Computational Neuroscience ESPRC Workshop
--Warwick
Information-Geometric
Studies on Neuronal Spike
Trains
Shun-ichi Amari
RIKEN Brain Science Institute
Mathematical Neuroscience Unit
Neural Firing
x1
x2
x3
xn
xi = 1, 0
p(x) = p( x1 , x2 ,..., xn ) : joint probability
ri = E[ xi ]
----firing rate
S = { p ( x1 , x2 ,..., xn )}
vij = Cov[ xi , x j ] ----covariance: correlation
higher-order correlations
orthogonal decomposition
Multiple spike sequence:
{xi (t ), i = 1,..., n; t = 1,..., T }
xi (t ) = 0,1
time t
neuron i
1
n
1
T
x1 (1)
x1 (T )
x2 (1)
x2 (T )
xn (1)
xn (T )
Information Geometry ?
S = {p (x; μ , )}
2
1
(x μ ) p (x; μ , ) =
exp 2
2
2
{p (x )}
S = {p (x; è )}
è = (μ , )
μ
Riemannian metric
Dual affine connections
Information Geometry
Systems Theory
Statistics
Combinatorics
Information Theory
Neural Networks
Information Sciences
Physics
Math.
AI
Riemannian Manifold
Dual Affine Connections
Manifold of Probability Distributions
Manifold of Probability Distributions
x = 1, 2, 3
{p ( x)}
p = ( p1 , p2 , p3 )
p3
p1
p1 + p2 + p3 = 1
M = {p (x; )}
p2
Manifold of Probability Distributions
M = {P (x, p )}
x = 1, 2,3
3
3
P (x, p ) = pi i (x )
( p3 = 1 p1 p2 )
i =1
M
p = ( p1 , p2 , p3 )
2
1
p1 + p2 + p3 = 1
3
i =
pi
12 + 22 + 32 = 1
1
2
1 = log
= log
2
p1
p3
p2
p3
= ( 1 , 2 )
Invariance
S = {p (x, )}
1. Invariant under reparameterization
p ( x, ) = p ( x, )
2.
D = i2 2
i
Invariant under different representation
2
y = y (x ), { p ( y, )} = { p ( x, )}
p (x, ) p (x, ) dx
| p ( y, ) p ( y, ) | dy
1
2
2
1
2
Two Structures
Riemannian metric
affine connection --- geodesic
g ij = E log p
log p j
i
Fisher information
ds 2 = gij ( )di d j =< d , d >
Affine Connection
covariant derivative
c X = Y
& & X=X(t)
geodesic X=X
s=
i
j
(
)
g
d
d
ij
minimal distance
straight line
Affine Connection
covariant derivative; parallel transport
XY ,
c X = Y
geodesic x& =x& x=x(t)
s=
i
j
g
d
d
(
)
ij
minimal distance : straight line
< X , Y >=< X , Y >
i
gij X Y
j
Duality: two affine connections
X , Y = X , Y
{S , g , , *}
< X , Y >= gij X iY j
*Y
X
Y
X
Riemannian geometry:
= (, )
Dual Affine Connections
*
( , )
e-geodesic
log r (x, t ) = t log p (x ) + (1 t )lg q (x ) + c (t )
m-geodesic
x& x& (t ) = 0
*x& x& (t ) = 0
r (x, t ) = tp (x ) + (1 t )q (x )
q (x )
p (x )
Information Geometry
-- Dually Flat Manifold
Convex Analysis
Legendre transformation
Divergence
Pythagorean theorem
I-projection
Dually Flat Manifold
1. Potential Functions
---convex
2. Divergence
(Bregman, Legendre transformation)
D [p : q]
3. Pythagoras Theorem
p
D [ p : q ]+ D [q : r ] = D [ p : r ]
r
4. Projection Theorem
5.Dual foliation
q
Pythagoras Theorem
correlations
p
D[p:r] = D[p:q]+D[q:r]
q
r
p,q: same marginals 1 ,2
r,q: same correlations
independent
p( x)
D[ p : r ] = p ( x) log
q( x)
x
estimation correlation
testing
invariant under firing rates
Projection Theorem
min D [P : Q ]
QM
Q=
m-geodesic
projection of
P
to
M
min D [Q : P ]
QM
Q = e-geodesic projection of
P to M
Multiple spike sequence:
{xi (t ), i = 1,..., n; t = 1,..., T }
xi (t ) = 0,1
time t
neuron i
1
n
1
T
x1 (1)
x1 (T )
x2 (1)
x2 (T )
xn (1)
xn (T )
spatial correlations
x = (x1 , L , xn )
p ( x ) = exp i xi + ij xi x j + LL+ 1L n x1
i< j
xn ( )
ri = E [xi ] = Prob {xi = 1}
rij = E xi x j = Prob {xi = x j = 1}
L
= (i , ij , L , 1L n )
r = (ri , rij , L , r1L n )
orthogonal structure
S = {p ( x )} : coordinates
Spatio-temporal correlations
correlated Poisson
p (x )
: temporally independent
correlated renewal process
p ( xt )
: firing rate ri (t )
modified
spatial correlations fixed
Two neurons:
{ p00 , p01 , p10 , p11}
x
1
x2 x3
firing rates: r1 , r2 ; r12
correlation—covariance?
