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.

Show MeSH

Related in: MedlinePlus

Bifurcation diagram.Bifurcation diagram of the reduced model for  as the bifurcation parameter (purely transmembrane dynamics) showing (a) the membrane potential of fixed points (FP) and limit cycles (LC), and (b) potassium concentrations. The fixed point continuation yields the black curves. Solid sections are fully stable, dashed sections are unstable. The stability of the fixed point changes in HBs and LPs. The initial physiological condition is marked by a black square. The limit cycle is represented by the extremal values of the dynamical variables during one oscillation. The continuation yields the green lines with the same stability convention for solid and dashed sections. The stability of the limit cycle changes either in a LP or in a period–doubling bifurcation (PD). In (b) the maximal and minimal extracellular potassium concentration of the limit cycle never differs by more than  mM. The values can hence not be distinguished on the scale of this figure and therefore only the maximal value is drawn. The bifurcations are marked by full circles and labelled by the type, i.e., HB, LP or , and a counter (cf. also the insets with blow–ups, in particular the rightmost one showing  and  on a very small horizontal scale). The vertical and diagonal arrows labelled ‘m’ and ‘r’ indicate the direction of extracellular potassium changes due to ion fluxes across the membrane (‘m’) and changes only due to , i.e., because of ion exchange with a reservoir (‘r’). Note that along the horizontal directions only the ICS potassium concentration changes by a precise mixture of fluxes across the membrane and ion exchange with a reservoir.
© Copyright Policy
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4256015&req=5

pcbi-1003941-g002: Bifurcation diagram.Bifurcation diagram of the reduced model for as the bifurcation parameter (purely transmembrane dynamics) showing (a) the membrane potential of fixed points (FP) and limit cycles (LC), and (b) potassium concentrations. The fixed point continuation yields the black curves. Solid sections are fully stable, dashed sections are unstable. The stability of the fixed point changes in HBs and LPs. The initial physiological condition is marked by a black square. The limit cycle is represented by the extremal values of the dynamical variables during one oscillation. The continuation yields the green lines with the same stability convention for solid and dashed sections. The stability of the limit cycle changes either in a LP or in a period–doubling bifurcation (PD). In (b) the maximal and minimal extracellular potassium concentration of the limit cycle never differs by more than mM. The values can hence not be distinguished on the scale of this figure and therefore only the maximal value is drawn. The bifurcations are marked by full circles and labelled by the type, i.e., HB, LP or , and a counter (cf. also the insets with blow–ups, in particular the rightmost one showing and on a very small horizontal scale). The vertical and diagonal arrows labelled ‘m’ and ‘r’ indicate the direction of extracellular potassium changes due to ion fluxes across the membrane (‘m’) and changes only due to , i.e., because of ion exchange with a reservoir (‘r’). Note that along the horizontal directions only the ICS potassium concentration changes by a precise mixture of fluxes across the membrane and ion exchange with a reservoir.

Mentions: Similar to the above explanation of slow backward buffering in the glia scheme, an extremely slow backward time scale follows naturally in diffusive coupling. For diffusion the potassium content is reduced at a time scale (48)if extracellular potassium is greater than . Backward diffusion, however, only occurs in the final recovery phase that sets in after the neuron has returned from the transient FES state and is repolarized. While is still far from the resting state level, is comparable to normal physiological conditions (see the below bifurcation diagrams in Figs. 2b and 3b) and hence the driving force during the final recovery phase is very small for a bath concentration close to the resting state level. Consequently backward diffusion is much slower than forward diffusion.


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 reduced model for  as the bifurcation parameter (purely transmembrane dynamics) showing (a) the membrane potential of fixed points (FP) and limit cycles (LC), and (b) potassium concentrations. The fixed point continuation yields the black curves. Solid sections are fully stable, dashed sections are unstable. The stability of the fixed point changes in HBs and LPs. The initial physiological condition is marked by a black square. The limit cycle is represented by the extremal values of the dynamical variables during one oscillation. The continuation yields the green lines with the same stability convention for solid and dashed sections. The stability of the limit cycle changes either in a LP or in a period–doubling bifurcation (PD). In (b) the maximal and minimal extracellular potassium concentration of the limit cycle never differs by more than  mM. The values can hence not be distinguished on the scale of this figure and therefore only the maximal value is drawn. The bifurcations are marked by full circles and labelled by the type, i.e., HB, LP or , and a counter (cf. also the insets with blow–ups, in particular the rightmost one showing  and  on a very small horizontal scale). The vertical and diagonal arrows labelled ‘m’ and ‘r’ indicate the direction of extracellular potassium changes due to ion fluxes across the membrane (‘m’) and changes only due to , i.e., because of ion exchange with a reservoir (‘r’). Note that along the horizontal directions only the ICS potassium concentration changes by a precise mixture of fluxes across the membrane and ion exchange with a reservoir.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003941-g002: Bifurcation diagram.Bifurcation diagram of the reduced model for as the bifurcation parameter (purely transmembrane dynamics) showing (a) the membrane potential of fixed points (FP) and limit cycles (LC), and (b) potassium concentrations. The fixed point continuation yields the black curves. Solid sections are fully stable, dashed sections are unstable. The stability of the fixed point changes in HBs and LPs. The initial physiological condition is marked by a black square. The limit cycle is represented by the extremal values of the dynamical variables during one oscillation. The continuation yields the green lines with the same stability convention for solid and dashed sections. The stability of the limit cycle changes either in a LP or in a period–doubling bifurcation (PD). In (b) the maximal and minimal extracellular potassium concentration of the limit cycle never differs by more than mM. The values can hence not be distinguished on the scale of this figure and therefore only the maximal value is drawn. The bifurcations are marked by full circles and labelled by the type, i.e., HB, LP or , and a counter (cf. also the insets with blow–ups, in particular the rightmost one showing and on a very small horizontal scale). The vertical and diagonal arrows labelled ‘m’ and ‘r’ indicate the direction of extracellular potassium changes due to ion fluxes across the membrane (‘m’) and changes only due to , i.e., because of ion exchange with a reservoir (‘r’). Note that along the horizontal directions only the ICS potassium concentration changes by a precise mixture of fluxes across the membrane and ion exchange with a reservoir.
Mentions: Similar to the above explanation of slow backward buffering in the glia scheme, an extremely slow backward time scale follows naturally in diffusive coupling. For diffusion the potassium content is reduced at a time scale (48)if extracellular potassium is greater than . Backward diffusion, however, only occurs in the final recovery phase that sets in after the neuron has returned from the transient FES state and is repolarized. While is still far from the resting state level, is comparable to normal physiological conditions (see the below bifurcation diagrams in Figs. 2b and 3b) and hence the driving force during the final recovery phase is very small for a bath concentration close to the resting state level. Consequently backward diffusion is much slower than forward diffusion.

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