Version 1.1 November 2nd, 2012 -------------------------------------------------------------------------- Hidden state and parameter estimation for a network model of sleep: The included set of files reproduce figures in: Sedigh-Sarvestani, Schiff, Gluckman,'Reconstructing Mammalian Sleep Dynamics with Data Assimiliation', PLoS Comp Biol, 2012 (In press). DOI:10.1371/journal.pcbi.1002788 The data assimilation framework uses the Diniz Behn (DB) and the Fleshner, Booth, Forger, Diniz Behn (FBFD) models of sleep: DB model from: Diniz Behn and Booth, J Neurophysiol 103:1937-1953, 2010. FBFD model from:Fleshner, Booth, Forger, Diniz Behn, Philos Transact A Math Phys Eng Sci 369(1952):3855-83, 2011. Implemented by Madineh Sedigh-Sarvestani, Steven Schiff and Bruce Gluckman. Contact Madineh for questions. (m.sedigh.sarvestani@gmail.com). -------------------------------------------------------------------------- The .zip file containes a folder for each figure and a folder to generate data from the DB and FBFD model which serves as the true data as well as the noise-added 'observations' throughout the figures. In the folder for each figure is a standalone .m file which will produce a figure similar to that in the paper. Many figure files share identical sub-functions, such as the UKF function, but these have been reproduced in each figure .m file for completeness. Parameter values are stored outside of these folders to assure universality across files. CI_eps.mat contains CI_eps, the default covariance inflation multiplier (gets multiplied by variance of variable) that gets applied for all UKF steps which use default CI matrix. Note: Figures 3 4 and 6 contain reconstruction of several days worth of data at 0.5 dT seconds sampling rate and therefore take longer than other figures to run, on the order of several hours. -------------------------------------------------------------------------- Brief description of each figure (see Sedigh-Sarvestani et al. for detailed methods and interpretation of results): Figure1: generates data from DB and FBFD models of sleep and plots short (DB) and long (FBFD) cycles. The DB model of sleep contains 5 nodes, two wake_active, one NREM-active, and one REM-active. Each node is described by its firing rate and output neurotransmitter concentration. The FBFD model of sleep is similar to the DB model, with the addition of the SCN as an additional node. The SCN causes the output of this model to have 24 hour periodicity entrained to the light-dark cycle. Figure 2: Uses noise-added F-LC (firing rate of wake-active LC variable) as an observable and reconstructs the remaining 11 variables of the DB model. Panels A and B show that reconstruction can be improved as a function of UKF covariance inflation. Figure 3: Creates the empirical observability (EOC) matrix. EOC(x,y) is a metric of how well variable x is reconstructed by observation of variable y (via UKF). The EOC can be used to gain intuition regarding the partial observability properties of the DB model. It can also be used (as we show in Figure 4) to optimize covariance inflation values to improve reconstruction. Figure 4: Optimizes covariance inflation parameter for variables delta and F_R (firing rate of REM-active group). Only these variables are considered because optimization of CI for other variables does not significantly improve reconstruction performance. Once optimized values are determined, the EOC for default, optimal CI_delta, and optimal CI_delat and F_R is plotted. Improvements in reconstruction after optimization of CI can be seen by comparing these EOC matrices. Figure 5: This is the first parameter estimation figure. The UKF model contains an arbitrary value for the parameter gALC that is different from the parameter used in the model that generated the data. UKF reconstruction is carried out iteratively with parameter estimation in 30 minute long windows. The method of parameter estimation is similar to a 'shooting method' approach. In each parameter estimation step, a group of test-trajectories are projected forward where each test-trajectry is generated from a model carrying a slightly different parameter. The trajectory with the smallest distance to xhat (reconstructed UKF estimate from last iteration) determines which parameter gets passed on for the next UKF iteration. In order to prevent divergence of test-trajectories, each trajectory is initilized at xhat at 1 minute intervals. Figure 6: This is the second parameter estimation figure. This is also the first figure wherein the UKF model (modified DB) significantly differs from the model that generated the observations (FBFD in this case). The DB model is slightly modified to include an additional input parameter pSCN into the sleep-wake nodes. The FBFD model, from which observations are generated, includes an SCN node with its own firing rate and neurotransmitter that results in 24 hour periodicity in the sleep-wake pattern. Using the same shooting-method parameter esimation approach, pSCN is estimated. Thus, even though the UK filter model lacked specific components of the underlying system (SCN periodicity), it was nonetheless able to accurately estimate this rhythm. Figure 7: This figure is tied to figure 8. Figure 7 produces observations by mapping overall sleep-state of the generated data set to first and second order statistics of the firing rate variables during each sleep-state. In this way, mapped firing rates for F_LC, F_DR, F_VLPO, and F_R are generated and used as variables in the UKF to reconstruct the remaining hidden variables. Although these SOV_mapped observables clearly lack certain details as compared to direct observation of firing rates, UKF reconstruction remains reasonably accurate. Figure 8: Figure 8 uses the observations generated in figure 7 to estimate unknown parameter gALC, much in the same way as was done for figure 5.