Limits...
Bifurcations of emergent bursting in a neuronal network.

Wu Y, Lu W, Lin W, Leng G, Feng J - PLoS ONE (2012)

Bottom Line: Here we present a general approach to mathematically tackle a complex neuronal network so that we can fully understand the underlying mechanisms.The approach enables us to uncover how emergent synchronous bursting can arise from a neuronal network which embodies known biological features.Surprisingly, the bursting mechanisms are similar to those found in other systems reported in the literature, and illustrate a generic way to exhibit emergent and multiple time scale oscillations at the membrane potential level and the firing rate level.

View Article: PubMed Central - PubMed

Affiliation: Centre for Computational Systems Biology and School of Mathematical Sciences, Fudan University, Shanghai, China.

ABSTRACT
Complex neuronal networks are an important tool to help explain paradoxical phenomena observed in biological recordings. Here we present a general approach to mathematically tackle a complex neuronal network so that we can fully understand the underlying mechanisms. Using a previously developed network model of the milk-ejection reflex in oxytocin cells, we show how we can reduce a complex model with many variables and complex network topologies to a tractable model with two variables, while retaining all key qualitative features of the original model. The approach enables us to uncover how emergent synchronous bursting can arise from a neuronal network which embodies known biological features. Surprisingly, the bursting mechanisms are similar to those found in other systems reported in the literature, and illustrate a generic way to exhibit emergent and multiple time scale oscillations at the membrane potential level and the firing rate level.

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(A) Establishment of a coordinate system on the half line  with the origin . Here,  is the equilibrium point and  is transversal to the vector field in the neighborhood of . Note that both  and  depend continuously on ; (B) Curves of the Poincaré map . Each intersection between the curves and the black line  corresponds to a fixed point of  as well as to a limit cycle of the system (8). For Hz, the curve has no intersection with the black line, so that there is no limit cycle. At higher values of , the curve moves upward; it first intersects with the black line at , where a single semi-stable limit cycle emerges. As  increases to Hz, two bifurcated limit cycles appears. Here, one cycle is stable characterized by the quantity  at one fixed point, and the other cycle is unstable with the quantity  at the other fixed point.
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pone-0038402-g006: (A) Establishment of a coordinate system on the half line with the origin . Here, is the equilibrium point and is transversal to the vector field in the neighborhood of . Note that both and depend continuously on ; (B) Curves of the Poincaré map . Each intersection between the curves and the black line corresponds to a fixed point of as well as to a limit cycle of the system (8). For Hz, the curve has no intersection with the black line, so that there is no limit cycle. At higher values of , the curve moves upward; it first intersects with the black line at , where a single semi-stable limit cycle emerges. As increases to Hz, two bifurcated limit cycles appears. Here, one cycle is stable characterized by the quantity at one fixed point, and the other cycle is unstable with the quantity at the other fixed point.

Mentions: When increasingly exceeds a critical value , the saddle-node bifurcation of limit cycles occurs. To demonstrate the existence of this bifurcation and verify the stability of the bifurcated limit cycle, we construct the Poincaré map of (8)/. Denote the equilibrium point by and setso that is a half line transversal to the vector field in the neighborhood of the equilibrium . Here we introduce a new coordinate system along , where is an origin and is a unit vector parallel to . Hence, becomes the coordinate of a point on if for some . Now, suppose that represents the solution of (8)/ with the initial point . Mathematically, it can be validated that there exists a number such that and for. In other words, is the point at which the trajectory intersects with for the first time after it departures from the initial point . Thus, the coordinate of can be uniquely determined through , and consequently the Poincaré map, denoted by , is established by for . Fig. 6B shows the curves of the constructed Poincaré map for different values of , where, clearly, each intersection between the curves and the black line is a fixed point of . When is smaller, has no fixed point for . When , it has a unique fixed point. Since the quantity at the two sides of the fixed point has different signs, this fixed point is attracting on the right side and repelling on the left. When becomes slightly larger than , two fixed points branch off: one is stable and the other is unstable. These stabilities can be derived from the sign of the above quantity at different fixed points. For example, when Hz, the quantities at the two fixed points are and respectively. Because the fixed points of correspond to limit cycles, the system (8) has a semi-stable limit cycle and the system (8) has two bifurcated limit cycles: the one with a larger amplitude is stable and the other in the interior is unstable. In the simulation, the two bifurcated limit cycles can be numerically observed (Fig. 5B).


