Biomedical Engineering Reference
In-Depth Information
9.3.3
An analysis of the spatial receptive field dynamics of a hippocam-
pal neuron
The receptive fields of neurons are dynamic in that their responses to relevant stimuli
change with experience. This plasticity, or experience-dependent change, has been
established in a number of brain regions. In the rat hippocampus the spatial receptive
fields of the CA1 pyramidal neurons evolve through time in a reliable manner as an
animal executes a behavioral task. When, as in the previous example, the experi-
mental environment is a linear track, these spatial receptive fields tend to migrate
and skew in the direction opposite the cell's preferred direction of firing relative to
the animal's movement, and increase in both maximum firing rate and scale [24, 25].
This evolution occurs even when the animal is familiar with the environment. As we
suggested in Section 9.3.2, this dynamic behavior may contribute to the failure of the
models considered there to describe the spike train data completely.
We have shown how the plasticity in neural receptive fields can be tracked on a
millisecond time-scale using point process adaptive filter algorithms [7, 13]. Central
to the derivation of those algorithms were the conditional intensity function (Equa-
tion (9.1)) and hte instantaneous log likelihood function (Equation (9.15)). We re-
view briefly the derivation of the point process adaptive filter and illustrate its ap-
plication by analyzing the spatial receptive field dynamics of a second pyramidal
neuron from the linear track experiment discussed in Section 9.3.2.
To derive our adaptive point process filter algorithm we assume that the q-dimen-
sional parameter q in the instantaneous log likelihood (Equation (9.15)) is time vary-
ing. We choose
K
large, and divide
K
,
so that there is at most one spike per interval. The adaptive parameter estimates will
be updated at
k
(
0
,
T
]
into
K
intervals of equal width
=
T
/
. A standard prescription for constructing an adaptive filter algo-
rithm to estimate a time-varying parameter is instantaneous steepest descent [17, 34]
. Such an algorithm has the form
e
∂
J
k
(
q
)
q
k
=
q
k
−
1
−
|
q
(9.38)
q
k
−
1
=
∂q
where q
k
is the estimate at time
k
,ande
is a positive learning rate parameter. If for continuous-valued observations
J
k
(
,
J
k
(
q
)
is the criterion function at
k
is
chosen to be a quadratic function of q then, it may be viewed as the instantaneous
log likelihood of a Gaussian process. By analogy, the instantaneous steepest descent
algorithm for adaptively estimating a time-varying parameter from point process ob-
servations can be constructed by substituting the instantaneous log likelihood from
Equation (9.15) for
J
k
(
q
)
q
)
in Equation (9.38). This gives
e
∂
l
k
(
q
)
q
k
=
q
k
−
1
−
|
(9.39)
q
k
−
1
q
=
∂q
which, on rearranging terms, gives the instantaneous steepest descent adaptive filter
algorithm for point process measurements
q
k
−
1
)
dN
q
k
−
1
)
.
e
∂log l
(
k
|
H
k
,
q
k
=
q
k
−
1
−
(
k
)
−
l
(
k
|
H
k
,
(9.40)
∂q
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