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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

Phase space plots.Phase space plots of the simulations in Fig. 5. As in Fig. 4 panels (a) and (b) contain plots of the membrane potentials, in panels (c) and (d) extracellular potassium is shown. (a) and (c) are for the reduced model, (b) and (d) for the detailed model. The trajectories of the reduced model are represented as red curves, those of the detailed model are magenta. The sections of the trajectories that belong to times before and during the stimulation are dashed. The fixed point curves from Fig. 4 are added on the plots as shaded lines whereas the fixed point continuations for the unbuffered models with dynamical chloride are slightly darker. The pair of arrows in the extracellular potassium plots indicates the direction of pure transmembrane (vertical) and pure buffering dynamics (diagonal).
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pcbi-1003941-g006: Phase space plots.Phase space plots of the simulations in Fig. 5. As in Fig. 4 panels (a) and (b) contain plots of the membrane potentials, in panels (c) and (d) extracellular potassium is shown. (a) and (c) are for the reduced model, (b) and (d) for the detailed model. The trajectories of the reduced model are represented as red curves, those of the detailed model are magenta. The sections of the trajectories that belong to times before and during the stimulation are dashed. The fixed point curves from Fig. 4 are added on the plots as shaded lines whereas the fixed point continuations for the unbuffered models with dynamical chloride are slightly darker. The pair of arrows in the extracellular potassium plots indicates the direction of pure transmembrane (vertical) and pure buffering dynamics (diagonal).

Mentions: The time series in Fig. 5 are useful to confirm that the neuron models we investigate have the desired phenomenology and indeed show SD-like dynamics. Yet the nature of the different phases of this ionic excitation process—the fast depolarization, the prolonged FES phase and the abrupt repolarization—remains enigmatic [12], [14], [29], [30]. In a phase space plot the picture becomes much clearer and the entire process can be directly related to the two stable branches, and , that we found for the closed and therefore pure transmembrane models in the previous subsection. In Fig. 6 the time series from Fig. 5 for a simulation time of 50 min are shown in the - and the -plane. The parts of the trajectories during the stimulation (pump interruption and potassium elevation) are dashed. In the chosen planes vertical lines belong to dynamics of constant potassium contents that can be understood in terms of the models we analyzed in the previous subsection. That is why Fig. 6 contains the fixed point curves from Fig. 4 as shaded lines as a guide to the eye. In Fig. 6c and 6d buffering dynamics is diagonal as indicated by the pair of arrows added to the plot.


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)

Phase space plots.Phase space plots of the simulations in Fig. 5. As in Fig. 4 panels (a) and (b) contain plots of the membrane potentials, in panels (c) and (d) extracellular potassium is shown. (a) and (c) are for the reduced model, (b) and (d) for the detailed model. The trajectories of the reduced model are represented as red curves, those of the detailed model are magenta. The sections of the trajectories that belong to times before and during the stimulation are dashed. The fixed point curves from Fig. 4 are added on the plots as shaded lines whereas the fixed point continuations for the unbuffered models with dynamical chloride are slightly darker. The pair of arrows in the extracellular potassium plots indicates the direction of pure transmembrane (vertical) and pure buffering dynamics (diagonal).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g006: Phase space plots.Phase space plots of the simulations in Fig. 5. As in Fig. 4 panels (a) and (b) contain plots of the membrane potentials, in panels (c) and (d) extracellular potassium is shown. (a) and (c) are for the reduced model, (b) and (d) for the detailed model. The trajectories of the reduced model are represented as red curves, those of the detailed model are magenta. The sections of the trajectories that belong to times before and during the stimulation are dashed. The fixed point curves from Fig. 4 are added on the plots as shaded lines whereas the fixed point continuations for the unbuffered models with dynamical chloride are slightly darker. The pair of arrows in the extracellular potassium plots indicates the direction of pure transmembrane (vertical) and pure buffering dynamics (diagonal).
Mentions: The time series in Fig. 5 are useful to confirm that the neuron models we investigate have the desired phenomenology and indeed show SD-like dynamics. Yet the nature of the different phases of this ionic excitation process—the fast depolarization, the prolonged FES phase and the abrupt repolarization—remains enigmatic [12], [14], [29], [30]. In a phase space plot the picture becomes much clearer and the entire process can be directly related to the two stable branches, and , that we found for the closed and therefore pure transmembrane models in the previous subsection. In Fig. 6 the time series from Fig. 5 for a simulation time of 50 min are shown in the - and the -plane. The parts of the trajectories during the stimulation (pump interruption and potassium elevation) are dashed. In the chosen planes vertical lines belong to dynamics of constant potassium contents that can be understood in terms of the models we analyzed in the previous subsection. That is why Fig. 6 contains the fixed point curves from Fig. 4 as shaded lines as a guide to the eye. In Fig. 6c and 6d buffering dynamics is diagonal as indicated by the pair of arrows added to the plot.

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