Supplementary Information for Mapping global sensitivity of cellular network dynamics: sensitivity heat maps and a global summation law.
David A. J. Rand · 2007
This is supplementary information for the paper Mapping global sensitivity of cellular network dynamics: sensitivity heat maps and a global summation law. by D. A. Rand. The latter paper is referred to henceforth as I. Suppose that the differential equation being considered is written as ẋ = f(t, x, k) where x ∈ R and the the set of parameters k1, . . . , ks is collected together into a parameter vector k ∈ R. The systems may depend upon other parameters but for the discussion here we assume that these other parameters are held fixed and only k1, . . . , ks are varied. As in I we introduce scaled parameters ηi = log ki. Suppose that the solutions of interest, which depends upon the parameters k is given by x(t) = g(t, k). Here we will concentrate on two types of dynamic solutions: periodic oscillations (i.e. limit cycles) and transcient signals. By the latter we mean solutions x(t) = g(t, k) of the equation ẋ = f(t, x, k) with a given initial condition x0 = g(0, k). The system starts in a given state x0 and is subject to a given perturbation caused by an incoming signal that is modeled in the time dependence of the right-hand side f(t, x, k) of the differential equation. Suppose that we denote the solution of the differential equation with initial condition x and parameters k by ξ(t, x, k). To determine the derivatives ∂ξ/∂xi and ∂ξ/∂kj we consider the variational equation ∂