Prapun B-Exam

Introduction
Sources of Variability and Threshold Derivation
Information-Theoretic Analysis of IF Neuron
Conclusion

Capacity Analysis of Neurons with Descending
Action Potential Thresholds

Prapun Suksompong

Electrical and Computer Engineering
Cornell University, Ithaca, NY 14853
ps92@cornell.edu

Final Examination for the Doctoral Degree (“B” Exam)
July 24, 2008

Prapun Suksompong Capacity Analysis of Neurons

Introduction
Conclusion

Outline

Introduction

Sources of Variability for the ISIs and Derivation of the Threshold


Conclusion


Introduction
Neuron Morphology
Integrate-and-Fire Neurons
Goal
Conclusion

Introduction
Neuron Morphology
Goal



Conclusion


Introduction
Neuron Morphology
Goal
Conclusion

Neuron Morphology

A neuron is the basic working unit of the nervous system.
Dendrite
A typical neuron has three Nucleus
Axon Hillock

functionally distinct parts, called Axon

dendrites, Axon
Axon from Terminals
soma, and another neuron Cell Body
(Soma)
Synapse
axon. Myelin
Sheath
Node of
Ranvier

The junction between two Synaptic Vesicle
Presynaptic
neurons is called a synapse. Axom
Terminal
Synaptic Cleft
Postsynaptic
Dendrite
Ion Channel


Introduction
Neuron Morphology
Goal
Conclusion

Action Potentials (Spikes)

Looking at a synapse, we refer to the sending neuron as the
presynaptic neuron and to the receiving neuron as the
postsynaptic neuron.

presynaptic postsynaptic

synapse
axon

The neuronal signals consist of short electrical pulses called action
potentials (APs) or spikes. A chain of APs emitted by a single
neuron is called a spike train.


Introduction
Neuron Morphology
Goal
Conclusion

Quantal Synaptic Failure (QSF)

presynaptic postsynaptic

synapse
axon

Synaptic failure: It is possible that an AP fails to get
“across” the synapse.
We may model a synapse as a Z -channel.
Spikes which successfully cross the synapse then propagate
down to soma.


Introduction
Neuron Morphology
Goal
Conclusion


Assumption: ∼ 104
pre-synaptic neurons.
True in cortex (higher brain
functions).


Introduction which are drawn here are in fact decreasing. We will return to t
Neuron Morphology
Goal
Conclusion


Descending Threshold

Ascending
Membrane
Potential

time
Spike Train
Spikes generated when the
membrane potentials hit the
time
thresholds.
• First jitter: Spike generation
Descending thresholds. • Poisson approximation


Introduction
Neuron Morphology
Goal
Conclusion

(Leaky) Integrate-and-Fire Model: LIF or IF

Let τ1 , τ2 , τ3 , . . . be the sequence of time that the spikes arrive at
the spike generating region. The membrane potential at time t is
then
X (t) = h (t, τm , Ym ) = Ym h(t − τm ).
m m

This is the “integrate” part of the integrate-and-ﬁre neuron.

X (t )
Ym is the weight for the mth
spike due to propagation
loss, synaptic strength,
synaptic failure, etc. Yi + 2 h ( t − τ i + 2 )
Yi h ( t − τ i ) Yi +1h ( t − τ i +1 )
h is the shape function.
τi τ i +1 τ i+2 time


Introduction
Neuron Morphology
Goal
Conclusion

Integrate-and-Fire Neurons (Con’t)

Constant bombardment of spikes
leads to increase in membrane
potential.
As soon as the membrane potential
reaches a critical value or
threshold, the neuron “ﬁres” an time (t)
action potential. Then, everything
resets.
Refractory period: The time after
a AP is produced, during which it
is impossible to generate another
Threshold
AP. time (t)
Set T (t) to be ∞ during this
period.

Introduction
Neuron Morphology
Goal
Conclusion

Integrate-and-Fire Neurons (Summary)

∼ 104 pre-synaptic neurons. Descending Threshold

Ascending
Membrane
Potential

time
Spike Train

time

• First jitter: Spike generation
Descending thresholds.
• Poisson approximation


Introduction
Neuron Morphology
Goal
Conclusion

Theoretical Approaches to Neuroscience

We use IF model, but more biologically-realistic models exist
(e.g. Hodgkin and Huxley [’52] model).
1. Too many parameters.
Physical measurements “fundamentally disturb cell properties”
2. Provide less insight.
Biological structures have evolved via natural selection to
operate optimally.
See the book Optima for Animals by R. McNeill Alexander.
What is the best strength for a bone?
At what speed should humans change from walking to
running?


