We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674–1697, [2012]). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model.

An important feature of many physiological systems is that they evolve on multiple scales. From a mathematical point of view, these systems are modeled as singular perturbation problems. It is the interplay of the dynamics on different temporal and spatial scales that creates complicated patterns and rhythms. Many important physiological functions are linked to time-dependent changes in the forcing which leads to nonautonomous behaviour of the cells under consideration. Transient dynamics observed in models of excitability are a prime example. Recent developments in canard theory have provided a new direction for understanding these transient dynamics. The key observation is that canards are still well defined in nonautonomous multiple scales dynamical systems, while equilibria of an autonomous system do, in general, not persist in the corresponding driven, nonautonomous system. Thus canards have the potential to significantly shape the nature of solutions in nonautonomous multiple scales systems. In the context of neuronal excitability, we identify canards of folded saddle type as firing threshold manifolds. It is remarkable that dynamic information such as the temporal evolution of an external drive is encoded in the location of an invariant manifold-the canard.

Neuronal excitability in the vertebrate brain is governed by the coordinated activity of both ligand- and voltage-gated ion channels. In the cerebellum, spontaneous action potential (AP) firing of inhibitory stellate cells (SCs) is variable, typically operating within the 5- to 30-Hz frequency range. AP frequency is shaped by the activity of somatodendritic A-type K + channels and the inhibitory effect of GABAergic transmission. An added complication, however, is that whole-cell recording from SCs induces a time-dependent and sustained increase in membrane excitability making it difficult to define the full range of firing rates. Here, we show that whole-cell recording in cerebellar SCs of both male and female mice augments firing rates by reducing the membrane potential at which APs are initiated. AP threshold is lowered due to a hyperpolarizing shift in the gating behavior of voltage-gated Na + channels. Whole-cell recording also elicits a hyperpolarizing shift in the gating behavior of A-type K + channels which contributes to increased firing rates. Hodgkin–Huxley modeling and pharmacological experiments reveal that gating shifts in A-type K + channel activity do not impact AP threshold, but rather promote channel inactivation which removes restraint on the upper limit of firing rates. Taken together, our work reveals an unappreciated impact of voltage-gated Na + channels that work in coordination with A-type K + channels to regulate the firing frequency of cerebellar SCs.

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