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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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The bifurcation behaviors in the mean field model.(A) Bifurcation diagram with  and with the variation of . Here, the asymptotical dynamics of the -component are taken into account. The black line and the dash line represent the stable and the unstable fixed points, respectively. For each , the blue and the red dots represent the eventually upper-and-lower boundaries of the stable and the unstable limit cycles in the -component. (B) The trajectories of the system (8) when  Hz and  (see also the phase orbit in Fig. 5F). The sharp peaks in the left plot and the sharp valleys in the right plot reflect the characteristics of the slow-fast dynamical system. (C) The bifurcation transition regulated by the input rate  with . The inner plot indicates the dynamics of the input rate with respect to time. We set Hz for (in seconds),  Hz for ,  Hz for  and  Hz for . (D) Network bursting dynamics in: (blue line) the SNN composed of 48 neurons and 12 dendrites. (red line) the ‘network’ replicated from the traces of voltage and store level in the mean field model with . Note that bursting events are recorded if the firing rate is >30 Hz in the SNN, while the burst frequency in the mean field model is the reciprocal of the period of the stable limit cycle.
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pone-0038402-g007: The bifurcation behaviors in the mean field model.(A) Bifurcation diagram with and with the variation of . Here, the asymptotical dynamics of the -component are taken into account. The black line and the dash line represent the stable and the unstable fixed points, respectively. For each , the blue and the red dots represent the eventually upper-and-lower boundaries of the stable and the unstable limit cycles in the -component. (B) The trajectories of the system (8) when Hz and (see also the phase orbit in Fig. 5F). The sharp peaks in the left plot and the sharp valleys in the right plot reflect the characteristics of the slow-fast dynamical system. (C) The bifurcation transition regulated by the input rate with . The inner plot indicates the dynamics of the input rate with respect to time. We set Hz for (in seconds), Hz for , Hz for and Hz for . (D) Network bursting dynamics in: (blue line) the SNN composed of 48 neurons and 12 dendrites. (red line) the ‘network’ replicated from the traces of voltage and store level in the mean field model with . Note that bursting events are recorded if the firing rate is >30 Hz in the SNN, while the burst frequency in the mean field model is the reciprocal of the period of the stable limit cycle.

Mentions: Interestingly, aside from the above two bifurcations, the other two bifurcations appear almost symmetrically and consecutively. When passes through Hz, the other subcritical Hopf bifurcation of the system (8) emerges with a positive FLC, (0.6262), which changes the stability of the originally-unstable equilibrium and brings an unstable limit cycle (Fig. 5F–G). Moreover, the amplitude of the bifurcated unstable limit cycle grows until increasingly approaches Hz, where the other saddle-node bifurcation occurs. This bifurcation leads to the coalescence and annihilation of the two limit cycles (Fig. 5H–I). The above-expatiated bifurcation procedure of the system (8) is illustrated in Fig. 7A.


Bifurcations of emergent bursting in a neuronal network.

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

The bifurcation behaviors in the mean field model.(A) Bifurcation diagram with  and with the variation of . Here, the asymptotical dynamics of the -component are taken into account. The black line and the dash line represent the stable and the unstable fixed points, respectively. For each , the blue and the red dots represent the eventually upper-and-lower boundaries of the stable and the unstable limit cycles in the -component. (B) The trajectories of the system (8) when  Hz and  (see also the phase orbit in Fig. 5F). The sharp peaks in the left plot and the sharp valleys in the right plot reflect the characteristics of the slow-fast dynamical system. (C) The bifurcation transition regulated by the input rate  with . The inner plot indicates the dynamics of the input rate with respect to time. We set Hz for (in seconds),  Hz for ,  Hz for  and  Hz for . (D) Network bursting dynamics in: (blue line) the SNN composed of 48 neurons and 12 dendrites. (red line) the ‘network’ replicated from the traces of voltage and store level in the mean field model with . Note that bursting events are recorded if the firing rate is >30 Hz in the SNN, while the burst frequency in the mean field model is the reciprocal of the period of the stable limit cycle.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3369873&req=5

pone-0038402-g007: The bifurcation behaviors in the mean field model.(A) Bifurcation diagram with and with the variation of . Here, the asymptotical dynamics of the -component are taken into account. The black line and the dash line represent the stable and the unstable fixed points, respectively. For each , the blue and the red dots represent the eventually upper-and-lower boundaries of the stable and the unstable limit cycles in the -component. (B) The trajectories of the system (8) when Hz and (see also the phase orbit in Fig. 5F). The sharp peaks in the left plot and the sharp valleys in the right plot reflect the characteristics of the slow-fast dynamical system. (C) The bifurcation transition regulated by the input rate with . The inner plot indicates the dynamics of the input rate with respect to time. We set Hz for (in seconds), Hz for , Hz for and Hz for . (D) Network bursting dynamics in: (blue line) the SNN composed of 48 neurons and 12 dendrites. (red line) the ‘network’ replicated from the traces of voltage and store level in the mean field model with . Note that bursting events are recorded if the firing rate is >30 Hz in the SNN, while the burst frequency in the mean field model is the reciprocal of the period of the stable limit cycle.
Mentions: Interestingly, aside from the above two bifurcations, the other two bifurcations appear almost symmetrically and consecutively. When passes through Hz, the other subcritical Hopf bifurcation of the system (8) emerges with a positive FLC, (0.6262), which changes the stability of the originally-unstable equilibrium and brings an unstable limit cycle (Fig. 5F–G). Moreover, the amplitude of the bifurcated unstable limit cycle grows until increasingly approaches Hz, where the other saddle-node bifurcation occurs. This bifurcation leads to the coalescence and annihilation of the two limit cycles (Fig. 5H–I). The above-expatiated bifurcation procedure of the system (8) is illustrated in Fig. 7A.

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