Introduction
Neuron Morphology
Goal
Conclusion

Information-Theoretic Optimization

Application of information theory has already found success in
many areas of neuroscience.
Barlow’s “economy of impulses”[’59, ’69]
Minimize redundancy.
Linsker’s InfoMax principle [’88, ’89]
Maximize the mutual information.
Levy and Baxter’s energy-eﬃcient coding [’96, ’02]
Maximize mutual information per unit energy expended.


Introduction
Neuron Morphology
Goal
Conclusion

Motivation and Goal

Integrate-and-Fire (IF) model is very popular.
The threshold function is a crucial element of the IF model.
Little amount of work exists on deriving the form of the
threshold curve.
In fact, using constant thresholding is also popular.
This leads to large jitter in the spike timing and hence
discourages the use of time coding.
Goal: Find (1) an expression for threshold curve under biologically
realistic constraints and (2) the optimal operating point of neuron
under such threshold.


Introduction
First Jitter: Spike generation
Second and Third Jitters
The Threshold Curve
Conclusion

Introduction

The Threshold Curve


Conclusion


Introduction
The Threshold Curve
Conclusion

Three sources of variability for inter-spike intervals

1. Spike generation
2. Spike propagation
3. Time-of-arrival estimation


Introduction
The Threshold Curve
Conclusion

First jitter: Spike generation
formula that governs how the membrane potential of this middle neuron rises given the
combined incoming rate λ .

λ1

λ2

⊕
λ3

λ = λ1 + λ2 + λ3 +

Now, of course, there is some jitter in the timing of the
paper today.
Large number of presynaptic neurons source of jitter comes from the fact that the in
The first allows Poisson
approximation for the superposed in fact some jitter in large. That assumption allows
has process.
presynaptic neurons are
them. Now, you may recall th

The membrane potential is governed by a ﬁltered Poisson to a Poisson p
.. the superposed spike trains … is close
process. spike train coming out of a single neuron is not a Poiss
a tractable formula that governs how the membrane po
Prapun Suksompong given the combined incoming rate λ .
Capacity Analysis of Neurons

Introduction
The Threshold Curve
Conclusion

Approximation for the first timing jitter

For fixed λ, different realizations of the membrane potential
correspond to different spiking times.

Membrane potential X ( t ) Filtered Poisson
σX approximation for
amount of variation
in vertical direction
[Parzen’62].
Linear approximation
Threshold T ( t ) for amount of
σ time variation in horizontal
Time
direction.


Introduction
The Threshold Curve
Conclusion
Membrane potential X ( t )
σX

Approximation for the ﬁrst timing jitter (con’t)
Threshold T ( t )
σ time
Time

H (τ ) T (τ ) c2 H2 (τ )
σtime (τ ) ≈ The figure here shows different realizations of the membrane potentials for a fixed
combined incoming rate λ . Here, we see that the randomness from the Poisson arrivals
T (τ ) h (τ ) − T (τ ) H (τ ) c1 H (τ )
causes fluctuation in the time that the membrane potentials hit the threshold. Under som
linear approximation, we can relate the jitter in the vertical direction to the one in
horizontal direction. This then gives us the formula for the magnitude of the timing jitte
as a function of the spike time τ .

h : shape function.
Membrane potential X ( t ) Here, h is the shape function which describes how the membrane potential changes in
e.g. exponential
response to a single input spike. For the usual leaky integrate-and-fire model, this h star
with some amplitude and then decay exponentially.

σX
h (t )

For conciseness, we define these two integrations which get used in the formula here.
t
H (t) = h (µ)dµ.
0
Threshold T ( t ) t
σ time H2 (t) = h2 (µ)dµ.
Time 0


Introduction
The Threshold Curve
Conclusion

Approximation for the ﬁrst timing jitter (con’t)

H (τ ) T (τ ) c2 H2 (τ )
σtime (τ ) ≈
T (τ ) h (τ ) − T (τ ) H (τ ) c1 H (τ )

c1 and c2 are constants
Membrane potential X ( t ) which depend on the
σX distribution of the weight
(Ym ) for each spike.
Recall:
X (t) = m Ym h(t − τm ).

