Noname manuscript No. (will be inserted by the editor) Jonathan Rubin · Martin Wechselberger Giant Squid - Hidden Canard: the 3D Geometry of the Hodgkin-Huxley Model the date of receipt and acceptance should be inserted later Abstract This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin- Huxley model, depending on parameters τ h ,τ 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 varia- tion of the parameters τ h ,τ n , and applied current I . As expected, we find that for fixed (τ h ,τ 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 τ h or τ 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. 1 Introduction The Hodgkin-Huxley (HH) model (Hodgkin and Huxley 1952) for the action potential of the space- clamped squid giant axon is defined by the following 4D vector field: C dV dt = I − I Na − I K − I L dm dt = φ[α m (V )(1 − m) − β m (V ) m] (1.1) dh dt = φ[α h (V )(1 − h) − β h (V ) h] dn dt = φ[α n (V )(1 − n) − β n (V ) n] . We use modern conventions such that the spikes of action potentials are positive, and the voltage ¯ V of the original HH model (Hodgkin and Huxley 1952) is shifted relative to the voltage V of this model by ¯ V =(V + 65). The first equation is obtained by applying Kirchhoff’s law to the space-clamped neuron, i.e. the transmembrane current is equal to the sum of intrinsic currents. C is the capacitance density in μF/cm 2 , V is the membrane potential in mV and t is the time in ms. The ionic currents on the right hand side are given by I Na = g na m 3 h(V − E Na ) , I K = g k n 4 (V − E K ) , I L = g l (V − E L ) (1.2) Department of Mathematics and Center for the Neural Basis of Cognition, University of Pittsburgh, PA, USA; email: rubin@math.pitt.edu, phone: 412-624-6157, fax: 412-624-8397 School of Mathematics and Statistics, University of Sydney, NSW, Australia Address(es) of author(s) should be given