Updating function discrete time dynamical system
Logical operations appear in the presence of latent state and discrete decisions, e.g.in the context of coupled models of gene regulation and signal transduction (Le Novère, 2015; Mc Adams and Shapiro, 1995).To achieve this, mechanistic mathematical models are developed which recast their essential properties (Kitano, 2002).These mathematical models—mostly ordinary differential equations (ODEs) (Klipp , 2005)—describe the temporal evolution of states of biological processes by accounting for continuous changes (e.g.Ordinary differential equation (ODE) models are frequently used to describe the dynamic behaviour of biochemical processes.Such ODE models are often extended by events to describe the effect of fast latent processes on the process dynamics.
In contrast, the interior-point algorithm in MATLAB can compute finite difference gradients using a sophisticated method for automatic step-size selection and is thereby applicable to all settings considered in this manuscript. For most applications, the minimization of the objective function is a non-convex optimization problem that possesses multiple local minima.
Of the toolboxes supporting parameter- and state-dependent trigger functions, only Sloppy Cell (Myers , 2007) allows for sensitivity analysis using symbolically derived forward sensitivity equations.
However, no existing toolbox supports sensitivity analysis for event-triggered observations which yield event-resolved data, e.g.
For the model of a spiking neuron, parameters are estimated solely from event-resolved data, in this case the time points of the after-spike resets.
In this section, we will introduce ODE models with discrete events and logical operations and formulate the respective sensitivity equations. Events are triggered at the roots of the trigger functions. The relation between elements of different subplots are indicated by arcs and arrows (Color version of this figure is available at The events must also be taken into account as they can induce jumps in the solutions to the sensitivity equations.