Statistical analysis of spike trains in neuronal networks · 2017. 1. 28. · Statistical analysis...

Statistical analysis of spike trains in neuronalnetworks

Bruno Cessac

NeuroMathComp Team,INRIA Sophia Antipolis,France.

Mathematical Modeling and Statistical Analysis inNeuroscience, 02-07-14

Bruno Cessac Statistical analysis of spike trains in neuronal networks

Visual system


Multi Electrodes Array

Figure : Multi-Electrodes Array.

Multi Electrodes Array

Encoding a visual scene

Encoding a visual scene

Do Ganglion cells act as independent encoders ?

Or do their dynamical (spatio-temporal) correlations play a role inencoding a visual scene (population coding) ?

Let us measure (instantaneous pairwise) correlations

E. Schneidman, M.J. Berry, R. Segev, and W. Bialek. "Weak pairwise correlations imply strongly correlated

network states in a neural population". Nature, 440(7087):1007-1012, 2006.

Constructing a statistical model handling measuredcorrelations

Assume stationarity.

Measure empirical correlations.

Select the probability distribution which maximizes theentropy and reproduces these correlations.

Spike events

Figure : Spike state.

Spike state

!k(n) 2 f 0; 1 g

Spike pattern

!(n) = (!k(n) )Nk=1

Spike block

!nm = f!(m)!(m + 1) : : : !(n) g

Raster plot

!def= !T0

Spike events

Figure : Spike pattern.

Spike state

!k(n) 2 f 0; 1 g

Spike pattern

!(n) = (!k(n) )Nk=1

Spike block

!nm = f!(m)!(m + 1) : : : !(n) g

Raster plot

!def= !T0

Spike events

Figure : Spike pattern.

Spike state

!k(n) 2 f 0; 1 g

Spike pattern

!(n) = (!k(n) )Nk=1

�1001

�

Spike block

!nm = f!(m)!(m + 1) : : : !(n) g

Raster plot

!def= !T0

Spike events

Figure : Spike block.

Spike state

!k(n) 2 f 0; 1 g

Spike pattern

!(n) = (!k(n) )Nk=1

Spike block

!nm = f!(m)!(m + 1) : : : !(n) g

Raster plot

!def= !T0

Spike events

Figure : Spike block.

Spike state

!k(n) 2 f 0; 1 g

Spike pattern

!(n) = (!k(n) )Nk=1

Spike block

!nm = f!(m)!(m + 1) : : : !(n) g�1 1 0 1 00 1 0 1 00 0 0 1 00 1 0 1 1

�

Raster plot

!def= !T0

Spike events

Figure : Raster plot/Spike train.

Spike state

!k(n) 2 f 0; 1 g

Spike pattern

!(n) = (!k(n) )Nk=1

Spike block

!nm = f!(m)!(m + 1) : : : !(n) g

Raster plot

!def= !T0


Let �(T )! be the empirical measure:

�(T )! [ f ] =

1

T

TXt=1

f � �t(!)

e.g. �(T )! [!i ] =

1T

PTt=1 !i (t): �ring rate;

�(T )! [!i!j ] =

1T

PTt=1 !i (t)!j(t).

Find the (stationary) probability distribution � thatmaximizes statistical entropy under the constraints:

�(T )! [!i ] = �(!i );�

(T )! [!i!j ] = �(!i!j)


There is a unique probability distribution which satis�es theseconditions.

This is the Gibbs distribution with potential:

H(!(0)) =NXi=1

hi!i (0) +NX

i ;j=1

Jij!i (0)!j(0)

Ising model

End of the story ?

End of the story ?

The Ising potential:

H(!(0)) =NXi=1

hi!i (0) +NX

i ;j=1

Jij!i (0)!j(0)

does not consider time correlations between neurons.It is therefore bad at predicting spatio-temporal patterns !

Which correlations ?

Spikes correlations seem to play a role in spike coding.

Although this statement depends on several assumption that could bias statistics

Stationarity;

Binning;

Stimulus dependence ?

Modulo these remarks, Maximum entropy seems to be a relevantsetting to study the role of spatio-temporal spike correlations in

retina coding.

Which correlations ?

Spikes correlations seem to play a role in spike coding.Although this statement depends on several assumption that could bias statistics

Stationarity;

Binning;

Stimulus dependence ?

Modulo these remarks, Maximum entropy seems to be a relevantsetting to study the role of spatio-temporal spike correlations in

retina coding.

OK. So let us consider spatio-temporal constraints.


Easy !

Euh... In fact not so easy.

H(!D0 ) =NXi=1

hi!i (0) +NX

i ;j=1

J(0)ij !i (0)!j(0)

+NX

i ;j=1

J(1)ij !i (0)!j(1)

+NX

i ;j ;k=1

J(2)ijk !i (0)!j(1)!k(2)

+????


Easy ! Euh... In fact not so easy.

H(!D0 ) =NXi=1

hi!i (0) +NX

i ;j=1

J(0)ij !i (0)!j(0)

+NX

i ;j=1

J(1)ij !i (0)!j(1)

+NX

i ;j ;k=1

J(2)ijk !i (0)!j(1)!k(2)

+????

