Limits...
Six types of multistability in a neuronal model based on slow calcium current.

Malashchenko T, Shilnikov A, Cymbalyuk G - PLoS ONE (2011)

Bottom Line: We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.We described a novel mechanism supporting the bistability of bursting and silence.This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia, United States of America.

ABSTRACT

Background: Multistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified.

Methodology/principal findings: Our study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.

Conclusions: We developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.

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Related in: MedlinePlus

The one-parameter () bifurcation diagrams corresponding to levels 3 and 4 in Figure 5. The diagrams show the evolution of equilibria and oscillatory regimes for two values of the leak reversal potential:  (3) and  (4) plotted against the bifurcation parameter . Labeling is the same as in Fig. 7. Numbers 3 and 4 correspond to the dashed lines 3 and 4 in Fig. 5. Here, the blue rectangles determine the range of bistability of the bursting and hyperpolarized equilibrium. In (3) the pink rectangle marks the range of the coexistence of bursting and the stable subthreshold oscillations. The critical values for the Andronov-Hopf bifurcation (AH2) and the saddle-node bifurcation  are very close to each other and marked by the single vertical line AH2.
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pone-0021782-g008: The one-parameter () bifurcation diagrams corresponding to levels 3 and 4 in Figure 5. The diagrams show the evolution of equilibria and oscillatory regimes for two values of the leak reversal potential: (3) and (4) plotted against the bifurcation parameter . Labeling is the same as in Fig. 7. Numbers 3 and 4 correspond to the dashed lines 3 and 4 in Fig. 5. Here, the blue rectangles determine the range of bistability of the bursting and hyperpolarized equilibrium. In (3) the pink rectangle marks the range of the coexistence of bursting and the stable subthreshold oscillations. The critical values for the Andronov-Hopf bifurcation (AH2) and the saddle-node bifurcation are very close to each other and marked by the single vertical line AH2.

Mentions: The dark blue curve AH1 marks the supercritical Androvov-Hopf (A-H) bifurcation of the depolarized equilibrium. To the left of AH1, the equilibrium is stable. To the right, it becomes unstable, giving rise to stable tonic spiking. The orbit of spiking loses stability at the period doubling bifurcation, the green curve PD. Followed further, the tonic spiking periodic orbit disappears through a homoclinic bifurcation of an equilibrium marked by the red curve H1. The A-H bifurcation curve (AH2) for hyperpolarized equilibrium is shown in light blue. On this curve, the two points AHG1 and AHG2 mark the Bautin bifurcations (‘’). They bound the section of the curve where the A-H bifurcation is supercritical and gives rise to the stable subthreshold oscillations. The outer sections, above AHG1 and below AHG2, mark the subcritical A-H bifurcation, giving rise to the saddle orbit. The range where the saddle orbit exists is bounded by the homoclinic bifurcation of the saddle equilibrium, the light brown curve H2. Passage through the supercritical section leads to the onset of stable subthreshold oscillations. These oscillations vanish through a saddle-node bifurcation of periodic orbits on the dashed black curve (Fig. 6). The area supporting bursting is obtained numerically (mapped in orange, light blue, and partially in pink) and its border is marked by ‘+’s. This border is bounded by the curves PD and H2. In the pink zone there coexist bursting and stable subthreshold oscillations (Fig. 8). The bright blue patch corresponds to the bistability of bursting and silence; it is bounded by the curves AH2, H2 and H1. The yellow area between the curves AH2 and PD corresponds to the coexistence of tonic spiking and the hyperpolarized silent regime. The dotted lines indicate the four levels of used in the diagrams: −0.048 V (1, Fig. 7-1), −0.04938 V (2, Fig. 7-2), −0.04958 V (3, Fig. 8-3), and −0.0505 V (4, Fig. 8-4). The dark brown curve, , corresponds to the saddle-node bifurcation at which the hyperpolarized equilibria disappear (Figs. 7 and 8). The green ‘’ locates a point of tri-stability ( = 15.4 nS,  = −0.0502 V), illustrated in Fig. 10. The red ‘’ locates a point of bistability ( = 15.70 nS,  = −0.0505 V), shown in Fig. 11.


Six types of multistability in a neuronal model based on slow calcium current.

