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The relationship between two fast/slow analysis techniques for bursting oscillations.

ABSTRACT: Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the two-fast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow.

SUBMITTER: Teka W

PROVIDER: S-EPMC3523400 | biostudies-other | 2012 Dec

REPOSITORIES: biostudies-other

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Project description:Experimental and theoretical studies demonstrate that neuronal gamma oscillations crucially depend on interneurons, but current models do not consider the diversity of known interneuron subtypes. Moreover, in CA1 of the hippocampus, experimental evidence indicates the presence of multiple gamma oscillators, two of which may be coordinated by differing interneuron populations. In this article, we show that models of networks with competing interneuron populations with different postsynaptic effects are sufficient to generate, within CA1, distinct oscillatory regimes. We find that strong mutual inhibition between the interneuron populations permits distinct fast and slow gamma states, whereas weak mutual inhibition generates mixed gamma states. We develop idealized firing rate models to illuminate dynamic properties of these competitive gamma networks, and reinforce these concepts with basic spiking models. The models make several explicit predictions about gamma oscillators in CA1. Specifically, interneurons of different subtype phase-lock to different gamma states, and one population of interneurons is silenced and the other active during fast and slow gamma events. Finally, mutual inhibition between interneuron populations is necessary to generate distinct gamma states. Previous experimental studies indicate that fast and slow gamma oscillations reflect different information processing modes, although it is unclear whether these rhythms are intrinsic or imposed. The models outlined demonstrate that basic architectures can locally generate these oscillations, as well as capture other features of fast and slow gamma, including theta-phase preference and spontaneous transitions between gamma states. These models may extend to describe general dynamics in networks with diverse interneuron populations.NEW & NOTEWORTHY The oscillatory coordination of neural signals is crucial to healthy brain function. We have developed an idealized neuronal model that generates distinct fast and slow gamma oscillations, a known feature of the rodent hippocampus. Our work provides a mechanism of this phenomenon, as well as a theoretical framework for future experiments concerning hippocampal gamma. It moreover offers a tractable model of competitive gamma oscillations that is generalizable across the nervous system.

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The relationship between two fast/slow analysis techniques for bursting oscillations.

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