Bifurcations of emergent bursting in a neuronal network.

Wu Y, Lu W, Lin W, Leng G, Feng J - PLoS ONE (2012)

(A) Establishment of a coordinate system on the half line  with the origin . Here,  is the equilibrium point and  is transversal to the vector field in the neighborhood of . Note that both  and  depend continuously on ; (B) Curves of the Poincaré map . Each intersection between the curves and the black line  corresponds to a fixed point of  as well as to a limit cycle of the system (8). For Hz, the curve has no intersection with the black line, so that there is no limit cycle. At higher values of , the curve moves upward; it first intersects with the black line at , where a single semi-stable limit cycle emerges. As  increases to Hz, two bifurcated limit cycles appears. Here, one cycle is stable characterized by the quantity  at one fixed point, and the other cycle is unstable with the quantity  at the other fixed point.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3369873&req=5

pone-0038402-g006: (A) Establishment of a coordinate system on the half line with the origin . Here, is the equilibrium point and is transversal to the vector field in the neighborhood of . Note that both and depend continuously on ; (B) Curves of the Poincaré map . Each intersection between the curves and the black line corresponds to a fixed point of as well as to a limit cycle of the system (8). For Hz, the curve has no intersection with the black line, so that there is no limit cycle. At higher values of , the curve moves upward; it first intersects with the black line at , where a single semi-stable limit cycle emerges. As increases to Hz, two bifurcated limit cycles appears. Here, one cycle is stable characterized by the quantity at one fixed point, and the other cycle is unstable with the quantity at the other fixed point.
Mentions: When increasingly exceeds a critical value , the saddle-node bifurcation of limit cycles occurs. To demonstrate the existence of this bifurcation and verify the stability of the bifurcated limit cycle, we construct the Poincaré map of (8)/. Denote the equilibrium point by and setso that is a half line transversal to the vector field in the neighborhood of the equilibrium . Here we introduce a new coordinate system along , where is an origin and is a unit vector parallel to . Hence, becomes the coordinate of a point on if for some . Now, suppose that represents the solution of (8)/ with the initial point . Mathematically, it can be validated that there exists a number such that and for. In other words, is the point at which the trajectory intersects with for the first time after it departures from the initial point . Thus, the coordinate of can be uniquely determined through , and consequently the Poincaré map, denoted by , is established by for . Fig. 6B shows the curves of the constructed Poincaré map for different values of , where, clearly, each intersection between the curves and the black line is a fixed point of . When is smaller, has no fixed point for . When , it has a unique fixed point. Since the quantity at the two sides of the fixed point has different signs, this fixed point is attracting on the right side and repelling on the left. When becomes slightly larger than , two fixed points branch off: one is stable and the other is unstable. These stabilities can be derived from the sign of the above quantity at different fixed points. For example, when Hz, the quantities at the two fixed points are and respectively. Because the fixed points of correspond to limit cycles, the system (8) has a semi-stable limit cycle and the system (8) has two bifurcated limit cycles: the one with a larger amplitude is stable and the other in the interior is unstable. In the simulation, the two bifurcated limit cycles can be numerically observed (Fig. 5B).

Bottom Line: Here we present a general approach to mathematically tackle a complex neuronal network so that we can fully understand the underlying mechanisms.The approach enables us to uncover how emergent synchronous bursting can arise from a neuronal network which embodies known biological features.Surprisingly, the bursting mechanisms are similar to those found in other systems reported in the literature, and illustrate a generic way to exhibit emergent and multiple time scale oscillations at the membrane potential level and the firing rate level.

View Article: PubMed Central - PubMed

Affiliation: Centre for Computational Systems Biology and School of Mathematical Sciences, Fudan University, Shanghai, China.

ABSTRACT
Complex neuronal networks are an important tool to help explain paradoxical phenomena observed in biological recordings. Here we present a general approach to mathematically tackle a complex neuronal network so that we can fully understand the underlying mechanisms. Using a previously developed network model of the milk-ejection reflex in oxytocin cells, we show how we can reduce a complex model with many variables and complex network topologies to a tractable model with two variables, while retaining all key qualitative features of the original model. The approach enables us to uncover how emergent synchronous bursting can arise from a neuronal network which embodies known biological features. Surprisingly, the bursting mechanisms are similar to those found in other systems reported in the literature, and illustrate a generic way to exhibit emergent and multiple time scale oscillations at the membrane potential level and the firing rate level.

Show MeSH