Threshold T ( t )
σ time
Time


Introduction
The Threshold Curve
Conclusion

Jitter in rate estimation

The only information contained in a Poisson process is its rate
λ.
Diﬀerent λ’s ⇒ diﬀerent spiking times τ ’s.

Descending Threshold
Spike times vary inversely with λ.

PSP for Large λ
T (τ )
λ(τ ) ≈
c1 H (τ )

PSP for Small λ Error in rate estimation:

1 T (τ ) c2 H2 (τ )
σλ (τ ) ≈ .
Time c1 H (τ ) c1 H (τ )
However, this spike time has some jitter, so the $lambda$ estimation also have some
error. We then go on and approximate this error:

The second jitter is the randomness in the length of time a spike takes to propa
Introduction
Sources of Variability and Threshold Derivationanother neuron. It is on the order of 10 microseconds. The third jitt
synapse on
time-of-arrival estimation error; that is, if this neuron tries to measure the inter
The Threshold Curve
Conclusion
interval, it needs to find out what time a spike arrives. We borrow some formu
Radar guys shown here because this is exactly the problem that they call the ra
Second and Third Jitters over N0 here. Iterror depends on the signal-to-noise the shape ofis show
ranging problem. The
this Es also depends on the bandwidth for
ratio which
the acti
potential. The amount of error here is about 10 microseconds as well.

Propagation Time.
Time-of-Arrival Estimation
Error (radar ranging
problem):
−1
2Es 2
4π 2 f .
N0

f 2 : Gabor-bandwidth.
ES : Signal energy.
N0
2 : Spectral height of the
Noise.
Small: < 10µs.

Introduction
The Threshold Curve
Conclusion

Deriving The Threshold Curves

Recall: Deviation in time:

H (τ ) T (τ ) c2 H2 (τ )
σtime (τ ) = .
T (τ ) h (τ ) − T (τ ) H (τ ) c1 H (τ )

We consider the thresholds which
1) preserve timing jitter σtime (τ ) ≡ σtime,0 , or
σtime (τ )
2) preserve relative timing jitter τ ≡ σ%time,0
across spiking times (or spiking frequencies) of interest.


Introduction
The Threshold Curve
Conclusion

Deriving The Threshold Curves

Recall:
H(τ ) T (τ )c2 H2 (τ )
(a) Deviation in time: σtime (τ ) = T (τ )h(τ )−T (τ )H(τ ) c1 H(τ ) .
1 T (τ )c2 H2 (τ )
(b) Deviation in λ estimation: σλ (τ ) = c1 H(τ ) c1 H(τ ) .
We consider the thresholds which
1) preserve timing jitter σtime (τ ) ≡ σtime,0 , or
σtime (τ )
2) preserve relative timing jitter τ ≡ σ%time,0 , or
3) preserve jitter in λ estimation σλ (τ ) ≡ σλ,0 , or
σλ (τ )
4) preserve relative jitter in λ estimation λ ≡ σ%λ,0 , or
5) preserve jitter in ln λ estimation
across spiking times (or spiking frequencies) of interest.


Introduction
The Threshold Curve
Conclusion

Deriving The Threshold Curves (con’t)

Constant timing jitter level:

h (t) 1 c2 H2 (t)
T (t) = T (t) − T (t) .
H (t) σtime,0 c1 H (t)

Constant relative-timing-jitter level:

h (t) 1 c2 H2 (t)
T (t) = T (t) − T (t) .
H (t) tσ%time,0 c1 H (t)

Preserve relative error in λ estimation:
H2 (t)
T (t) = c0 .
H (t)


Introduction
The Threshold Curve
Conclusion

The diﬀerential equations are Bernoulli equations of the form

T (t) = T (t) P (t) − T (t)Q (t) .

They can be reduced to linear equation by introducing
v (t) = T (t) which gives

1 1
v (t) = P (t) v (t) − Q (t) .
2 2
Linear! The solution is
t
1
v (t) = v (t0 ) φ (t, t0 ) − φ (t, τ ) Q (τ )dτ,
2
t0

t
1
2
P(τ )dτ
where φ (t, s) = e s .