Correlations of Neural Firing
{p (x , x )}
1
x1
x2
2
2
{p00 , p10 , p01 , p11}
r1 = p1 = p10 + p11
r2 = p1 = p01 + p11
p11 p00
= log
p10 p01
1
firing rates
correlations
{( r1 , r2 ), }
orthogonal coordinates
Independent Distributions
x1 , x2 = 0,1
S = { p ( x1 , x2 )}
M = {q ( x1 )q ( x2 )}
Orthogonal Coordinates:
correlation fixed
scale of correlation
firing rates
r’
r
two neuron case
r1 , r2 , r 12 ; 1 , 2 , 12
r12 (1 + r12 r1 r2 )
p00 p11
= log
12 = log
p01 p10
(r1 r12 )(r2 r12 )
r12 = f (r1 , r2 , )
r12 (t ) = f (r1 (t ), r2 (t ), )
Decomposition of KL-divergence
correlations
D[p:r] = D[p:q]+D[q:r]
p
q
p,q: same marginals 1 ,2
r,q: same correlations
p( x)
D[ p : r ] = p ( x) log
q( x)
x
r
independent
pairwise correlations
covariance: cij = rij ri rj
not orthogonal
independent distributions
rij = ri rj ,
rijk = ri rj rk , L
How to generate correlated spikes?
(Niebur, Neural Computation [2007])
higher-order correlations
Orthogonal
higher-order correlations
= (i , ij ; L , 1L n )
r = (ri , rij ;
L , r1L n )
tractable models of
distributions M = { p(x)}
n
Full model: 2 1 parameters
Homogeneous model: n parameters
Boltzmann machine: only pairwise
correlations
Mixture model
Models of Correlated Spike Distributions (1)
Reference
y (t )
0110001011
x%
1 (t )
1001011001
x%2 (t )
0101101101
x%3 (t )
Models of Correlated Spike Distributions (2)
xi (t ) = x%i (t )o y (t )
additive interaction
eliminating interaction
Niebur replacement
y (t )
0110001011
x%
1 (t )
%
x 2 (t )
1001001101
1100010011
Mixture Model
y (t ) :
0110001011 mixing sequence
p1 ( x; u ) = p (xi , ui ) when y = 1
p2 ( x; u ) = p (xi , vi ) when y = 0
p ( x, u ) : x = 1 with probability u
p ( x) = p ( y ) p ( x | y )
Mixture model
p ( x; u ) = p (xi , ui )
: independent distribution
p ( x; u, v , m ) = mp ( x; u ) + mp ( x; v )
u = (u1 , L , un )
m = 1 m
v = (v1 , L , vn )
M n = {p ( x; u, v , m )}, S n = {p ( x )}
firing rates
ri = mui + mvi
rij = mui u j + mvi v j
rijk = mui u j uk + mvi v j vk
L
Conditional distributions
and mixture
y = 1, 2,3,...
p (x | y ) : Hebb assembly y
p ( x) = p ( y ) p ( x | y )
State transition of Hebb assemblies
mixture and covariances
p ( x , t ) = (1 t ) pu ( x ) + tpv ( x )
cij (t ) = (1 t )cij (u ) + tcij (v ) + t (1 t )(ui vi )(u j v j )
Extra increase (decrease) of covariances
within Hebb assemblis---- increase
between Hebb assemblies ---decrease
Temporal correlations
{x(t)}, t =1, 2, …T
Prob{x(1),..., x(T )}
Poisson sequence:
Markov chain
Renewal process
Orthogonal parameter of
correlation ---- Markov case
p ( xt +1 | xt )
aij = Pr{xt +1 = i | xt = j}
Pr{xt } = Pr( x1 )p ( xt +1 | xt )
r1 = Pr{xt = 1}
2
1
c = r11 r
a11a00
= log
a10 a01
Spatio-temporal correlation
correlated Poisson
T
p ({xt }) = p (xt )
t =1
Correlated renewal process
Pr( xt ) = k (t t ') : t ' last spike time
Pr( xi ,t ) = k (t t ') p ( xi | x)
Population and Synfire
x1
x2
xn
Neurons
xi = 1(ui )
ui = Gaussian
E[ui u j ] = Population and Synfire
ui = wij s j h
x1
xn
xi = 1[ui ]
ui = (1 ) i + h
i , N (0, 1)
E[ui u j ] = 2
E[ui ] = 1
s
pi = Prob {i neurons fire at the same time }
i
r=
n
q(r , ) = e
Pr = Pr{nr neurons fire}
nH (r )
e
nz ( )
d
2
z ( ) =
r log F (1 r )log (1 F )
2n
1
F = F (a h ) =
2
a h
0
e
t2
2
dt
2 1
1
2
q (r , ) = c exp[
{F ( ) h} ]
2(1 )
2 1
p (x, ) = exp{ i xi + ij xi x j + ijk xi x j xk + ...}
i1i2 ...ik = O(1/ n k 1 )
Synfiring
p ( x ) = p ( x1 ,..., xn )
1
r = xi
n
q (r )
q(r )
r
Bifurcation
xi : independent---single delta peak
pairwise correlated
Pr
higher-order correlation
r
Semiparametric Statistics
M = {p (x, , r )}
p ( x, ) = r ( x )
y
x
y =x
yi = i + i
'
x
=
+
i
i
i
p (x, y; ) = p (x, y; , )r ( )d mle, least squre, total least square
Linear Regression: Semiparametrics
(x1 , y1 )
(x2 , y2 )
M
(xn , yn )
y = x
xi = i + i
yi = i + i , '
i
'
i
N (0, 2
)
Statistical Model
1
2
2
1
p (x, y , ) = c exp (x ) ( y ) 2
2
p (xi , yi , i ) : , 1 ,L , n
p (x, y ) = p (x, y , )Z ( )d
semiparametric
Least squares?
