Stochastic Properties Stochastic Properties of Neural Coincidence of Neural Coincidence Detector cellsDetector cells
Ram Krips and Miriam FurstRam Krips and Miriam Furst
TOCTOC
Neural ProcessingNeural Processing Stochastic AnalysisStochastic Analysis Auditory ExamplesAuditory Examples Boundary EvaluationBoundary Evaluation
Spiking informationSpiking information
Data within the Data within the brain travels in the brain travels in the form of neural form of neural spiking trains.spiking trains.
The information is The information is encoded both in the encoded both in the rate and timing of rate and timing of the spiking events.the spiking events.
The signal is The signal is stochastic in naturestochastic in nature
Neural CellsNeural Cells
The receivers/processors and The receivers/processors and transmitters of the spiking transmitters of the spiking information within the brains are information within the brains are the neural cellsthe neural cells
Common functionalities Common functionalities associated are:associated are:– Timing analysis Timing analysis – MemoryMemory– Signal generationSignal generation
Statistical Models of Statistical Models of Spiking BehaviourSpiking Behaviour The stochastic behavior of neural The stochastic behavior of neural
cells can be described as NHPP.cells can be described as NHPP. Considering the discharge history, Considering the discharge history,
a more general form of a more general form of representation is obtained: self representation is obtained: self excitatory models such as excitatory models such as renewal or doubly stochastic.renewal or doubly stochastic.
NHPP Model NHPP Model DefinitionsDefinitions
Poisson process is a pure birth Poisson process is a pure birth process:process:In an interval dt only one arrival with In an interval dt only one arrival with
probabilityprobability Number of arrivals N(t) in a finite Number of arrivals N(t) in a finite
interval of length t obeys:interval of length t obeys:
non-overlapping intervals are non-overlapping intervals are independent.independent.
The inter arrival times are The inter arrival times are independent and obey the independent and obey the Exponential distribution:Exponential distribution:
( )t dt
00
!
T
nT
t dt
OR
t dt
P n n en
0
T
t dt
e
Neural Cells ModelsNeural Cells Models
I&FI&F
•SimplificationSimplification•Mathematical Mathematical InsightInsight•More More assumptionsassumptionsWith regards to the modelWith regards to the model
•No mathematical No mathematical UnderstandingUnderstanding•Not suited for Not suited for Large scale Large scale simulationsimulation
CDCD
Coincidence Detection Coincidence Detection CellsCells Coincidence detection (CD) is one of Coincidence detection (CD) is one of
the common ways to describe the the common ways to describe the functionality of a single neural cell.functionality of a single neural cell.
CorrelationCorrelation There are several type of such cells:There are several type of such cells:
– Excitatory Inhibitory (EI)Excitatory Inhibitory (EI)– Excitatory Excitatory (EE)Excitatory Excitatory (EE)– CumulativeCumulative
Neural mechanisms – Neural mechanisms – EE Type cellsEE Type cells
Input 1
Input 2
(1) ( )E It
( )EE t
E
(2) ( )E It
I
E
_
I_
Spikes when inputs coincide.Spikes when inputs coincide.
EE
Input E
Input E
1 2 2 1
t t
EE
t t
t t t dt t t dt
1 2max( , )r r
EE FormulationEE Formulation
( ) ( ) ( ) ( ) ( ) ( )0; 0 ,0E I E I E Ip q p q p qEI t t or t t t T t T
0 0
( ) ( ) ( , )E I
NN
E EI Ii j
P EI P n i P n j P EI n i n j
( ) ( ) ( ) ( )
0
(0 , ) ( ) ( )T
E I E Ip q E p E qI I
P t t n i n j P t t n i P t t t n j dt
( ) ( ) * * 0
0
(0 , ) ( ) ( ') 'T t
E Ip q E EI I
Et I
P t t n i n j t t dt dt
0
( ) exp ( ) ( ') 'T t
E It
P EI t t dt dt
Neural mechanisms – Neural mechanisms – EI Type cellsEI Type cells Spikes with excitatory input Spikes with excitatory input
unless inhibited.unless inhibited.
