Computational Neuroscience ESPRC Workshop--Warwick

Information-GeometricStudies on Neuronal SpikeTrains

Shun-ichi AmariShun-ichi AmariRIKEN Brain Science Institute

Mathematical Neuroscience Unit

Neural Firing

1x 2x3

xn

x

higher-order correlations

orthogonal decomposition

1 2( ) ( , ,..., ) : joint probabilityn

p p x x x=x

[ ]i ir E x=

[ , ]ij i jv Cov x x=

----firing rate

----covariance: correlation

1,0i

x =

1 2{ ( , ,..., )}n

S p x x x=

( ) 0,1i

x t =

time tneuron i

1 T

1 ( )11x

1( )x T

( )21x ( )2

x T

n ( )1n

x ( )nx T

Multiple spike sequence:

{ ( ), 1,..., ; 1,..., }i

x t i n t T= =

( ){ } ( )( )

2

2

1; , ; , exp

22

xS p x p x

μμ μ= =

Information GeometryInformation Geometry ? ?

( ){ }p x

μ

( ){ };S p x= è

Riemannian metric

Dual affine connections

( , )μ=è

Information GeometryInformation Geometry

Systems Theory Information Theory

Statistics Neural Networks

Combinatorics PhysicsInformation Sciences

Riemannian Manifold

Dual Affine Connections

Manifold of Probability Distributions

Math. AI

Manifold of Probability DistributionsManifold of Probability Distributions

( )1 2 3 1 2 3

1, 2,3 { ( )}

, , 1

x p x

p p p p p p

=

= + + =

3p

2p

1p

p

( ){ };M p x=

Manifold of Probability DistributionsManifold of Probability Distributions

( ){ },

1, 2,3

P x

x

=

=

M p

3

21

( ) ( ) ( )3

3 1 2

1

, 1i i

i

P x p x p p p=

= =p

3

21

M

( )1 2 3

1 2 3

, ,

1

p p p

p p p

=

+ + =

p

2 2 2

1 2 31

i ip=

+ + =

( )

11

3 1 2

22

3

log

,

log

p

p

p

p

=

=

=

InvarianceInvariance ( ){ },S p x=

1. Invariant under reparameterization1. Invariant under reparameterization

( )( , ) ,p x p x=

2. 2. Invariant under different representationInvariant under different representation

( ) ( ), { , } { ( , )}y y x p y p x= = ( ) ( )2

1 2

2

1 2

, ,

| ( , ) ( , ) |

p x p x dx

p y p y dy

2 2

i iD =

Two StructuresTwo Structures

Riemannian metricRiemannian metric

affine connection --- geodesicaffine connection --- geodesic

( )2,ij i jds g d d d d= =< >

Fisher informationFisher information

log logij

i j

g E p p=

Affine Connection

covariant derivative

geodesic X=X X=X(t)

( )

c

i j

ij

X Y

s g d d

=

=

minimal distance

& &

straight line

Affine Connectioncovariant derivative; parallel transport

,

geodesic =x x=x(t)

:

, ,

( )

X c

i j

i

i j

ij

j

Y X Y

x

s g d

X Y X

d

Y g X Y< >=< >

=

=

minimal distance straight line

& &

Duality: two affine connectionsDuality: two affine connections

, , ,i j

ijX Y X Y X Y g X Y= < >=

Riemannian geometry:Riemannian geometry: =

X

Y

X

Y

*

{ , , , *}S g

Dual Affine ConnectionsDual Affine Connections

e-geodesice-geodesic

m-geodesicm-geodesic

( ) ( ) ( ) ( ) ( )log , log 1 lgr x t t p x t q x c t= + +

( ) ( ) ( ) ( ), 1r x t tp x t q x= +

( ),

( )q x

( )p x

*( , )

( ) 0

* ( ) 0

x

x

x t

x t

=

=

&

&

&

&

Information GeometryInformation Geometry

-- Dually Flat Manifold-- Dually Flat Manifold

Convex Analysis

Legendre transformation

Divergence

Pythagorean theorem

I-projection

Dually Flat ManifoldDually Flat Manifold

1. Potential Functions1. Potential Functions

---convex (Bregman, Legendre transformation)

