Biomedical Engineering Reference
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of the previous spike. The IP, IG and IIG models are inhomogeneous analogs of
the simple Poisson (exponential), gamma and inverse Gaussian models discussed in
Section 9.3.1. These inhomogeneous models allow the spiking activity to depend on
a temporal covariate, which, in this case, is the position of the animal as a function
of time.
The parameters for all three models, the IP, IG and the IIG can be estimated from
the spike train data by maximum likelihood [3, 8]. The log likelihoods for these two
models have the form
J
Â
j
log
L
(
q
|
N
0:
T
)=
log
p
(
u
j
|
u
j
−
1
,
q
)
(9.37)
=
1
To compute the spatial smoothing estimate of the conditional intensity function, we
proceed as in [8]. We divide the 300 cm track into 4.2 cm bins, count the number of
spikes per bin, and divide the count by the amount of time the animal spends in the
bin. We smooth the binned firing rate with a six-point Gaussian window with a stan-
dard deviation of one bin to reduce the effect of running velocity [12]. The smoothed
spatial rate function is the spatial conditional intensity estimate. The spatial smooth-
ing procedure yields a histogram-based estimate of
for a Poisson process be-
cause the estimated spatial function makes no history dependence assumption about
the spike train. The IP, IG and IIG models were fit to the spike train data by maxi-
mum likelihood whereas the spatial rate model was computed as just described. As
in Section 9.3.1 we use the KS plots to compare directly goodness-of-fit of the four
models of this hippocampal place cell spike train.
The smoothed estimate of the spatial rate function and the spatial components
of the rate functions for the IP, IG and IIG models are shown in
Figure 9.6.
The
smoothed spatial rate function most closely resembles the spatial pattern of spiking
seen in the raw data (
Figure 9.5
).
While the spiking activity of the neuron is confined
between approximately 50 and 150 cms, there is, on nearly each upward pass along
the track, spiking activity between approximately 50 to 100 cms, a window of no
spiking between 100 to 110 or 125 cms and then, a second set of more intense spiking
activity between 125 to 150 cms. These data features are manifested as a bimodal
structure in the smoothed estimate of the spatial rate function. The first mode is
10 spikes/sec and occurs at 70 cms, whereas the second mode is approximately 27
spikes/sec and occurs at approximately 120 cms. The spatial components of the IP
and IG models were identical. Because of Equation (9.34), this estimate is unimodal
and suggests a range of non-zero spiking activity which is slightly to the right of that
estimated by the smoothed spatial rate function. The mode of the IP/IG model fits is
20.5 spikes/sec and occurs at approximately 110 cms. The IIG spatial component is
also unimodal by virtue of Equation (9.34). It has its mode of 20.5 at 114 cms. This
model significantly overestimates the width of the place field as it extends from 0 to
200 cm. The scale parameter, s, is 23 cm for the IG model and 43 cm for the IIG
model.
For only the IP and the smoothed spatial rate models do the curves in Figure 9.6
represent a spatial rate function.
l
(
t
)
For the IG and IIG models the rate function, or
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