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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

Time series.Time series for three types of oscillatory dynamics in the bath coupled reduced model. In the left panels (a), (c) and (e) the membrane potential and the three Nernst potentials are shown. Ion concentrations are shown in the right panels (b), (d) and (f). The color code is as in Fig. 5. (a) and (b), (c) and (d), and (e) and (f) are simulations for ,  and , respectively. The dynamics is typical for (a) and (b) seizure–like activity, (c) and (d) tonic firing, (e) and (f) periodic SD. Note the different time scales of SLA, tonic firing and period SD and also the different oscillation amplitudes in the ionic variables.
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pcbi-1003941-g007: Time series.Time series for three types of oscillatory dynamics in the bath coupled reduced model. In the left panels (a), (c) and (e) the membrane potential and the three Nernst potentials are shown. Ion concentrations are shown in the right panels (b), (d) and (f). The color code is as in Fig. 5. (a) and (b), (c) and (d), and (e) and (f) are simulations for , and , respectively. The dynamics is typical for (a) and (b) seizure–like activity, (c) and (d) tonic firing, (e) and (f) periodic SD. Note the different time scales of SLA, tonic firing and period SD and also the different oscillation amplitudes in the ionic variables.

Mentions: Depending on the level of the bath concentration, we find three qualitatively different types of oscillatory dynamics that are shown in Fig. 7. The top row (a) shows the time series of seizure-like activity for . It is characterized by repetitive bursting and low amplitude ion oscillations. The other types of oscillatory dynamics are tonic firing at with almost constant ion concentrations (Fig. 7b) and periodic SD at with large ionic amplitudes (Fig. 7c). We see that SLA and periodic SD exhibit slow oscillations of the ion concentrations and fast spiking activity, which hints at the toroidal nature of these dynamics. Below we will relate SLA and periodic SD to torus bifurcations of the tonic firing limit cycle.


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 three types of oscillatory dynamics in the bath coupled reduced model. In the left panels (a), (c) and (e) the membrane potential and the three Nernst potentials are shown. Ion concentrations are shown in the right panels (b), (d) and (f). The color code is as in Fig. 5. (a) and (b), (c) and (d), and (e) and (f) are simulations for ,  and , respectively. The dynamics is typical for (a) and (b) seizure–like activity, (c) and (d) tonic firing, (e) and (f) periodic SD. Note the different time scales of SLA, tonic firing and period SD and also the different oscillation amplitudes in the ionic variables.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g007: Time series.Time series for three types of oscillatory dynamics in the bath coupled reduced model. In the left panels (a), (c) and (e) the membrane potential and the three Nernst potentials are shown. Ion concentrations are shown in the right panels (b), (d) and (f). The color code is as in Fig. 5. (a) and (b), (c) and (d), and (e) and (f) are simulations for , and , respectively. The dynamics is typical for (a) and (b) seizure–like activity, (c) and (d) tonic firing, (e) and (f) periodic SD. Note the different time scales of SLA, tonic firing and period SD and also the different oscillation amplitudes in the ionic variables.
Mentions: Depending on the level of the bath concentration, we find three qualitatively different types of oscillatory dynamics that are shown in Fig. 7. The top row (a) shows the time series of seizure-like activity for . It is characterized by repetitive bursting and low amplitude ion oscillations. The other types of oscillatory dynamics are tonic firing at with almost constant ion concentrations (Fig. 7b) and periodic SD at with large ionic amplitudes (Fig. 7c). We see that SLA and periodic SD exhibit slow oscillations of the ion concentrations and fast spiking activity, which hints at the toroidal nature of these dynamics. Below we will relate SLA and periodic SD to torus bifurcations of the tonic firing limit cycle.

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