2. Divergence2. Divergence [ ]:D p q

3. Pythagoras Theorem3. Pythagoras Theorem

[ ] [ ] [ ]: : :D p q D q r D p r+ =

4.4. Projection TheoremProjection Theorem

5.Dual foliation5.Dual foliation

p

r

q

Pythagoras Theorem

p

q

r

D[p:r] = D[p:q]+D[q:r]

p,q: same marginals

r,q: same correlations

1 2, independent

correlations

( )[ : ] ( ) log

( )x

p xD p r p x

q x=

estimation correlationtesting

invariant under firing rates

Projection Theorem

[ ]min :Q M

D P Q

Q = m-geodesic

projection of P to M

[ ]min :Q M

D Q P

Q = e-geodesic projection of P to M

( ) 0,1i

x t =

time tneuron i

1 T

1 ( )11x

1( )x T

( )21x ( )2

x T

n ( )1n

x ( )nx T

Multiple spike sequence:

{ ( ), 1,..., ; 1,..., }i

x t i n t T= =

spatial correlationsspatial correlations

( )1, ,

nx x= Lx

( ) ( )1 1exp

i i ij i j n n

i j

p x x x x x<

= + + +L

LLx

[ ] { }

{ }Prob 1

Prob 1

i i i

ij i j i j

r E x x

r E x x x x

= = =

= = = =

( )

( )1

1

, , ,

, , ,

i ij n

i ij nr r r

=

=

L

L

L

Lr

L

orthogonal structure

( ){ }S p= x : coordinates

Spatio-temporal correlationsSpatio-temporal correlations

correlated Poisson

( )p x : temporally independent

correlated renewal process

( )tp x : firing rate ( )ir t modified

spatial correlations fixed

firing rates:

correlation—covariance?

1x

2x

3x

00 01 10 11{ , , , }p p p p

1 2 12, ;r r r

Two neurons:

Correlations of Neural Firing

( ){ }{ }

1 2

00 10 01 11

1 1 10 11

2 1 01 11

,

, , ,

p x x

p p p p

r p p p

r p p p

= = +

= = +

11 00

10 01

logp p

p p=

1x 2

x

2

1

1 2{( , ), }r r

orthogonal coordinates

firing rates

correlations

1 2{ ( , )}S p x x=

1 2, 0,1x x =

1 2{ ( ) ( )}M q x q x=

Independent Distributions

Orthogonal Coordinates:

firing rates

r

correlation fixedscale of correlation

r’

two neuron casetwo neuron case

( )

( )( )

( )

( ) ( ) ( )( )

1 2 12 1 2 12

12 12 1 200 11

12

01 10 1 12 2 12

12 1 2

12 1 2

, , ; , ,

1log log

, ,

, ,

r r r

r r r rp p

p p r r r r

r f r r

r t f r t r t

+= =

=

=

Decomposition of KL-divergence

D[p:r] = D[p:q]+D[q:r]

p,q: same marginals

r,q: same correlations

1 2,

p

q

rindependent

correlations

( )[ : ] ( ) log

( )x

p xD p r p x

q x=

pairwise correlations

ij ij i jc r r r=

independent distributions

,,ij i j ijk i j kr r r r r r r= = L

How to generate correlated spikes?

(Niebur, Neural Computation [2007])

higher-order correlations

covariance: not orthogonal

Orthogonalhigher-order correlations

( )

( )1

1

;

;

,

,

,

,

i i n

i n

j

i jrr r

=

=

L

L

L

Lr

tractable models ofdistributions

Full model: parameters

Homogeneous model: n parameters

Boltzmann machine: only pairwise

correlations

Mixture model

2 1n

{ ( )}M p= x

Models of Correlated Spike Distributions (1)

Reference 0110001011 ( )y t

( )1x t%

( )2x t%

( )3x t%

1001011001

0101101101

Models of Correlated Spike Distributions (2)

