We investigate the interaction of an excitable system with a slow oscillation. Under robust and general assumptions compatible with the more stringent assumptions usually made about excitable systems, we show that such a coupled system can display bursting, i.e. a stable solution in which some variable undergoes rapid oscillations followed by a period of quiescence, with both oscillation and quiescence continually repeated. Under a further weak condition, the bursting is "parabolic", i.e. the local frequency of the fast oscillation increases and then decreases within a burst. The technique in this paper involves nonlinear changes of coordinates which transform the equations into ones which are closely related to Hill's equation.

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Abstract. Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) have been observed both experimentally and numerically in various prototypical systems in the natural sciences. In the present article, we propose a mathematical model problem which, though analytically simple, exhibits a wide variety of MMO patterns upon variation of a control parameter. One characteristic feature of our model is the presence of three distinct time-scales, provided a singular perturbation parameter is sufficiently small. Using geometric singular perturbation theory and geometric desingularization, we show that the emergence of MMOs in this context is caused by an underlying canard phenomenon. We derive asymptotic formulae for the return map induced by the corresponding flow, which allows us to obtain precise results on the bifurcation (Farey) sequences of the resulting MMO periodic orbits. We prove that the structure of these sequences is determined by the presence of secondary canards. Finally, we perform numerical simulations that show good quantitative agreement with the asymptotics in the relevant parameter regime.

Hippocampal networks of excitatory and inhibitory neurons that produce ␥-frequency rhythms display behavior in which the inhibitory cells produce spike doublets when there is strong stimulation at separated sites. It has been suggested that the doublets play a key role in the ability to synchronize over a distance. Here we analyze the mechanisms by which timing in the spike doublet can affect the synchronization process. The analysis describes two independent effects: one comes from the timing of excitation from separated local circuits to an inhibitory cell, and the other comes from the timing of inhibition from separated local circuits to an excitatory cell. We show that a network with both of these effects has different synchronization properties than a network with either excitatory or inhibitory type of coupling alone, and we give a rationale for the shorter space scales associated with inhibitory interactions.When neurons communicate over some distance, there are conduction delays between the firing of the presynaptic neuron and the receipt of the signal at the postsynaptic cell. It is also known that cells can synchronize over distances of at least several millimeters, over which conduction delays can be significant. This raises the question of how cells can synchronize in spite of the delays. Traub et al. (1) and Whittington et al. (2) suggested that the fine structure of the spiking of some of the cells may play a part in the synchronization process for the ␥ frequency rhythm, found in hippocampal and neocortical systems during states of sensory stimulation. (For references, see ref. 2.) More specifically, for some models of cortical structure, they noted that the ability to synchronize in the presence of delays is correlated with the appearance of spike doublets in the inhibitory cells. The doublets appear in slice preparations when there is strong stimulation at separated sites (1, 2). In this paper, we analyze a mechanism for such synchronization, using a simplified version of equations of Traub and colleagues.The timing of spikes within a doublet is shown to encode information about phases of local circuits in a previous cycle; the model shows how the circuit can use this information in an automatic way to bring nonsynchronous local circuits closer to synchrony. There are two independent effects in the model. The first is the response of the inhibitory (I) cells to excitation from more than one local circuit. The I-cells may produce more than one spike, whose relative timing depends on strength of excitation and recovery properties of the cell after the firing of a first spike; the latter can include effects of afterhyperpolarization or self-inhibition in a local circuit. The second effect is the response of the excitatory (E) cells to the multiple inhibitory spikes they receive from within their local circuit or other circuits. The maximal inhibition received by an E-cell can depend on the times and sizes of the inhibitory postsynaptic potentials it receives, and this affects the time un...