcbi-1003941-g009: Bifurcation diagram.Different representations of the bifurcation diagram of Fig. 8. Panel (a) shows the extracellular sodium concentration and includes an inset around TR4 and PD. Panel (b) presents the potassium gain/loss.

Mentions: In Fig. 9 we look at the same bifurcation diagram in the - and the -plane. While in Fig. 8 most of the ionic phase space structure is hidden, because for fixed points and limit cycles, the -presentation in Fig. 9a provides further insights into the ion dynamics. We see that the stable fixed point branch before HB1 has extracellular sodium concentrations close to the physiological value . The stable branch after HB4, however, has an extremely reduced extracellular sodium level and is indeed FES-like. The stable limit cycles between PD and TR4 and between TR3 and TR2, and also SLA are rather close to the physiological sodium level. On the other hand, periodic SD is an oscillation between FES and normal physiological conditions, which is an expected confirmation of the findings from the previous section.


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.Different representations of the bifurcation diagram of Fig. 8. Panel (a) shows the extracellular sodium concentration and includes an inset around TR4 and PD. Panel (b) presents the potassium gain/loss.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g009: Bifurcation diagram.Different representations of the bifurcation diagram of Fig. 8. Panel (a) shows the extracellular sodium concentration and includes an inset around TR4 and PD. Panel (b) presents the potassium gain/loss.
Mentions: In Fig. 9 we look at the same bifurcation diagram in the - and the -plane. While in Fig. 8 most of the ionic phase space structure is hidden, because for fixed points and limit cycles, the -presentation in Fig. 9a provides further insights into the ion dynamics. We see that the stable fixed point branch before HB1 has extracellular sodium concentrations close to the physiological value . The stable branch after HB4, however, has an extremely reduced extracellular sodium level and is indeed FES-like. The stable limit cycles between PD and TR4 and between TR3 and TR2, and also SLA are rather close to the physiological sodium level. On the other hand, periodic SD is an oscillation between FES and normal physiological conditions, which is an expected confirmation of the findings from the previous section.

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.

Show MeSH
Related in: MedlinePlus