Malashchenko T, Shilnikov A, Cymbalyuk G - PLoS ONE (2011)

The one-parameter () bifurcation diagrams corresponding to levels 3 and 4 in Figure 5. The diagrams show the evolution of equilibria and oscillatory regimes for two values of the leak reversal potential:  (3) and  (4) plotted against the bifurcation parameter . Labeling is the same as in Fig. 7. Numbers 3 and 4 correspond to the dashed lines 3 and 4 in Fig. 5. Here, the blue rectangles determine the range of bistability of the bursting and hyperpolarized equilibrium. In (3) the pink rectangle marks the range of the coexistence of bursting and the stable subthreshold oscillations. The critical values for the Andronov-Hopf bifurcation (AH2) and the saddle-node bifurcation  are very close to each other and marked by the single vertical line AH2.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3140973&req=5

pone-0021782-g008: The one-parameter () bifurcation diagrams corresponding to levels 3 and 4 in Figure 5. The diagrams show the evolution of equilibria and oscillatory regimes for two values of the leak reversal potential: (3) and (4) plotted against the bifurcation parameter . Labeling is the same as in Fig. 7. Numbers 3 and 4 correspond to the dashed lines 3 and 4 in Fig. 5. Here, the blue rectangles determine the range of bistability of the bursting and hyperpolarized equilibrium. In (3) the pink rectangle marks the range of the coexistence of bursting and the stable subthreshold oscillations. The critical values for the Andronov-Hopf bifurcation (AH2) and the saddle-node bifurcation are very close to each other and marked by the single vertical line AH2.
Mentions: The dark blue curve AH1 marks the supercritical Androvov-Hopf (A-H) bifurcation of the depolarized equilibrium. To the left of AH1, the equilibrium is stable. To the right, it becomes unstable, giving rise to stable tonic spiking. The orbit of spiking loses stability at the period doubling bifurcation, the green curve PD. Followed further, the tonic spiking periodic orbit disappears through a homoclinic bifurcation of an equilibrium marked by the red curve H1. The A-H bifurcation curve (AH2) for hyperpolarized equilibrium is shown in light blue. On this curve, the two points AHG1 and AHG2 mark the Bautin bifurcations (‘’). They bound the section of the curve where the A-H bifurcation is supercritical and gives rise to the stable subthreshold oscillations. The outer sections, above AHG1 and below AHG2, mark the subcritical A-H bifurcation, giving rise to the saddle orbit. The range where the saddle orbit exists is bounded by the homoclinic bifurcation of the saddle equilibrium, the light brown curve H2. Passage through the supercritical section leads to the onset of stable subthreshold oscillations. These oscillations vanish through a saddle-node bifurcation of periodic orbits on the dashed black curve (Fig. 6). The area supporting bursting is obtained numerically (mapped in orange, light blue, and partially in pink) and its border is marked by ‘+’s. This border is bounded by the curves PD and H2. In the pink zone there coexist bursting and stable subthreshold oscillations (Fig. 8). The bright blue patch corresponds to the bistability of bursting and silence; it is bounded by the curves AH2, H2 and H1. The yellow area between the curves AH2 and PD corresponds to the coexistence of tonic spiking and the hyperpolarized silent regime. The dotted lines indicate the four levels of used in the diagrams: −0.048 V (1, Fig. 7-1), −0.04938 V (2, Fig. 7-2), −0.04958 V (3, Fig. 8-3), and −0.0505 V (4, Fig. 8-4). The dark brown curve, , corresponds to the saddle-node bifurcation at which the hyperpolarized equilibria disappear (Figs. 7 and 8). The green ‘’ locates a point of tri-stability ( = 15.4 nS,  = −0.0502 V), illustrated in Fig. 10. The red ‘’ locates a point of bistability ( = 15.70 nS,  = −0.0505 V), shown in Fig. 11.

Bottom Line: We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.We described a novel mechanism supporting the bistability of bursting and silence.This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia, United States of America.

ABSTRACT

Background: Multistability of oscillatory and silent regimes is a ubiquitous phenomenon exhibited by excitable systems such as neurons and cardiac cells. Multistability can play functional roles in short-term memory and maintaining posture. It seems to pose an evolutionary advantage for neurons which are part of multifunctional Central Pattern Generators to possess multistability. The mechanisms supporting multistability of bursting regimes are not well understood or classified.

Methodology/principal findings: Our study is focused on determining the bio-physical mechanisms underlying different types of co-existence of the oscillatory and silent regimes observed in a neuronal model. We develop a low-dimensional model typifying the dynamics of a single leech heart interneuron. We carry out a bifurcation analysis of the model and show that it possesses six different types of multistability of dynamical regimes. These types are the co-existence of 1) bursting and silence, 2) tonic spiking and silence, 3) tonic spiking and subthreshold oscillations, 4) bursting and subthreshold oscillations, 5) bursting, subthreshold oscillations and silence, and 6) bursting and tonic spiking. These first five types of multistability occur due to the presence of a separating regime that is either a saddle periodic orbit or a saddle equilibrium. We found that the parameter range wherein multistability is observed is limited by the parameter values at which the separating regimes emerge and terminate.

Conclusions: We developed a neuronal model which exhibits a rich variety of different types of multistability. We described a novel mechanism supporting the bistability of bursting and silence. This neuronal model provides a unique opportunity to study the dynamics of networks with neurons possessing different types of multistability.

Show MeSH
Related in: MedlinePlus