Stochastic Properties of Neural Coincidence Detector cells Ram Krips and Miriam Furst.

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