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State-Space Analysis of Time-Varying Higher-Order Spike Correlation for Multiple Neural Spike Train Data
Figure 1
Geometric view of recursive filtering in subspace 
.
Each point in this figure represents a probability distribution, 
, of an 
-tuple binary variable, 
. The underlying time-dependent model is represented by white circles in the space of 
. The dashed lines indicate projections of the underlying models to the model subspace, 
. The maximum a posteriori (MAP) estimates of the underlying models projected on subspace 
were obtained recursively: Starting from the MAP estimate at time 
(filter estimate, red circle), the model at time 
is predicted based on the prior knowledge of the state transition, Eq. 8 (blue arrow, prediction; black cross, a predicted distribution). The maximum likelihood estimate (MLE, black circle) for the spike data at time 
derived by Eq. 4 is expected to appear near the projection point of the underlying model at time 
in 
. The filter distribution at time 
is obtained by correcting the prediction by the observation of data at time 
(black arrow). The filter estimation at time 
is used for predicting the model at time 
and so on. This recursive procedure allows us to retain past information while tracking the underlying time-dependent model based on the current observation.
doi: https://doi.org/10.1371/journal.pcbi.1002385.g001