Biomedical Engineering Reference
In-Depth Information
where
λ
= 10 is the mean, as well as the variance (Ross
2007
). The histogram of
100 trials from a single Poisson neuron with a rate of 10 spikes/s is shown in
Fig.
4.3
b. To test if such a spike histogram from an unknown distribution can be
approximated by a Poisson spiking neuron, one can as a irst step check if the
ratio of the spike count variance and its mean is close to one. This ratio is called
the Fano factor (Gabbiani and Cox
2010
).
Sensory neurons change their spiking rate based on the speciic external stimuli
presented to an animal. This change in spike rate can be quantiied using a tuning
curve as shown by the grey area curve in Fig.
4.3
c. The horizontal axis is the
parameter of the stimulus, in our example the sound source direction, while the
vertical axis is the average iring rate of the neuron responding to that stimulus
parameter.
For our Poisson neuron, this would correspond to a 0
°
sound source and coin-
cides with the peak iring rate of the neuron. We model the tuning curve with
the form
2
2
−
(
ITD
−
A
sin(
wq
))/(
2
s
)
r
(
ITD
)
=
r
e
,
n
= …
1
, ,,
N
(4.9)
n
g
n
max
where
r
max
is the peak iring rate and
θ
n
is the
n
th neuron's preferred direction. For
our Poisson neuron, this would be 10 spikes/s and 0
°
respectively. The parameters
A
, ω and
σ
g
are the same as in Eq. (
4.5
); hence, Eq. (
4.9
) is proportional to Eq. (
4.5
).
We shall assume that the population of neurons responsible for sound localization is
homogeneous except that neurons have varying preferred directions,
θ
n
, as shown
by the black curves in Fig.
4.3
c.
The
N
neurons in the population have preferred direction
θ
n
sampled from the
distribution
1
2
2
−
qs
/(
2
)
p
()
q
=
e
,
(4.10)
p
2
ps
p
which is exactly the same as the prior in Eq. (
4.6
). Using the neuronal population
vector of the form
1
N
ˆ
()
∑
(4.11)
qq
=
u
()
q
k
nn
N
n
=
1
to decode the estimated sound source direction, we get results shown by the grey
crossed curve in Fig.
4.3
d. In Eq. (
4.11
),
θ
is the true sound source,
k
n
is the iring
rate from a single trial of neuron
n
with tuning function given in Eq. (
4.9
) and
N
= 400 is the number of neurons.
Notice that the curve looks strikingly similar to Fig.
4.2
d. This is no coincidence,
as our neural implementation can be shown to converge to the Bayesian estimate as
the number of neurons
N
→
∞
(Fischer and Peña
2011
). Note also that in Eq. (
4.9
),
A
sin(ω
θ
) is the mean ITD, according to Eq. (
4.4
). Thus, a simple averaging
