An efficient approach to inverse sensitivity problems
The forward sensitivity problem for a parametrized system of differential equations seeks to determine the distribution of outputs of the system corresponding to a known distribution of inputs (parameters and initial conditions). The inverse sensitivity problem seeks the reverse, to determine the distribution of possible inputs corresponding to an observed distribution of outputs. The map between input and output spaces is deterministic and the likelihood for observing a given output provided with a specific input is a delta function. An output distributionarises, not due to the presence of stochasticity, but, for example, by taking observations from a heterogeneous population of individuals (or specimens or cells). Recently, Loos et al (2018), measured distributions of protein levels in cultures of adult sensory neurons at a sequence of fixed times after activation, rather than measurements from single neurons. These distributions suggested the presence of two distinct subpopulations of neurons. For highly parametrized systems, the inverse problem is frequently underdetermined. We formulate the underdetermined inverse sensitivity problem, review some of its known properties, and discuss recent developments for its efficient computation. We then present directions for future research.
Loos et al. 2018, Cell Systems 6, 593—603.