EI
Input E
Input I
1t
EI E I
t
t t d
Er
EI FormulationEI Formulation
( ) ( ) ( ) ( )
1
( ) 0
,0 ,
E
N ME I E I
E I p q p q E In m n
P EI P n
P n n P n m P t t t t n n n m
( ) ( ) * * 0
0
(0 , )T t
E Ip q E I E I
E It
P t t n n n m t t dt dt
0 0
1 0
( )! ! !
E E I E
n m n nN M NI
nn m n n
P EI e e e en m n n
00
1 ' '0
0
( )!
T t
E I
tE E
n t t dt dtNT T T
n
TP EI e e e
n
Complex CellsComplex Cells
1
1m i
tM
CDEI E Ii t
t d
Complex CellsComplex Cells
Input 1
Input 2
Input M
...
Inhibitory input
N LE
' ' '
'
'
1 ( ') ' ( ) ( ') 'N L
L LN L
L
t tN
j l jCDEL L I l Ij I j It t
j lj I
t dt t t dt
Cumulative Type CellsCumulative Type Cells
Spikes if the number of excitatory events Spikes if the number of excitatory events during during exceeds inhibitory by P exceeds inhibitory by P
Input 1
Input 2
Input M
...
Inhibitory inputs
M N PE I
max( )r
1 1 ( ') ' 1 ( ') ' ( ) ( ') 'j j j
NLM N PP P
P
t t t
I E l El Ij I j E j EE Et t t
j lj E
t dt t dt t t dt
2 2 ( ) 2 ( )1
( ) ( ) 1 ( )N P t
E K I KtK
t t t
2 ( ) ( ) 1 ( ') ' ( ) ( ') 'j j
NLN K PK P P K
K P
t t
E K E l El Ij E j EE E t t
j lj E
t t dt t t dt
' '' ''
2 ( )' 1
1 ( ') ' ( ) ( ') 'j l j
NLN LL L
L
t tM
I K I I IL K l Ij I j II I t t
j lj I
t dt t t dt
' '' ''
3'
( ) 1 ( ') ' ( ) ( ') 'j l j
NLN LL L
L
t tN
E E EL N P l Ij E j IE E t t
j lj E
t t dt t t dt
1 2 3, ,( ) ( ) ( )M N PCD E It t t
EI
E1
EI
EI
...
...
EN
E2
I1
I2
IN
...
NPEE
E1
EI
EN
E2
IM
...
NP KEE
... 1MKEE
I2
I1
E1
EM
E2
...
MP NEE
EI Cells Signal EI Cells Signal SeparationSeparation
EI
S+N
N
Signal separation ability is considered as Signal separation ability is considered as most important in tasks such as cocktail most important in tasks such as cocktail party, BMLD. party, BMLD.
