Parabolic bursting in an excitable system coupled with a slow oscillation
SIAM Journal on Applied Mathematics
Oscillator death in systems of coupled neural oscillators
SIAM Journal on Applied Mathematics
Analysis of neural excitability and oscillations
Methods in neuronal modeling
The simulation of large-scale neural networks
Methods in neuronal modeling
Synchronization of pulse-coupled biological oscillators
SIAM Journal on Applied Mathematics
Synchrony in excitatory neural networks
Neural Computation
Weakly connected neural networks
Weakly connected neural networks
A modular network for legged locomotion
Physica D
Phase equations for relaxation oscillators
SIAM Journal on Applied Mathematics
Neuronal Networks of the Hippocampus
Neuronal Networks of the Hippocampus
Type i membranes, phase resetting curves, and synchrony
Neural Computation
The Combined Effects of Inhibitory and Electrical Synapses in Synchrony
Neural Computation
Synaptic and intrinsic determinants of the phase resetting curve for weak coupling
Journal of Computational Neuroscience
Stability of two cluster solutions in pulse coupled networks of neural oscillators
Journal of Computational Neuroscience
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Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study.We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.