2
L ( ) = ( yi xi ) min
y
x
yi
1
,
n
xi
mle,
i
i
TLS
(y
i
Neyman-Scott
xi )( yi + xi ) = 0
: ˆ =
x y
x
i
i
2
i
Semiparametric statistical model
x1 , x2 ,L
p ( x, , Z )
Estimating function
y ( x, )
E , Z y (x, ) = 0
E ' , Z y (x, ) 0
Estimating equation
y (xi , ) = 0
ˆ
u ( x , y ; , Z ) =
log p
= sE s L
v (x, y; , Z ) = E f ( ) s = k {s (x, y; )}
u
I
(x, y, ) = u E u s = k (x + y )( y x )
Fibre bundre
function space
r
z ( ) N (μ , 2
)
u I (x, , Z ) = (x + y + c )( y x )
c = μ 2
f (x, ; c ) = (x + y + c )( y x )
2
c = μ 1
μ = xi
n
2
1
= xi2 μ n
2
( )
2
Poisson process
Poisson Process: Instantaneous firing rate is constant over
time.
dt
For every small time window dt,
generate a spike with probability dt.
T
Cortical Neuron
(Softky & Koch, 1993)
Poisson Process
p(T ) = e T
Poisson process
cannot explain interspike interval
distributions.
Gamma distribution
Gamma Distribution: Every -th spike of the Poisson process is left.
( ) 1 T
q(T; , ) =
T e
.
( )
T
=1 (Poisson)
=3
Two parameters : Firing rate
: Irregularity
Gamma distribution
r ) 1
(
f (T ) =
T exp {r T }
( )
=1
: Poisson
: regular
Integrate-and fire
Markov model
Irregularity is unique to
individual neurons.
Regularlarge
t
t
t
Irregularsmall
Irregularity varies among
)
neurons.
We assume that is independent of
time.
Information geometry of
estimating function
•Estimating function y(T,):
N
y(T ; ) = 0
l
l =1
E[y(T,)]=0
•Maximum likelihood Method:
N
d
log p(T1 )L p(TN ) = u(Tl ; ) = 0
d
l =1
How to obtain an estimating
function y:
d log p(T;ê , k )
u
T;ê
k
(
,
)
dê
Score log p(T;ê , k )
function
v (T;ê , k ) s
k ( î )
(See poster for details)
< uv >
y = u
v
< vv >
temporal correlations
x (1) x (2 )L x (N )
independent with firing probability r (t )
spike counts: Poisson
ISI: exponential
renewal:
r (t ) = k (t ti ) : last
spike
ISI distribution
f (t )
T
f (T ) = ck (T )exp k (t )dt 0
Estimation of by estimating functions
No estimating function exists if the neighboring firing rates are
different.
m() consecutive observations must have the same firing rate.
Example:
m=2
l-th
set:
Model: p(T1 , T2 ; , k ) = q(T1 ; , )q(T2 ; , )k ( )d
0
(T1l ,T2l )
Estimating function:
(E[y]=0)
1
y=
N
l
,m
log
l =1
T1 T2
(T
1
l
y = log
l
+ T2
)
l 2
+ 2ö ( 2ê ) 2ö ( ê ) = 0
T1T2
+ 2 ö ( 2 ê ) 2ö ( ê )
2
(T1 +T 2 )
Case of m=1
(spontaneous discharge)
Firing rate continuously
changes and is driven by
Gaussian noise.
: Correlation
can be approximated if
the firing rate changes
slowly.
Estimating function (m=2):
1
y=
N
slow
l
,m
log
l =1
T1 T2
(T
1
l
l
+ T2
)
l 2
+ 2ö ( 2 ê ) 2ö ( ê ) = 0