( ) ( ) ( )i ix t x t y t= % o

additive interaction

eliminating interaction

Niebur replacement

( )y t

( )1

2 ( )

x t

x t

%

%

0110001011

1001001101

1100010011

Mixture Model

( ):y t 0110001011 mixing sequence

( ) ( )1; , when 1

i ip p x u y= =x u

( ) ( )2; , when 0

i ip p x v y= =x u

( , ) : 1 with probability p x u x u=

( ) ( ) ( | )p p y p y=x x

Mixture modelMixture model

( ) ( )

( ) ( ) ( )

; ,

; , , ; ;

i ip p x u

p m mp mp

=

= +

x u

x u v x u x v

( )

( )1

1

, , 1

, ,

n

n

u u m m

v v

= =

=

L

L

u

v

( ){ } ( ){ }; , , ,n n

M p m S p= =x u v x

: independent distribution

firing ratesfiring rates

ij i j i jr mu u mv v= +

ijk i j k i j kr mu u u mv v v= +

i i ir mu mv= +

L

Conditional distributionsand mixture

1,2,3,...

( | ) : Hebb assembly

( ) ( ) ( | )

y

p y y

p p y p y

=

=

x

x x

State transition of Hebb assembliesmixture and covariances

( ) ( ) ( ) ( ), 1u v

p t t p tp= +x x x

( ) ( ) ( ) ( ) ( )( )( )1 1ij ij ij i i j j

c t t c u tc v t t u v u v= + +

Extra increase (decrease) of covariances

within Hebb assemblis---- increase between Hebb assemblies ---decrease

Temporal correlations {x(t)}, t =1, 2, …T

Poisson sequence:

Markov chain

Renewal process

Prob{ (1),..., ( )}x x T

Orthogonal parameter ofcorrelation ---- Markov case

1

1

1 1

( | )

Pr{ | }

Pr{ } Pr( ) ( | )

t t

ij t t

t t t

p x x

a x i x j

x x p x x

+

+

+

= = =

=

1

2

11 1

11 00

10 01

Pr{ 1}

log

tr x

c r r

a a

a a

= =

=

=

Spatio-temporal correlation

correlated Poisson

1({ }) ( )

T

t tt

p p=

=x x

Correlated renewal process

,

Pr( ) ( ') : ' last spike time

Pr( ) ( ') ( | )

t

i t i

x k t t t

x k t t p x

=

= x

Neurons

1x

nx

( )1i i

x u=

Gaussian [ ]i i j

u E u u= =

2x

Population and Synfire

Population and Synfire

hswu jiji =

[ ]ii

ux 1=

(1 )i i

u h= +

( ), 0, 1i

N

s

1x

nx

2

[ ]

[ ] 1

i j

i

E u u

E u

=

=

{ } timesame at the fire neurons Prob ipi=

Pr{ neurons fire}r

ir P nr

n= =

( ) ( )( , )

nH r nzq r e e d=

( ) ( ) ( )FrFrn

z = 1 log 1 log 2

2

( ) dt 2

1 2

0

2t

ha

ehaFF ==

1 22 1( , ) exp[ { ( ) } ]

2(1 ) 2 1q r c F h=

1 2

1

...

( , ) exp{ ...}

(1/ )k

i i ij i j ijk i j k

k

i i i

p x x x x x x

O n

= + + +

=

x

Synfiring

( )

1( ) ( ,..., )

1

n

i

p p x x

r x q rn

=

=

x

( )q r

r

Bifurcation

r

rP

ix : independent---single delta peak

pairwise correlated

higher-order correlation

Semiparametric StatisticsSemiparametric Statistics

( ){ }( ) ( )

, ,

,

M p x r

p x r x

=

=

y x=

'

i i i

i i i

y

x

= +

= +

mle, least squre, total least square

( ) ( ) ( ), ; , ; ,p x y p x y r d=

x

y

Linear Regression: Semiparametrics

( )

( )

( )

1 1

2 2

,

,

,n n

x y

x y

x y

M

'

i i i

i i i

x

y

= +

= +

( )' 2, 0,

i iN

y x=

Statistical Model

( ) ( ) ( )

( )

( ) ( ) ( )

2 2

1

1 1, , exp

2 2

, , : , , ,

, , ,

i i i n

p x y c x y

p x y

p x y p x y Z d

=

=

L

semiparametric

Least squares?