Introduction
The Threshold Curve
Conclusion

Comparison between derived thresholds

Constant
timing jitter

Constant relative
timing jitter

Exponential

Heavy-tail
Linear

6 7 8
Time [ms]


Introduction
The Threshold Curve
Conclusion

Summary

Analyze and quantify three sources of timing jitter

Predict shape of threshold curves

Constant
timing jitter

Constant relative
timing jitter

Exponential

Heavy-tail
Linear

6 7 8
Time [ms]


Introduction
OPT1: Maximization of Mutual Information
OPT2: Mutual Information per Unit Energy Cost
Rate Matching
Conclusion

Introduction


Rate Matching

Conclusion


Introduction
Rate Matching
Conclusion

Conditional density Q(t|λ) = fτ |Λ (t|λ)

We have formula(s) for the threshold curve T (t).
Assumption: λ stays constant during each ISI.
Given Poisson input intensity λ, can ﬁnd the conditional
density Q(t|λ) = fτ |Λ (t|λ).
τ = g (Λ)+jitter.

Λ λ τ

τ |Λ |λ


Introduction
Rate Matching
Conclusion

Optimization 1: Mutual Information

OPT1:
sup I (Λ; τ )
where
fΛ,τ (Λ, τ )
I (Λ; τ ) = E log
fΛ (Λ)fτ (τ )
and the supremum is taken over all possible fΛ (λ).
Blahut-Arimoto Algorithm (BAA)


Introduction
Rate Matching
Conclusion

Exponential “Constant-Jitter” “Constant-Relative-Jitter”
Threshold Threshold Threshold
-3 -3 -3
x 10 x 10 x 10
6 6 6

4 4 4

f ()
f ()

f ()
2 2 2

0 0 0
0 500 1000 1500 2000 0 500 1000 1500 2000 0 500 1000 1500 2000
  

0.8 0.8 0.8

0.6 0.6 0.6
f(t)
f(t)

f(t)
0.4 0.4 0.4

0.2 0.2 0.2

0 0 0
4 6 8 10 12 14 4 5 6 7 8 4 5 6 7 8 9
t t t

C = 5.457 C = 4.931 C = 5.109
(a) (b) (c)


Introduction
Rate Matching
Conclusion

Capacity-achieving input densities look similar.
−3
x 10
6
exponential
constant jitter
5 constant relative jitter

4
fΛ(λ)

3

2

1

0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
λ


Introduction
Rate Matching
Conclusion

Input-Intensity Density Approximation

BAA does not provide any
insight.
Our simpler formula:

σ0 (c1 H (t))3
fΛ (λ) ≈
d T (t) c2 H2 (t)
t=g (λ)

where g (λ) = E [τ |λ].


Introduction
Rate Matching
Conclusion


−3

BAA does not provide any 6
x 10

exponential

insight. 5
constant jitter
constant relative jitter
approximation

Our simpler formula: 4

fΛ(λ)
(c1 H (t))3
3
σ0
fΛ (λ) ≈
d T (t) c2 H2 (t) 2

t=g (λ) 1

where g (λ) = E [τ |λ]. 0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
λ


Introduction
Rate Matching
Conclusion


BAA does not provide any exponential
constant jitter
insight. constant relative jitter
approximation

Our simpler formula:

fΛ(λ)
−3
10

σ0 (c1 H (t))3
fΛ (λ) ≈
d T (t) c2 H2 (t)
t=g (λ)

where g (λ) = E [τ |λ]. 1
10 10
2

λ
10
3


Introduction
Rate Matching
Conclusion

Approximation Strategy

Assume that g (Λ) is uniform and then ﬁnd the corresponding fΛ .
-3 -3 -3
x 10 x 10 x 10
6 6 6

4 4 4

f ()
f ()

f ()
2 2 2

0 0 0
0 500 1000 1500 2000 0 500 1000 1500 2000 0 500 1000 1500 2000
  

0.8 0.8 0.8

0.6 0.6 0.6
f(t)
f(t)

f(t)
0.4 0.4 0.4

0.2 0.2 0.2

0 0 0
4 6 8 10 12 14 4 5 6 7 8 4 5 6 7 8 9
t t t

C = 5.457 C = 4.931 C = 5.109
(a) (b) (c)


Introduction
Rate Matching
Conclusion

Approximation Strategy
Assume that g (Λ) is uniform and then ﬁnd the corresponding fΛ .

�� 
�� ��

�� 

*
Convolution

For invertible function g , the pdf of Z = g (Λ) is given by
d −1 1
fZ (z) = g (z) fΛ g −1 (z) = fΛ (λ) ,
dz |g (λ)|
where z = g (λ).

