Limits...
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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Phase space plots.Phase space plots of the simulations (a) for SLA and (b) periodic SDs from Fig. 7. Only extracellular potassium is shown. The limit cycle and fixed point curves from Figs. 2 and 4 are superimposed to the plots as shaded lines whereas the limit cycle and fixed point from Fig. 2 (dynamical chloride) are darker. The limit cycle and fixed point are not graphically distinguished, but comparison with Fig. 2 should avoid confusion.
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pcbi-1003941-g010: Phase space plots.Phase space plots of the simulations (a) for SLA and (b) periodic SDs from Fig. 7. Only extracellular potassium is shown. The limit cycle and fixed point curves from Figs. 2 and 4 are superimposed to the plots as shaded lines whereas the limit cycle and fixed point from Fig. 2 (dynamical chloride) are darker. The limit cycle and fixed point are not graphically distinguished, but comparison with Fig. 2 should avoid confusion.

Mentions: Last we consider the dynamics of SLA and periodic SD in a phase space projection. In Fig. 10 the trajectories for SLA and periodic SD are plotted in the -plane together with the underlying fixed point and limit cycles from the transmembrane model (cf. Fig. 4). The periodic SD trajectory has a very similar shape to the single SD excursion from Fig. 6 and is clearly guided by the stable fixed point branches and . On the other hand SLA is a qualitatively very different phenomenon. Rather than relating to the FES branch, it is an oscillation between physiological conditions and those stable limit cycles that exist for moderately elevated extracellular potassium concentrations. The ion concentrations remain far from FES. So SLA and SD are not only related to distinct bifurcations, though of similar toroidal nature and branching from the same limit cycle, but they are also located far from each other in the phase space. This completes our phase space analysis of local ion dynamics in open neuron systems.


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 (a) for SLA and (b) periodic SDs from Fig. 7. Only extracellular potassium is shown. The limit cycle and fixed point curves from Figs. 2 and 4 are superimposed to the plots as shaded lines whereas the limit cycle and fixed point from Fig. 2 (dynamical chloride) are darker. The limit cycle and fixed point are not graphically distinguished, but comparison with Fig. 2 should avoid confusion.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g010: Phase space plots.Phase space plots of the simulations (a) for SLA and (b) periodic SDs from Fig. 7. Only extracellular potassium is shown. The limit cycle and fixed point curves from Figs. 2 and 4 are superimposed to the plots as shaded lines whereas the limit cycle and fixed point from Fig. 2 (dynamical chloride) are darker. The limit cycle and fixed point are not graphically distinguished, but comparison with Fig. 2 should avoid confusion.
Mentions: Last we consider the dynamics of SLA and periodic SD in a phase space projection. In Fig. 10 the trajectories for SLA and periodic SD are plotted in the -plane together with the underlying fixed point and limit cycles from the transmembrane model (cf. Fig. 4). The periodic SD trajectory has a very similar shape to the single SD excursion from Fig. 6 and is clearly guided by the stable fixed point branches and . On the other hand SLA is a qualitatively very different phenomenon. Rather than relating to the FES branch, it is an oscillation between physiological conditions and those stable limit cycles that exist for moderately elevated extracellular potassium concentrations. The ion concentrations remain far from FES. So SLA and SD are not only related to distinct bifurcations, though of similar toroidal nature and branching from the same limit cycle, but they are also located far from each other in the phase space. This completes our phase space analysis of local ion dynamics in open neuron systems.

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