( ) ( )

( )( )

2

2ˆmin :

1,

mle, TLS

0

Neyman-Scott

i i

i i

i

ii

i i

i i i i

x yL y x

x

yy

n x x

y x y x

= =

+ =

( )

( ) ( )

( )

( )

'

1 2

,

,

, , , ,

, , 0

, 0

ˆ, 0

Z

Z

i

x x p x Z

y x E y x

E y x

y x

=

=

Semiparametric statistical model

Estimating function

Estimating equation

L

( )

( ) ( )

( ){ }( )

( )( )

, ; , log

, ; ,

, ;

, ,I

u x y Z p

sE s

v x y Z E f s

k s x y

u x y u E u s

k x y y x

=

=

=

=

=

= +

L

Fibre bundreFibre bundre

function spacefunction space

r

( ) ( )( ) ( )( )

2

2

,

, ,I

z N

u x Z x y c y x

c

μ

μ

= + +

=

( ) ( )( )

( )

2

222 2

, ;

1

1

i

i

f x c x y c y x

c

xn

xn

μ

μ

μ

= + +

=

=

=

Poisson process

Poisson Process: Instantaneous firing rate is constant over

time. dtFor every small time window dt,

generate a spike with probability dt.

T

Cortical Neuron Poisson Process

Poisson processPoisson process

cannot explain inter-cannot explain inter-

spike intervalspike interval

distributions.distributions.

TeTp =)(

(Softky & Koch, 1993)

Gamma distribution

Gamma Distribution: Every -th spike of the Poisson process is left.

: Firing rate

: Irregularity.T

)(

)(),q(T;

T1= e Two parameters

=1 (Poisson)

=3

T

Gamma distributionGamma distribution

( )( )( )

{ }1exp

rf T T r T=

1=

Integrate-and fire

Markov model

: Poisson

: regular

Irregularity is unique to

individual neurons.

t

t

t

Regular large

Irregular small

Irregularity varies among

neurons.)

We assume that We assume that is independent of is independent oftime.time.

Information geometry of

estimating function•Estimating function y(T, ):

(See poster for details)

)(

),(log),(

),(log),(

îk

kT;êpkT;êv

dê

kT;êpdkT;êu

How to obtain an estimatingfunction y:

•Maximum likelihood Method:

vvv

vuuy

><

><=

Scorefunctions

E[y(T, )]=0

0);(1

==

N

l

lTy0);()()(log

1

1 ===

N

l

lN TuTpTpd

dL

temporal correlationstemporal correlations

( ) ( ) ( )1 2x x x NL

independent with firing probability ( )r t

spike counts: Poisson

ISI: exponential

renewal:

( ) ( )ir t k t t= : last

spikeISI distribution ( )f t

( ) ( ) ( )0

exp

T

f T ck T k t dt=

Estimation of

by estimating functionsNo estimating function exists if the neighboring firing rates aredifferent.

m( ) consecutive observations must have the same firing rate.

Estimating function:

(E[y]=0)

Example:Example:m=2m=2

( ))(2)2(2log

2

21

21 êöêöTT

TTy +

+=

l-thset:

( )0)(2)2(2log

1

12

21

21 =++

==

êöêöTT

TT

Ny

,m

lll

ll

=0

2121 )(),;(),;(),;,( dkTqTqkTTp

)( 21

ll,TT

Model:

Case of m=1

(spontaneous discharge)

Firing rate continuously

changes and is driven by

Gaussian noise.: Correlation

( )0)(2)2(2log

1

12

21

21 =++

==

êöêöTT

TT

Ny

,m

lll

ll

can be approximated if can be approximated ifthe firing rate changesthe firing rate changesslowly.slowly.Estimating function (m=2):

slow