A Closed-loop Framework to Discriminate Models Using Optimal Control

Abstract

Predicting the response of an observed system to a known input is a fruitful first step to accurately control the system's dynamics. Despite the recent advances in fully data-driven algorithms, the most interpretable way to reach this goal is through mechanistic mathematical modeling. Here, we leverage optimal control and propose a closed-loop iterative method to choose among a set of candidate models the one that most accurately predict an observed system. We assume that one has control over an input of the observed system and access to measurements of its response. Our approach is to identify the input control that maximally discriminates the response of the candidate models, allowing us to determine which model is best by comparing such responses with the observed data. We demonstrate our proposed framework in numerical simulations before applying it during an electrophysiology experiment, successfully discriminating between different models for photocurrents produced via opsin dynamics.

Figures (9)

• Figure 1: Schematic illustration of the closed-loop algorithm for model discrimination. For the first iteration, the initial control input $u(t)$ is arbitrarily chosen and produces the initial measurement dataset from the reference system. One then performs the parameter fitting of both candidate models ($k=1,2$). Using the fitted parameters, one maximizes the performance measure $\mathcal{J}$ over the control input $u(t)$ . Finally, the obtained control input is used to actuate the reference system and the second iteration starts. This process continues until the models are successfully discriminated.
• Figure 2: Schematic diagram of induction and measurement of photocurrents under voltage clamp. The left portion of the figure illustrates the experimental setting including hardware and biological components, the middle portion illustrates the flow of information between the hardware and software during a closed-loop experiment, and the right portion summarizes the software used. The graph in the lower-middle portion is an example of a photocurrent, which is our observable, under the application of the box input to control the LED intensity, indicated in the top-left.
• Figure 3: Depiction of 3-, 4-, and 6-state models. Blue, red, green and gray discs represent open, closed, intermediary and desensitized states respectively. Arrows show transitions and how they are parameterized and depend on control. In particular, dashed arrows indicate light-sensitive transitions; solid arrows represent light-insensitive transitions.
• Figure 4: Box control input and parameter fitting (I), i.e. the standard procedure, compared to our procedure (II). (a) Control profiles. (b) Corresponding responses of the 3-state (green) and 4-state (orange) models reference model (dashed black), which in this case is the 3-state model with tuned parameters.
• Figure 5: (a) Evolution of mismatch with the reference trajectory (solid) and of maximal parameter increment (dotted) for the 3-state (green) and 4-state (orange) models. (b) Optimal control at the end of the procedure. (c) Corresponding responses of both models compared with the reference model (dashed black), which in this case is the 4-state model with tuned parameters.