Abstract
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.
Highlights
Excitable neurons are typically at rest, but can fire action potentials in response to certain forms of stimulation
These models are amenable to analysis using geometric singular perturbation theory (GSPT) [9, 14] with the specific aim of giving predictions of model dynamics based on singular limit observations
Complex pattern generation in such slow/fast systems is almost exclusively related to loss of normal hyperbolicity of invariant critical manifolds, which is associated with bifurcation sets in the fast subsystem
Summary
Excitable neurons are typically at rest, but can fire action potentials in response to certain forms of stimulation. The interaction of ionic currents acting on different timescales is responsible for the creation of action potentials in neurons This time-scale feature leads to mathematical models of neurons often referred to as singular perturbation problems. Under variation of a particular parameter, the position of a folded saddle canard varies explaining the excitability properties of the model This geometric observation becomes important when we try to understand changes in neural dynamics. The famous Hodgkin–Huxley model of the squid giant axon [12] incorporates temperature changes through a Q10-temperature factor that increases the speed of the ion channel gates with increasing temperature If such environmental changes are encoded in different speeds of slow ion channels, we expect this to be reflected in a change of the firing threshold manifold of the neuron. Identifying the cause of different neural dynamics through the specific position of a canard in a singularly perturbed system becomes a valuable diagnostic tool to understand this phenomenon, and it is the focus of this work
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