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We consider a model for GFP expression after transfection and a model for spiking neurons and demonstrate that we can improve computational efficiency and robustness of parameter estimation by using sensitivity equations for systems with events.Moreover, we demonstrate that, by using event-outputs, it is possible to consider event-resolved data, such as time-to-event data, for parameter estimation with ODE models.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).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.For these models we evaluate the optimizer efficiency and convergence using sensitivity based gradients as well as finite difference based gradients.Ordinary differential equation (ODE) models are frequently used to describe the dynamic behaviour of biochemical processes.
For models without events, gradient based optimization schemes perform well for parameter estimation, when sensitivity equations are used for gradient computation.
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.
In this manuscript we present the governing equations for state and output sensitivities for ODE models with discrete events and logical operations.
The equations have been derived in the context of hybrid models by Barton (1998) and Rozenvasser (1967), but were neither broadly adopted nor evaluated in the systems biology community.