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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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Time series.Time series for single SD excursions in (a), (c) the reduced and in (b), (d) the detailed model. In the reduced model SD is triggered by an interruption of the pump activity for about 10 sec (shaded region). In the detailed model the extracellular potassium concentration is increased by  mM after 20 sec (vertical line). In (a) and (b) the time series of the membrane potentials (black lines) are shown. Nernst potentials for all ion species are included to the diagrams as a reference. Ion dynamics are shown in (c) and (d) where extracellular ion concentrations are in lighter color.
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pcbi-1003941-g005: Time series.Time series for single SD excursions in (a), (c) the reduced and in (b), (d) the detailed model. In the reduced model SD is triggered by an interruption of the pump activity for about 10 sec (shaded region). In the detailed model the extracellular potassium concentration is increased by mM after 20 sec (vertical line). In (a) and (b) the time series of the membrane potentials (black lines) are shown. Nernst potentials for all ion species are included to the diagrams as a reference. Ion dynamics are shown in (c) and (d) where extracellular ion concentrations are in lighter color.

Mentions: For the simulations in Fig. 5 we have interrupted the pump activity for about 10 sec in the reduced model, and we have elevated the extracellular potassium concentration by mM in the detailed model to trigger SD. Both stimulation types work for both models, but only the two examples are shown. The phase of pump interruption (Fig. 5a and 5c) is indicated by the shaded region in the plots, the time of potassium elevation is marked by the vertical grey line. The dynamics of the two models is very similar: in response to the stimulation the neuron strongly depolarizes and remains in that depolarized state for about 70 sec (Fig. 5a and 5b). After that the neurons repolarize abruptly and asymptotically return to their initial state. In addition to the membrane potential (black curve) the potential plots also contain the Nernst potentials for sodium (red line), potassium (blue line) and chloride (green line) that change with the ion concentrations according to the definition of the Nernst potentials in Eq. (24). In Fig. 5c and 4d we see that the potential dynamics goes along with great changes in the ion concentrations. In particular, extracellular potassium is strongly increased in the depolarized phase. These conditions are very similar to the type of FES states discussed in the previous subsection. The recovery of ion concentrations sets in with the abrupt repolarization, but it is a very slow asymptotic process that is not shown in Fig. 5.


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)

Time series.Time series for single SD excursions in (a), (c) the reduced and in (b), (d) the detailed model. In the reduced model SD is triggered by an interruption of the pump activity for about 10 sec (shaded region). In the detailed model the extracellular potassium concentration is increased by  mM after 20 sec (vertical line). In (a) and (b) the time series of the membrane potentials (black lines) are shown. Nernst potentials for all ion species are included to the diagrams as a reference. Ion dynamics are shown in (c) and (d) where extracellular ion concentrations are in lighter color.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g005: Time series.Time series for single SD excursions in (a), (c) the reduced and in (b), (d) the detailed model. In the reduced model SD is triggered by an interruption of the pump activity for about 10 sec (shaded region). In the detailed model the extracellular potassium concentration is increased by mM after 20 sec (vertical line). In (a) and (b) the time series of the membrane potentials (black lines) are shown. Nernst potentials for all ion species are included to the diagrams as a reference. Ion dynamics are shown in (c) and (d) where extracellular ion concentrations are in lighter color.
Mentions: For the simulations in Fig. 5 we have interrupted the pump activity for about 10 sec in the reduced model, and we have elevated the extracellular potassium concentration by mM in the detailed model to trigger SD. Both stimulation types work for both models, but only the two examples are shown. The phase of pump interruption (Fig. 5a and 5c) is indicated by the shaded region in the plots, the time of potassium elevation is marked by the vertical grey line. The dynamics of the two models is very similar: in response to the stimulation the neuron strongly depolarizes and remains in that depolarized state for about 70 sec (Fig. 5a and 5b). After that the neurons repolarize abruptly and asymptotically return to their initial state. In addition to the membrane potential (black curve) the potential plots also contain the Nernst potentials for sodium (red line), potassium (blue line) and chloride (green line) that change with the ion concentrations according to the definition of the Nernst potentials in Eq. (24). In Fig. 5c and 4d we see that the potential dynamics goes along with great changes in the ion concentrations. In particular, extracellular potassium is strongly increased in the depolarized phase. These conditions are very similar to the type of FES states discussed in the previous subsection. The recovery of ion concentrations sets in with the abrupt repolarization, but it is a very slow asymptotic process that is not shown in Fig. 5.

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