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Fig. 15 | The Journal of Mathematical Neuroscience

Fig. 15

From: Data Assimilation Methods for Neuronal State and Parameter Estimation

Fig. 15

Bursting model bifurcation diagrams and trajectories. The bifurcation diagrams (top row) depict stable fixed points (red), unstable fixed points (black), stable limit cycles (blue), and unstable limit cycles (green) of the fast subsystem \((V,n)\) with bifurcation parameter z. The gray curves are the projection of the 3-D burst trajectory (V, second row; n, third row; Ca, fourth row) onto the \((V,z)\) plane, where z is a function of Ca. (A) During the quiescent phase of the burst, Ca and therefore z are decreasing and the trajectory slowly moves leftward along the lower stable branch of fixed points until reaching the saddle-node bifurcation or “knee”, at which point spiking begins. During spiking, Ca and z are slowly increasing and the trajectory oscillates while traveling rightward until the stable limit cycle is destroyed at a homoclinic bifurcation and spiking ceases. (B) During the quiescent phase of the burst, z is decreasing and the trajectory moves leftward along the branch of stable fixed points with small-amplitude decaying oscillations until reaching the Hopf bifurcation, at which point the oscillations grow in amplitude to full spikes. During spiking, z is slowly increasing and the trajectory oscillates while traveling rightward until the stable limit cycle is destroyed at a saddle-node of periodic orbits bifurcation and spiking ceases

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