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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

Fixed point continuation.Fixed point continuations for a range of impermeant intracellular chloride concentrations  in (a), (b) the –plane and (c), (d) the –plane. The black curves are the stable FES branches that lose their stability in Hopf bifurcations (black circles). Starting from the leftmost fixed point curves the fixed  values are 8, 12, 16, 20, 24, 28 and 32 mM for the reduced model and 9, 13, 17, 21, 25, 29 and 33 mM for the detailed model. The Hopf bifurcations for different chloride concentrations lead to the blue Hopf line. As a reference the fixed point curves from Figs. 2 and 3 are also included in the diagram and drawn in grey.
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pcbi-1003941-g004: Fixed point continuation.Fixed point continuations for a range of impermeant intracellular chloride concentrations in (a), (b) the –plane and (c), (d) the –plane. The black curves are the stable FES branches that lose their stability in Hopf bifurcations (black circles). Starting from the leftmost fixed point curves the fixed values are 8, 12, 16, 20, 24, 28 and 32 mM for the reduced model and 9, 13, 17, 21, 25, 29 and 33 mM for the detailed model. The Hopf bifurcations for different chloride concentrations lead to the blue Hopf line. As a reference the fixed point curves from Figs. 2 and 3 are also included in the diagram and drawn in grey.

Mentions: For each value of we perform a fixed point continuation as in Figs. 2 and 3 which yields similarly folded s-shaped curves. The result is shown in Fig. 4. For our analysis of SD it is only relevant where ends. That is why the plot does not contain the whole fixed point curve, but only and a part of the unstable branch for a selection of values. As a reference the diagrams also contain the fixed point curves from Figs. 2 and 3 which include chloride dynamics. The FES branches in Fig. 4 end in Hopf bifurcations. The bifurcation points for different chloride concentrations yield the blue Hopf line. It marks the threshold for recovery from FES when dynamics of chloride and is slow.


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)

Fixed point continuation.Fixed point continuations for a range of impermeant intracellular chloride concentrations  in (a), (b) the –plane and (c), (d) the –plane. The black curves are the stable FES branches that lose their stability in Hopf bifurcations (black circles). Starting from the leftmost fixed point curves the fixed  values are 8, 12, 16, 20, 24, 28 and 32 mM for the reduced model and 9, 13, 17, 21, 25, 29 and 33 mM for the detailed model. The Hopf bifurcations for different chloride concentrations lead to the blue Hopf line. As a reference the fixed point curves from Figs. 2 and 3 are also included in the diagram and drawn in grey.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g004: Fixed point continuation.Fixed point continuations for a range of impermeant intracellular chloride concentrations in (a), (b) the –plane and (c), (d) the –plane. The black curves are the stable FES branches that lose their stability in Hopf bifurcations (black circles). Starting from the leftmost fixed point curves the fixed values are 8, 12, 16, 20, 24, 28 and 32 mM for the reduced model and 9, 13, 17, 21, 25, 29 and 33 mM for the detailed model. The Hopf bifurcations for different chloride concentrations lead to the blue Hopf line. As a reference the fixed point curves from Figs. 2 and 3 are also included in the diagram and drawn in grey.
Mentions: For each value of we perform a fixed point continuation as in Figs. 2 and 3 which yields similarly folded s-shaped curves. The result is shown in Fig. 4. For our analysis of SD it is only relevant where ends. That is why the plot does not contain the whole fixed point curve, but only and a part of the unstable branch for a selection of values. As a reference the diagrams also contain the fixed point curves from Figs. 2 and 3 which include chloride dynamics. The FES branches in Fig. 4 end in Hopf bifurcations. The bifurcation points for different chloride concentrations yield the blue Hopf line. It marks the threshold for recovery from FES when dynamics of chloride and is slow.

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