Neural Decoding Using Generative BMI Models - Brain-Machine Interface Engineering - page 147

Kalman filter). The most general case is called the particle filter where the system and observation
models can be nonlinear and there are no modeling constrains imposed on the posterior density.
The particle filter became very popular recently with the availability of fast processors and efficient
algorithms, such as Monte Carlo integration, where a set of samples drawn from the posterior dis-
tribution of the model parameters is used to approximate the integrals by sums [ 25 ]. Alternatively,
the sum of Gaussian models can be used, or more principally graphical models, which take advan-
tage of known dependencies in the data to simplify the estimation of the posterior density.

5.3 KalMaN FIlTER
Our discussion of generative models will begin with the most basic of the Bayesian approaches: the
Kalman filter. The Kalman filter has been applied in BMI experimental paradigms by several groups
[ 7 , 16 , 27-29 ]. This approach assumes a linear relationship between hand motion states and neural
firing rates (i.e., continuous observations obtained by counting spikes in 100-msec windows), as
well as Gaussian noise in the observed firing activity. The Kalman formulation attempts to estimate
the state, x ( t ), of a linear dynamical system as shown in Figure 5.3 . For BMI applications, we define
the states as the hand position, velocity, and acceleration, which are governed by a linear dynamical
equation as shown in ( 5.12 )

T

(5.12)

x ( ) t = [ HP ( ) t HV ( ) t HA ( )] t

where hP , hV , and ha are the hand position, velocity, and acceleration vectors, 1 respectively. The
Kalman formulation consists of a generative model for the data specified by a linear dynamic equa-
tion for the state in ( 5.13 )

(5.13)

x ( t + 1 = ) Ax ( ) t + u ( ) t

where u ( t ) is assumed to be a zero-mean Gaussian noise term with covariance U . The observation
model also called the output mapping (from state to spike trains) for this BMI linear system is
simply

(5.14)

z ( ) t = Cx ( ) t + v ( ) t

where v ( t ) is the zero-mean Gaussian measurement noise with covariance V and z is a vector con-
sisting of the neuron firing patterns binned in nonoverlapping (100 msec) windows. In this specific
formulation, the output-mapping matrix C has dimension N ´9 . Alternatively, we could have

1 The state vector is of dimension 9 + N ; each kinematic variable contains an x , y , and z component plus the dimen-
sionality of the neural ensemble.