Introduction
Rate Matching
Conclusion

OPT2: Energy-Eﬃcient Neuron

Brains consume 20% of energy consumption for adults and
60% for infant [Laughlin and Sejnowski’03].
Suppose neuron spends
1 unit of energy per ms when it is idle, and
e unit of energy per ms when AP is produced.
e >> 1.
If the time to the next spike is τ = t, the energy expended is

bo (t) = 1 × (t − ∆) + e × ∆ = t + (e − 1)∆ = t + r .

where ∆ is the time used to produce a spike.
The value of r depends on the type of neurons under
consideration.


Introduction
Rate Matching
Conclusion

Optimization 2: I/E

OPT2:
I (Λ; τ )
sup
E [bo (τ )]
where the supremum is taken over all possible fΛ (λ).
I (Λ;τ )
Jimbo-Kunisawa algorithm (JKA) maximizes E[b(Λ)] .
b is a function of input.
Our bo is a function of output.
We deﬁne b(λ) = E [bo (τ )|Λ = λ] and apply JKA.
Because bo (τ ) = τ + r , we have

b(λ) = E [τ |λ] + r = g (λ) + r .


Introduction
Rate Matching
Conclusion


Can use the same technique as in OPT1 to do approximation
of input-intensity density.
In stead of uniform density, consider bounded exponential
density of the form
γ
f (t; γ, α, β) = e −γt 1[α,β] (t) .
e −γα − e −γβ


Introduction
Rate Matching
Conclusion


Can use the same technique as in OPT1 to do approximation
of input-intensity density.
Result:

(c1 H (t))3
fΛ (λ) ≈ σ0 f (t; γ, g (b), g (a)) ,
T (t) c2 H2 (t)
t=g (λ)

where f (t; γ, g (b), g (a)) is the bounded exponential pdf with
support on the interval [g (b), g (a)] and parameter γ.


Introduction
Rate Matching
Conclusion

−3
x 10
6
exponential exponential
constant jitter constant jitter
5 constant relative jitter 0.5 constant relative jitter
approximation

4 0.4

fΛ(λ)

fτ(t)
3 0.3

2 0.2

1 0.1

0 0
0 500 1000 1500 2000 4 6 8 10 12 14
λ t [ms]

exponential
constant jitter
approximation
fΛ(λ)

−3
fτ(t)

10

−1
10

exponential
constant jitter

1 2 3 4 5 6 7 8 9 10
10 10 10
λ t [ms]


Introduction
Rate Matching
Conclusion

Free Parameters - Revisited

Recall, for example, the differential equation that define our
“constant-relative-jitter” threshold:

h (t) 1 c2 H2 (t)
T (t) = T (t) − T (t) .
H (t) tσ%time,0 c1 H (t)

There are a couple of parameters which we want to revisit.
The constants c1 and c2 .
Embedded in them is the effect of QSF.
σtime,0
σ%time,0 = t0 . What value should we set σtime,0 to be?
> 10µs.


Introduction
Rate Matching
Conclusion

By scaling the unit of the voltage, we can make c1 = 1.
The scaling makes
1 1
c2 ∝ =
psuccess 1 − pfailure
where pfailure is the QSF probability.
pfailure depends on the type of neurons under consideration.
Let σtime,0 = σ1 and play with it.


Introduction
Rate Matching
Conclusion

Average Rates

From our optimization (either OPT1 or OPT2), we get the optimal
input-intensity density fΛ (λ) and spiking-time density fτ (t) for each
value of σ1 .
¯
Each value of σ1 gives average arrival rate λin and the average
¯ out .
spiking rate λ
¯
λout = 1 . E[τ ]
¯
λin = E [Λ]?


Introduction
Rate Matching
Conclusion

Average Aﬀerent Rate
High value of Λ corresponds to low value of τ .
Neuron experiences large Λ value for short amount of time.
¯
λin should be lower than E [Λ].
Let (Λi , τi ) be the pair of input-intensity and the length of the
ISI associated with the ith spike.
The number of input spikes during the ith ISI is a Poisson
random variable Ni with mean Λi τi .
The long-term average input rate is then
k 1 k
i=1 Ni k i=1 Ni E [Λ1 τ1 ]
k
= 1 k
→ .
i=1 τi i=1 τi
E [τ1 ]
k

¯ E[Λτ ]
λin = E[τ ] .


