Model-order reduction of ion channel dynamics using approximate bisimulation

We show that in the context of the Iyer et al. (IMW) 67-variable cardiac myocycte model, it is possible to replace the detailed 13-state probabilistic subsystem of the sodium channel dynamics with a much simpler Hodgkin-Huxley (HH)-like two-state abstraction, while only incurring a bounded approximation error. We then extend our technique to the 10-state subsystem of the fast-recovering calcium-independent potassium channel. The basis of our results is the construction of an approximate bisimulation between the HH-type abstraction and the corresponding detailed ion channel model, both of which are input-controlled (voltage in this case) CTMCs. The construction of the appropriate approximate bisimulation, as well as the overall result regarding the behavior of this modified IMW model, involves: (1) Identification of the voltage-dependent parameters of the m and h gates in the HH-type models via a two-step fitting process, carried out over more than 20,000 representative observational traces of the detailed sodium and potassium ion channel models. (2) Proving that the distance between observations of the detailed models and their respective abstraction is bounded. (3) Exploring the sensitivity of the overall IMW model to the HH-type approximations. Our extensive simulation results experimentally validate our findings, for varying IMW-type input stimuli.