Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales

Many models of physiological processes have the feature that one or more state variables evolve much faster than the other variables. Classic examples are neural activities such as bursting and spiking, and intracellular calcium signalling [1]. In many of these models, the time scale separation becomes apparent in the form of a small dimensionless parameter (often denoted by ε) after non-dimensionalisation of the model that brings it into a standard slow-fast form:awhere x ∈ ？k denotes the fast (dimensionless) state variables, z ∈ ？l denotes the slow (dimensionless) state variables, μ ∈ ？m denotes (dimensionless) parameters of the model, prime denotes differentiation with respect to the fast (dimensionless) time scale t, and ε ？ 1. Such a model has an equivalent representation on the slow time scale τ = εt, obtained by rescaling time and given bywhere the overdot denotes differentiation with respect to the slow time scale τ. Models with this feature are called singularly perturbed systems and one can exploit the separation of time scales in the analysis of these (k + l)-dimensional models by splitting the system into the k-dimensional fast subsystem obtained in the singular limit ε → 0 of (1) and known as the layer problem, and the one-dimensional slow subsystem obtained in the singular limit ε → 0 of (2) and known as the reduced problem. The aim is to make predictions about the dynamics in the full model based on what is seen in the lower-dimensional fast and slow subsystems. Geometric singular perturbation theory (GSPT) [2-9] forms the mathematical foundation behind this approach and it is a well-established tool in the analysis of many multiple time scales problems in the biosciences (see, e.g., [1,10-12]).Perhaps the best-known instance of the use of GSPT in this way is the analysis of the famous Hodgkin-Huxley (HH) model of the (space-clamped) squid giant axon [13] by FitzHugh [14,15]. The HH model is a four-dimensional conductance-based model in which two stat