Introduction
Rate Matching
Conclusion

Rate Matching

On average, our neuron in consideration is bombarded with an
¯
average input-intensity λin .
Suppose our neuron is receiving input from n other neurons.
Then, on average, each of the sending neurons ﬁre at a rate
1¯
of n λin .
Our neuron should also generate spikes at this rate.
Therefore, we must have
1¯ ¯
λin = λout .
n


Introduction
Rate Matching
Conclusion

Rate Matching (Con’t)

Average spiking rate [spikes/s]
200
input
150 output

100

50

0
10 20 30 40 50 60 70 80 90 100
σ1 [microsec]

5.4
Capacity [bits]

5.2

5

4.8

10 20 30 40 50 60 70 80 90 100
σ1 [microsec]


Introduction
Rate Matching
Conclusion


Capacity value increases as
we increase the noise level

Average spiking rate [spikes/s]
200

σ1 . 150
input
output

100
Larger value of σ1 implies 50

threshold decays slower This 0
10 15 20 25 30 35 40 45 50 55 60
σ1 [microsec]
gives larger support for the
spiking time. 5.8

Capacity [bits]
5.6
Noise is small. Eﬀect of 5.4

increasing the support of the 5.2
10 15 20 25 30 35 40 45 50 55 60
output is stronger than the σ1 [microsec]

eﬀect of increased noise.


Introduction
Rate Matching
Conclusion

60

55
(c) max I/E, r = 100
50
Optimal rate [spikes per second]

45

40
(b) max I/E, r = 400
35

30

25 exponential
constant relative jitter (a) max I
20

15
1 1.5 2 2.5 3 3.5 4 4.5 5
c2


Introduction
Rate Matching
Conclusion


60

55
(c) max I/E, r = 100
Firing rate decreases as the 50

Optimal rate [spikes per second]
failure probability increases. 45

Smaller value of r 40
(b) max I/E, r = 400
35
corresponds to higher rate.
30

As r → ∞, the rate 25 exponential
constant relative jitter (a) max I
converges to the max-I case. 20

15
1 1.5 2 2.5 3 3.5 4 4.5 5
c2


Introduction
Rate Matching
Conclusion

Optimal Threshold: c2 = 5 and r = 400
800

700 exponential
constant relative timing jitter

600

500

400

300

200

100

0
0 20 40 60 80 100 120 140 160 180 200
Time [ms]


Introduction
Conclusion

Introduction



Conclusion


Introduction
Conclusion

Contributions

1. Quantify the amount of timing jitter in neuron.
2. Construct threshold functions.
3. Provide optimal operating points for neurons which are close
to experimentally observed values.
Formulas to approximate the optimal input-intensity densities.


Introduction
Conclusion

References I

P. Suksompong and T. Berger.
Jitter Analysis of Timing Codes for Neurons with Descending
Action Potential Thresholds.
ISIT, 2006.
Capacity Analysis of Neurons with Descending Action
Potential Thresholds.
In preparation for special issue of IEEE Tran. on Info. Theory,
2009.


Introduction
Conclusion

References II

Energy-Eﬃcient Neurons with Descending Action Potential
Thresholds.
In preparation for Journal of Comp. Neuroscience, 2009.
T. Berger and W.B. Levy.
Encoding of excitation via dynamic thresholding.
Society for Neuroscience, 2004.
W.B. Levy and R.A. Baxter.
Energy-Eﬃcient Neuronal Computation via Quantal Synaptic
Failures.
Journal of Neuroscience, 2002.


Introduction
Conclusion

References III

W.B. Levy and R.A. Baxter.
Energy Eﬃcient Neural Codes
Neural Computation, 1996.
Patrick Crotty and William Levy.
Biophysical limits on axonal transmission rates in axons.
CNS, 2005.
E. Parzen.
Stochastic Processes.
Holden Day, 1962.
H. Vincent Poor.
An introduction to signal detection and estimation (2nd ed.).
Springer-Verlag New York, Inc., New York, NY, USA, 1994.

Introduction
Conclusion

References IV

Dale Purve et al.
Neuroscience (3rd ed.).
Sinauer Associates Inc., Sunderland, MA USA, 1997.


Introduction
Conclusion

THE
END


Prapun B-Exam

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