The neural basis for sleep regulation — Data assimilation from animal to model

Abstract

Sleep is important for normal brain function, and sleep disruption is comorbid with many neurological diseases. There is a growing mechanistic understanding of the neurological basis for sleep regulation that is beginning to lead to mechanistic mathematically described models. It is our objective to validate the predictive capacity of such models using data assimilation (DA) methods. If such methods are successful, and the models accurately describe enough of the mechanistic functions of the physical system, then they can be used as sophisticated observation systems to reveal both system changes and sources of dysfunction with neurological diseases and identify routes to intervene. Here we report on extensions to our initial efforts [1] at applying unscented Kalman Filter (UKF) to models of sleep regulation on three fronts: tools for multi-parameter fitting; a sophisticated observation model to apply the UKF for observations of behavioral state; and comparison with data recorded from brainstem cell groups thought to regulate sleep.

Optimization of an unscented Kalman filter for an embedded platform

“…In UKFs, the computation of the sigma points X i,k requires a square root of the matrix Pxx,k . In the prediction step, the sigma points are propagated through the nonlinear dynamics in Eq (1). The predicted mean xk+1 and predicted covariance Pxx,k+1 are found from the forward-iterated sigma points:…”

“…Next, the forward-iterated sigma points Xi,k+1 are transformed into the observation space using Eq (2): Ỹi,k+1 = g( Xi,k+1 , t k+1 ). 1 The Ỹi 's are used to find the predicted mean and covariance transformed into the observation space:…”

“…Numerical integration is used to advance the sigma points from t k to t k+1 using the dynamics in Eq (1). However, the integrator can also be used to forecast the state well beyond the next anticipated observation time by integrating the sigma points to a given forecast horizon.…”

“…Among nonlinear Kalman filters, the unscented Kalman filter (UKF) has gained widespread use due to its excellent performance and its deterministic, gradient-free implementation. Through the UKF, physiological states that are difficult or costly to measure can be estimated by the assimilation of more easily acquired or less costly observables [1]. Model states and parameters can be tracked to gain insight into physiological phenomena.…”

Using Data Assimilation for Quantitative Electroencephalography Analysis

“…In [ 9 ], the correlation between cortical dynamics and seizures was analyzed to gain a better understanding of epilepsy. In terms of using DA for data modeling, most of the work is related to signal improvement for brain-computer interfaces (BCI) [ 10 , 11 ], and to study the brain during certain conditions like sleeping [ 12 ]. Finally, for the goal of eliminating noise in brain data, some examples of how DA has been used can be found in [ 13 , 14 , 15 ].…”