Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

We give a geometric analysis of canard solutions in three-dimensional singularly perturbed systems with a folded two-dimensional critical manifold. By analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold-curve. We classify these singularities, called canard points, as folded saddles, folded nodes, and folded saddle-nodes. We prove the existence of canard solutions in the case of the folded saddle. We show the existence of canards in the folded node case provided a generic non-resonance condition is satisfied and in a subcase of the folded saddle-node. The proof is based on the blow-up method.

We give a geometric analysis of canards of folded node type in singularly perturbed systems with twodimensional (2D) folded critical manifold using the blow-up technique. The existence of two primary canards is known provided a nonresonance condition µ / ∈ N is satisfied, where µ = λ1/λ2 denotes the ratio of the eigenvalues of the associated folded singularity of the reduced flow. We show that, due to resonances, bifurcation of secondary canards occurs. We give a detailed geometric explanation of this phenomenon using an extension of Melnikov theory to prove a transcritical bifurcation of canards for odd µ. Furthermore, we show numerically the existence of a pitchfork bifurcation for even µ and a novel turning point bifurcation close to µ ∈ N. We conclude the existence of [(µ − 1)/2] secondary canards away from the resonances. Finally, we apply our results to a network of HodgkinHuxley neurons with excitatory synaptic coupling and explain the observed slowing of the firing rate of the synchronized network due to the existence of canards of folded node type.

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Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory-the blow-up techniqueand from delayed Hopf bifurcation theory-complex time path analysis-to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/wayout function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.Crown

This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin-Huxley model, depending on parameters tau( h ), tau( n ) that scale the rates of change of the gating variables. After reducing the model to an appropriate nondimensionalized form featuring one fast and two slow variables, we use geometric singular perturbation theory to analyze the model's dynamics under systematic variation of the parameters tau( h ), tau( n ), and applied current I. As expected, we find that for fixed (tau( h ), tau( n )), the model undergoes a transition from excitable, with a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO's), featuring slow action potential generation, arise for an intermediate range of I values, if tau( h ) or tau( n ) is sufficiently large. Our analysis explains in detail the geometric mechanisms underlying these results, which depend crucially on the presence of two slow variables, and allows for the quantitative estimation of transitional parameter values, in the singular limit. In particular, we show that the subthreshold oscillations in the observed MMO patterns arise through a generalized canard phenomenon. Finally, we discuss the relation of results obtained in the singular limit to the behavior observed away from, but near, this limit.

The existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically. It is shown that near a singular periodic orbit a return map can be defined which has a one-dimensional slow manifold with a stable invariant foliation. Under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter. Additionally the existence of an invariant torus is proved for the periodically forced van der Pol oscillator. The analysis is based on methods from geometric singular perturbation theory. The blow-up method is used to analyze the dynamics near the fold-curves.

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