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Furthermore, we introduce an objective function that allows the consideration of event-triggered observations through mechanistic models.All relevant methods are implemented in the open-source software toolbox Advanced Matlab Interface for CVODES and IDAS (AMICI, originally presented by Fröhlich , 2005) and automatically handles events, which are not natively supported by CVODES or IDAS.The corresponding nonlinear optimization problems can be solved using local and global optimization schemes (Egea , 2013).For ODE models with events for which no sensitivity equations are available, numerical differentiation has to be employed to assess the gradient of the objective function with respect to the parameters. Beyond gradient computation, sensitivity equations can be used to gain insight into model properties (Dai , 2009).By providing a user-friendly, modular implementation in the toolbox AMICI, the developed methods are made publicly available and can be integrated in other systems biology toolboxes.
For models without events, gradient based optimization schemes perform well for parameter estimation, when sensitivity equations are used for gradient computation.
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.
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.
The methods and their implementation are evaluated using two examples: A model for GFP expression after transfection which includes the instantaneous release of m RNA molecules and a model for a spiking neuron, in which the after-spike reset of the membrane potential is instantaneous.