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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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Bifurcation diagram.Bifurcation diagram of the bath coupled reduced model for –variation. Color and line style conventions for fixed points and limit cycles are Figs. 2 and 3: black and green lines are fixed point and limit cycles, solid and dashed line styles mean stable and unstable sections. Stable solution on invariant tori are blue. They were obtained by direct simulations. The fixed point changes stability in HBs and LPs. The bifurcation types limit cycle undergoes are , period–doubling (PD) and torus bifurcation (TR). Some physiologically irrelevant unstable limit cycles are omitted (cf. text). Panel (a) shows the membrane potential, panel (b) shows the extracellular potassium concentration. (b) does not contain the limit cycle, because it can hardly be distinguished from the fixed point line.
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pcbi-1003941-g008: Bifurcation diagram.Bifurcation diagram of the bath coupled reduced model for –variation. Color and line style conventions for fixed points and limit cycles are Figs. 2 and 3: black and green lines are fixed point and limit cycles, solid and dashed line styles mean stable and unstable sections. Stable solution on invariant tori are blue. They were obtained by direct simulations. The fixed point changes stability in HBs and LPs. The bifurcation types limit cycle undergoes are , period–doubling (PD) and torus bifurcation (TR). Some physiologically irrelevant unstable limit cycles are omitted (cf. text). Panel (a) shows the membrane potential, panel (b) shows the extracellular potassium concentration. (b) does not contain the limit cycle, because it can hardly be distinguished from the fixed point line.

Mentions: Fig. 8 shows the bifurcation diagram for variation in the -plane and in the -plane. In addition to fixed points (black) and limit cycles (green) also quasiperiodic torus solutions (blue) are contained in the diagram. In comparison to Fig. 2 this model contains a new type of bifurcation, namely a torus bifurcation (TR). A torus bifurcation is a secondary Hopf bifurcation of the radius of a limit cycle in which an invariant torus is created. If this torus is stable, nearby trajectories will be asymptotically bound to its surface. However, we cannot follow such solutions with standard continuation techniques, because these require an algebraic formulation in terms of the oscillation period. This is not possible for torus solutions, because on a torus the motion is quasiperiodic, i.e., characterized by two incommensurate frequencies. We can hence only track the stable solutions by integrating the equations of motion and slowly varying . It is due to this numerically expensive method that in this section we will only analyze oscillatory dynamics of the reduced HH model with time-dependent ion concentrations.


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.Bifurcation diagram of the bath coupled reduced model for –variation. Color and line style conventions for fixed points and limit cycles are Figs. 2 and 3: black and green lines are fixed point and limit cycles, solid and dashed line styles mean stable and unstable sections. Stable solution on invariant tori are blue. They were obtained by direct simulations. The fixed point changes stability in HBs and LPs. The bifurcation types limit cycle undergoes are , period–doubling (PD) and torus bifurcation (TR). Some physiologically irrelevant unstable limit cycles are omitted (cf. text). Panel (a) shows the membrane potential, panel (b) shows the extracellular potassium concentration. (b) does not contain the limit cycle, because it can hardly be distinguished from the fixed point line.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g008: Bifurcation diagram.Bifurcation diagram of the bath coupled reduced model for –variation. Color and line style conventions for fixed points and limit cycles are Figs. 2 and 3: black and green lines are fixed point and limit cycles, solid and dashed line styles mean stable and unstable sections. Stable solution on invariant tori are blue. They were obtained by direct simulations. The fixed point changes stability in HBs and LPs. The bifurcation types limit cycle undergoes are , period–doubling (PD) and torus bifurcation (TR). Some physiologically irrelevant unstable limit cycles are omitted (cf. text). Panel (a) shows the membrane potential, panel (b) shows the extracellular potassium concentration. (b) does not contain the limit cycle, because it can hardly be distinguished from the fixed point line.
Mentions: Fig. 8 shows the bifurcation diagram for variation in the -plane and in the -plane. In addition to fixed points (black) and limit cycles (green) also quasiperiodic torus solutions (blue) are contained in the diagram. In comparison to Fig. 2 this model contains a new type of bifurcation, namely a torus bifurcation (TR). A torus bifurcation is a secondary Hopf bifurcation of the radius of a limit cycle in which an invariant torus is created. If this torus is stable, nearby trajectories will be asymptotically bound to its surface. However, we cannot follow such solutions with standard continuation techniques, because these require an algebraic formulation in terms of the oscillation period. This is not possible for torus solutions, because on a torus the motion is quasiperiodic, i.e., characterized by two incommensurate frequencies. We can hence only track the stable solutions by integrating the equations of motion and slowly varying . It is due to this numerically expensive method that in this section we will only analyze oscillatory dynamics of the reduced HH model with time-dependent ion concentrations.

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