Burst suppression is a state of profound brain inactivation that appears in several drug-induced and pathological conditions. We have formulated the problem of analyzing burst suppression as a dynamic signal processing question and presented a state-space model to characterize its temporal evolution. The observation model is a binomial process (1
) and the state equation is a Gaussian random walk model (3
). We introduced the concept of the BSP (2
) as a principled way to define the instantaneous probability of the EEG being suppressed.
We estimated the state and model parameters by modifying the approximate EM algorithm for state-space estimation for binary and point processes developed in [32
] to include a gamma prior distribution on the inverse of the state variance (21–22). Our approach allows us to estimate the BSP on a second-to-second time scale and to make formal statistical comparisons of burst suppression activity at different time points by computing confidence intervals and/or empirical Bayes posterior probabilities. We illustrated our new BSP algorithms in comparison to the BSR algorithms in one experimental and one clinical application.
Our state-space model approach offers several advantages over current methods for analyzing burst suppression. First, the state-space model provides a clear definition of the BSP as the instantaneous probability of being suppressed (2
). Second, the BSR does not have a principled method for selecting the bandwidth and degree of overlap for its filters. We used 15-s and 60-s windows to compute the BSR because these non-overlapping filters had been used in previous reports [15
]. In addition, we computed the BSR estimates with overlap to provide BSR updates that matched the 1-s updates we computed from our BSP algorithms. We further constructed both symmetric and one-sided versions of both BSR algorithms to compare directly with our BSP smoothing and filter algorithms respectively. Although BSR estimates computed in 4-s intervals have also been reported [24
], we did not show analyses with those estimates because they did not differ appreciably from the binary time-series. As we demonstrated, it is possible to compute smoother (rougher) estimates of the BSR by taking longer (shorter) computation windows and/or by not allowing (allowing) adjacent windows to overlap. A bandwidth selection procedure may offer one solution to guide these decisions [40
Third, our state-space framework addresses these issues by using the state-model to impose a temporal continuity constraint on the relation of the BSP values at nearby time points. The state-space variance
governs the degree of smoothing in the BSP estimates. Larger (smaller) values of
allow for less (more) smoothness in the BSP time course. Placing a prior distribution on
places a constraint on the degree of smoothness that can be imposed by this parameter.
Although our BSP algorithm uses an empirical Bayes’ procedure to choose
, the algorithm’s local prediction-and-correction scheme is another important feature that helps explain its good performance. Equation (13)
shows that the update xi|i
is computed based on the previous update xi−1|i−1
so that when the update interval is small, xi−1|i−1
gives a good guess of where the next state estimate is likely to be. This is the algorithm’s prediction term. The binomial innovations term, bi
, is the difference between the number of suppression events that is observed and the number that would be expected in the current observation interval based on the current estimate of pi|i
. This term is bounded between −n
. The left extreme occurs if the BSP is close to one and no suppression is observed, whereas the right extreme occurs if the BSP is close to 0 and n
suppressions are observed. These rare events provide the maximum possible innovation or local correction to the new state estimate. The closer (more distant) the observed number of suppression events is from the prediction, the less (greater) the correction that is made to xi−1|i−1
to compute xi|i
, which is the gain in the BSP filter, governs how much the innovation is weighted in computing the new update. Because
is the one-step prediction variance, the greater (less) this variance is, the greater (less) the innovation is weighted. Under our Gaussian approximation to the state, the FIS algorithm (15–17) provides an approximately optimal strategy for computing from the filter estimates state estimates that depend on all of the binary observations [32
]. These local adaptive features of the BSP algorithms, which are characteristic of Kalman filter-like algorithms [41
], are another reason that these BSP algorithms could be expected to perform better than the BSR methods that use only elementary filtering strategies. We have previously demonstrated that our binary smoothing algorithm performs better than ad hoc smoothing methods [33
], and as well as or better than more elaborate smoothing algorithms that have an automatic bandwidth selection criterion [42
Fourth, a significant benefit of our framework is the ability to use the model to make statistical inferences about the character of burst suppression being studied. This is especially important in studies such as the rat example in which key questions are measuring the second-to-second arousal effect of physostigmine on burst suppression and comparing the level of burst suppression before and after drug administration. Our state-space modeling framework provides an empirical Bayes’ estimate of the joint posterior distribution of the BSP estimates across time. Using Monte Carlo methods we can easily compute confidence statements () and posterior probabilities () for any functions of interest. For example, if in the physostigmine experiment we wanted to compare BSP in a pre-treatment interval with the BSP in a post-treatment interval, we can use the Monte Carlo algorithm to make pairwise comparisons of points chosen at random from the two intervals. The posterior probability that the BSP on the pre-treatment interval is greater than the BSP on the post-treatment interval is the fraction on the pairwise comparisons in which the pre-treatment point exceeded the post-treatment point. Because our inferences are based on an estimate of the joint posterior distribution of the state variables, we obviate the problems of multiple comparisons that are common to hypothesis-testing approaches in multivariate analyses.
We demonstrated how local 95% confidence intervals can be computed from the BSR estimates using the well-known Gaussian approximation to the binomial [32
]. Such intervals have not been previously reported. Because these confidence intervals, unlike those computed from our BSP algorithm, are local they do not use all of the data. Moreover, because they assume that the observations are independent the BSR confidence intervals understate the uncertainty in the BSR estimates. In contrast, the BSP algorithm reports wider confidence intervals because the state-space formulation models the temporal dependence in the binary time-series.
Finally, our last example shows that given estimates of the model parameters, the BSP filter algorithm could be combined with a thresholding and segmenting algorithm to track burst suppression in patients in real-time. In this situation the model parameter estimation would be conducted either off line or on a slower time scale than that for updating the BSP estimates. Possible applications of such a real-time BSP algorithm include tracking burst suppression during surgery [13
] as well as monitoring the state of medically-induced coma in patients in the intensive care unit [8
]. We were unable to compare our BSP filter algorithm directly with the BSR algorithms used in current depth-of-anesthesia monitors because they are proprietary [13
]. Our results suggest that our algorithm should compete favorably with these procedures.
In summary our state-space paradigm for the analysis of burst suppression could be applied in research and clinical analyses of this important brain state. Studies using our paradigm to track burst suppression in real-time in the operating room and in the ICU, to study the efficacy of physostigmine and other stimulants in inducing emergence from burst suppression, to track the state of the brain in postasphytic neonates, to track the state of burst suppression in patients receiving anesthetics for maintenance of medical coma [43
] will be the topics of future reports.