CONTINUOUS-TIME FILTERS FOR STATE ESTIMATION FROM POINT PROCESS MODELS OF NEURAL DATA

Abstract

Neural spike trains, the primary communication signals in the brain, can be accurately modeled as point processes. For many years, significant theoretical work has been done on the construction of exact and approximate filters for state estimation from point process observations in continuous-time. We have previously developed approximate filters for state estimation from point process observations in discrete-time and applied them in the study of neural systems. Here, we present a coherent framework for deriving continuous-time filters from their discrete-counterparts. We present an accessible derivation of the well-known unnormalized conditional density equation for state evolution, construct a new continuous-time filter based on a Gaussian approximation, and propose a method for assessing the validity of the approximation following an approach by Brockett and Clark. We apply these methods to the problem of reconstructing arm reaching movements from simulated neural spiking activity from the primary motor cortex. This work makes explicit the connections between adaptive point process filters for analyzing neural spiking activity in continuous-time, and standard continuous-time filters for state estimation from continuous and point process observations.

Figure 1

Sample arm movement with simulated spiking activity of two neurons, one in red and the other in green, overlaid. (A) (x, y) coordinates of movement trajctory, (B) x-velocity versus time, (C) y-velocity versus time.

Figure 2

Decoding Analysis. (A) True receptive field properties of each neuron, and the sum term, $\sum {\lambda }^i$. This sum term has a tuning depth that is much smaller than any individual neuron and can be well approximated by a linear function of the movement speed, suggesting that the Gaussian approximation in the filter algorithm will be good. (B) True arm movement profile (blue) and filter estimates (green) for a single arm movement. Velocity tracking remains accurate, while position estimation suffers from integrated errors.