Mathematical Models in Neuroscience: Approaches to Experimental Design and Reliable Parameter Determination

Overparametrization of models in natural sciences, including neuroscience, is a problem that is widely recognized but often not addressed in experimental studies. The systematic reduction of complex models to simpler ones for which the parameters may be reliably estimated is based on asymptotic model reduction procedures taking into account the presence of vastly different time scales in the natural phenomena being studied. The steps of the reduction process, which are reviewed here, include basic model formulation (e.g., using the law of mass action applied routinely for problems in neuroscience, biological and chemical kinetics, and other fields), model non-dimensionalization using characteristic scales (of times, species concentrations, etc.), application of an asymptotic algorithm to produce a reduced model, and analysis of the reduced model (including suggestions for experimental design and fitting the reduced model to experimental data). In addition to the review of some classical results and basic examples, we illustrate how the approach can be used in a more complex realistic case to produce several reduced kinetic models for N-methyl-D-aspartate receptors, a subtype of glutamate receptor expressed on neurons in the brain, with models applied to different experimental protocols. Simultaneous application of the reduced models to fitting the data obtained in a series of specially designed experiments allows for a stepwise estimation of parameters of the original conventional model which is otherwise overparameterized with respect to the existing data.