Biomedical Engineering Reference
In-Depth Information
analogue of the Kalman filter performs the best, because it provides an adjustable step size to update
the state, which is estimated from the covariance information. However, this method still assumes
that the posterior density of the state vector given the discrete observation is Gaussian distributed,
which is rather unlikely. A Monte Carlo sequential estimation algorithm for point processes has
been recently proposed to infer the kinematic information directly from neural spike trains [
7
].
Given the neural spike train, the posterior density of the kinematic stimulus can be estimated at
each time step without the Gaussian assumption. The preliminary simulations have shown a better
velocity reconstruction from exponentially tuned neural spike trains.
These spike-based BMI methods require preknowledge of the neuron receptive properties,
and an essential stationary assumption is used when the receptive field is estimated from a block of
data, which may not account for changes in response of the neural ensemble from open to closed loop
experiments [
8
]. All the encoding methods effectively model the PDF (often called the intensity
function) of spike firings. PDF estimation is a difficult problem that is seldom attempted because it
requires lots of data and stationary conditions. An alternative methodology that effectively bypasses
this requirement is the use of maximum likelihood methods, assuming a specific PDF. Unfortunately,
the extension of these methods to multineuron spike trains is still based on the assumption of spike
independence, which does not apply when neurons are part of neural assemblies.
6.1 adaPTIVE FIlTERINg FoR PoINT PRoCESSES wITh a
gaUSSIaN aSSUMPTIoN
One can model a point process by using a Bayesian approach to estimate the system state by evalu-
ating the posterior density of the state given the discrete observation [
6
]. This framework provides
a nonlinear time-series probabilistic model between the state and the spiking event [
10
].
Given an observation interval
(0,
N t
of events (e.g., spikes) can be mod-
eled as a stochastic inhomogeneous Poisson process characterized by its conditional intensity func-
tion
λ
( | ( ), ( ),
T
]
, the number
( )
, that is, the instantaneous rate of events, defined as
t
x
θ
Z
t
t
( ))
t
lim
Pr(
N t
(
+ − =
∆
t
)
N t
( )
1
| ( ), ( ),
t
t
Z
(
t
))
x
θ
λ
( | ( ), ( ),
t
t
θ
t
Z
( ))
t
=
x
∆
t
(6.1)
∆
t
→
0
where
Z
is the history
of all the states, parameters, and the discrete observations up to time
t
. The relationship between
the single-parameter Poisson processl
l
, the state
x
( )
t
is the system state,
θ
(
t
is the parameter of the adaptive filter, and
( )
x
(
t
)
, and the parameter
θ
( )
is a nonlinear model
t
represented by
λ
( | ( ), ( ))
t
x
t
θ
=
t
f
( ( ), ( ))
x
t
θ
t
(6.2)
