Biomedical Engineering Reference
In-Depth Information
1
/t
if
−
t/
2
≤
t
≤
t/
2
w(t)
=
.
0
Otherwise
If the slide is continuous rather than discrete then we can rewrite the frequency estimate as
f(t)
=
w(τ)s(t
−
τ)dτ
(9.1)
where the window,
w
remains stationary and the spike train is moved. When presented in this way, the
window is sometimes called a
kernel
.
It is also possible to use windows with different shapes that will weight some spikes more than
others. For example, the Gaussian kernel is defined as
exp
τ
2
2
σ
w
1
√
2
πσ
w
=
−
w(t)
(9.2)
and so spikes closer to the evaluation time are more heavily weighted in the frequency estimate.
Thus far we have assumed that the entire spike train is known, and so times both before and after
the evaluation time may be used. These kernels are called
noncausal
because they involve spikes that
will occur after the evaluation time. In any real-time application, only spikes that occurred before the
evaluation time are known and therefore a
causal
window must be used. In principle, the same window
shapes can be used, but instead of evaluating at the center of the kernel the evaluation is performed at
the right edge.
9.3 TUNINGCURVES
While Sec. 8.4 described how an extracellular stimulus may induce neural firing, the previous sections
of this chapter described how to characterize a neural response. The combination of an controlled input
(stimulation) and recorded output (neural firing rate) enables a transfer function to be created that
characterizes a network of neurons. This type of transfer function is typically called a
tuning curve
and is
represented graphically as a plot of a stimulus parameter versus firing rate.The stimulus parameter can be
any number of possible input parameters. As there are too many possible types of stimuli we will broadly
classify them as either direct electrical stimulation or indirect stimulation. Direct electrical stimulation
can take the form of cortical or sub-cortical stimulation through electrodes or even magnetic stimulation
(see Sec. 10.3.2). The stimulus parameters that may be varied are the stimulus strength, duration, or
pacing rate. Indirect stimulation may take the form of light, smell, taste, touch, sound, hormones, or
drugs injected into the blood stream.
Once the empirical data are collected, it is often the case that an analytic function, called a
tuning
function
, is fit to the data. There are many mathematical forms for tuning functions but two deserve
special mention here. First are a group of Gaussian-like curve that have a maximum firing rate at some
particular value of the stimulus parameter with a lesser response for deviations from that value. Networks
of neurons that have a Gaussian tuning curve have a sensitivity to a particular range of the stimulus.
Second are a group of sigmoid-like functions that do not respond for some large range of stimuli, then
rapidly transition to a strong response as the stimulus is changed. In other words, networks of neurons
that have a sigmoidal tuning curve exhibits threshold behavior.