2 4 6 8
200
250
300
Time [mSec]
Spi
king
rat
e [n
orm
aliz
ed]
200 300 400 500 6000
0.05
0.1
Frequency [Hz]
|fft(
resp
onse
)|
[nor
mal
ized
]
EE Cells spontaneous EE Cells spontaneous raterate The spontaneous rate of cells that The spontaneous rate of cells that
results from external noise results from external noise reduced at higher levelsreduced at higher levels
EE Cells Harmonic EE Cells Harmonic Signals EnhancementSignals Enhancement Harmonic signals are most desirable in Harmonic signals are most desirable in
mammalsmammals
EE
S
S
Neural NetworksNeural Networks
Input Signal
Input NoiseEE
EI
EI
Input Signal
Input Noise
Auditory Lateralization Auditory Lateralization CuesCues Interaural Time Interaural Time
delay – The sound delay – The sound reaches the closest reaches the closest ear before the other ear before the other
Interaural Level Interaural Level delay – The sound delay – The sound at the closest ear is at the closest ear is louderlouder
Auditory cues analysis Auditory cues analysis - ITD- ITD
-200 -150 -100 -50 0 50 100 150 2000
50
100
150
200
250
relative phase [deg]
spik
ing
rate
10 uSec EE
10 uSec EI
100 uSec EE100 uSec EI
200 uSec EE
200 uSec EI
Auditory cues analysis Auditory cues analysis - ILD- ILD
0.1 1 100.1
1
10
100
1000
Relative Amplitude
Sp
ike
Ra
te [s
p/s
ec]
1 mSec EE1 mSec EI
3 mSec EE
3 mSec EI
10 mSec EE10 mSec EI
Auditory signals Auditory signals analysis Pitchanalysis Pitch
0 50 100 150 200 250 300 350 400 450 5000
20
40
60
80
100
120
140
160
Frequency [Hz]
Mea
n in
stan
tane
ous
rate
[sp
ikes
/sec
]
200mSec delay
20 uSec EE 20 uSec EI
2.000000e+002 uSec EE
2.000000e+002 uSec EI
500 uSec EE500 uSec EI
pitch
F
EE
100 200 300 400 500
50
100
150
200
250
300
pitch
F
EI
100 200 300 400 500
50
100
150
200
250
300 0
20
40
60
80
100
120
140
160
CD
Delay
Before going on…Before going on…
We have presented the We have presented the mathematical building blocks for mathematical building blocks for CD cells and networks analysisCD cells and networks analysis
Before going on to building Before going on to building networks we will develop another networks we will develop another tool that allows us to evaluate the tool that allows us to evaluate the quality of the processor formed:quality of the processor formed:
Bound evaluationBound evaluation
Overall Localization Overall Localization Performance - MAAPerformance - MAA Minimal Audible Angle is a Minimal Audible Angle is a
common test for evaluating common test for evaluating human localization ability human localization ability ..
MethodologyMethodology
The first point of stochastic The first point of stochastic behaviour is at the auditory behaviour is at the auditory nerve.nerve.
An optimal neural response was An optimal neural response was consideredconsidered
Ambiguity in Sound Ambiguity in Sound LateralizationLateralization For 1 kHz, the phase difference between For 1 kHz, the phase difference between
signals arriving at right and left ears is signals arriving at right and left ears is 180180oo. It is impossible to distinguish . It is impossible to distinguish between the possibility of the sound between the possibility of the sound arriving from the right or left speaker.arriving from the right or left speaker.
Frequency: 1kHzWavelength: 30cmHead size: 15cm
Frequency: 2kHzWavelength: 15cmHead size: 15cm
Bounds EvaluationBounds Evaluation
12*
*0
,1
,
T r tCRLB dt
r t
2 1 1 1 1ˆT
A A
1TB A A
1 2
*1
1 10 0 0 0
ln ,...,,..., ... ,1
n
T T TNn
p n p nn t t t
P t tA P t t dt dt p L
1 2
1
1 1*0 0 0 0 1
,..., 1,..., ... ,
1,...,n
T T TNn i
ij n j nn t t t n
P t t i LB P t t dt dt
j LP t t
1,..., L
MAA evaluation using MAA evaluation using CRLB and BBLB for NHPPCRLB and BBLB for NHPP
Going into the Brain - Going into the Brain - ITDITD CRLB for single neuron.CRLB for single neuron.
-100 -50 0 50 100
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Rel phase [ ]
CR
LB [
nor
mal
ized
]
1 2 3 4 5 6
35
40
45
50
55
60
65
Opt
imal
pha
se [
]
Sin amplitude
SummarySummary
Analytical tools for analysis and Analytical tools for analysis and evaluation of CD cells and evaluation of CD cells and networks were introduced.networks were introduced.
Validity demonstrated comparing Validity demonstrated comparing to biological findingsto biological findings