Two "small" problems.

Handling temporality and memory.



Ising model considers successive times as independent



Probability of characteristic spatio-temporal patterns

The probability of a spike pattern ....depends on the network history (transition probabilities).



Probability of characteristic spatio-temporal patterns

Given a set of hypotheses on transition probabilities there exists amathematical framework to solve the problem.

Handling memory.

Markov chains

Variable length Markov chains

Chains with complete connections

: : :

Gibbs distributions.

Mathematical setting

Probability distribution on (bi-in�nite) rasters:� [!nm ] ;8m < n 2 Z

Conditional probabilities with memory depth D:Pn�!(n)

��!n�1n�D

�.

Generating arbitrary depth D blocks probabilities:

��!m+Dm

�= Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�� [!nm ] =

Qnl=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;

8m < n 2 ZChapman-Kolmogorov relation


Probability distribution on (bi-in�nite) rasters:� [!nm ] ;8m < n 2 ZConditional probabilities with memory depth D:Pn�!(n)

��!n�1n�D

�.


��!m+Dm

�= Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�� [!nm ] =

Qnl=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;




��!n�1n�D

�.

�1 1 0 1 j 00 1 0 1 j 00 0 0 1 j 00 1 0 1 j 1

�


��!m+Dm

�= Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�� [!nm ] =

Qnl=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;




��!n�1n�D

�.


��!m+Dm

�= Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�� [!nm ] =

Qnl=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;




��!n�1n�D

�.


��!m+Dm

�=

Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�� [!nm ] =

Qnl=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;




��!n�1n�D

�.


��!m+Dm

�= Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�

� [!nm ] =Qn

l=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;




��!n�1n�D

�.


��!m+Dm

�= Pm+D

�!(m + D)

��!m+D�1m

��!m+D�1m

�� [!nm ] =

Qnl=m+D Pl

h!(l)

��!l�1l�D

i��!m+D�1m

�;



� [!nm ] =nY

l=m+D

Pl

h!(l)

��!l�1l�D

i�h!m+D�1m

i; 8m < n 2 Z

�l

�!ll�D

�= logPl

h!(l)

��!l�1l�D

i

� [!nm ] = expnX

l=m+D

�l

�!ll�D

��h!m+D�1m

i

�h!nm j!m+D�1

m

i= exp

nXl=m+D

�l

�!ll�D

�

Gibbs distribution

8� � Zd ; �(f S g j @�) = 1

Z�;@�e��H

�;@�( fS g )

f (�) = � 1

�lim�"1

1

j�j logZ�;@�

(free energy density)

Gibbs distribution

Gibbs distribution

8m; n; �h!nm j!m+D�1

m

i= exp

nXl=m+D

�l

�!ll�D

�

(normalized potential)

Gibbs distribution

8m < n; A <� [!nm ]

expPn

l=m+DH�!ll�D

�exp�(n �m)P(H)

< B

(non normalized potential)

Gibbs distribution

P(H) is called "topological pressure" and is formalyequivalent to free energy density.

Does not require time-translation invariance (stationarity).

In the stationary case (+ assumptions) a Gibbs state is alsoan equilibrium state.

sup�2Minv

h(�) + �(H) = h(�) + �(H) = P(H)

.

Gibbs distribution

This formalism allows to handle the spatio-temporal case

H(!D0 ) =NXi=1

hi!i (0) +NX

i ;j=1

J(0)ij !i (0)!j(0)

+NX

i ;j=1

J(1)ij !i (0)!j(1)

+NX

i ;j ;k=1

J(2)ijk !i (0)!j(1)!k(2) + : : :

even numerically.J.C. Vasquez, A. Palacios, O. Marre, M.J. Berry II, B. Cessac, J. Physiol. Paris, , Vol 106, Issues 34, (2012).

H. Nasser, O. Marre, and B. Cessac, J. Stat. Mech. (2013) P03006.

H. Nasser, B. Cessac, Entropy (2014), 16(4), 2244-2277.

Gibbs distribution

This formalism allows to handle the spatio-temporal case

H(!D0 ) =NXi=1

hi!i (0) +NX

i ;j=1

J(0)ij !i (0)!j(0)

+NX

i ;j=1

J(1)ij !i (0)!j(1)

+NX

i ;j ;k=1

J(2)ijk !i (0)!j(1)!k(2) + ?????

even numerically.J.C. Vasquez, A. Palacios, O. Marre, M.J. Berry II, B. Cessac, J. Physiol. Paris, , Vol 106, Issues 34, (2012).

H. Nasser, O. Marre, and B. Cessac, J. Stat. Mech. (2013) P03006.

H. Nasser, B. Cessac, Entropy (2014), 16(4), 2244-2277.

Two small problems.

Exponential number of possible terms.

Contrarily to what happens usually in physics, we do not knowwhat should be

the right potential.

Can we have a reasonable idea of what could be the spike statisticsby studying a neural network model ?

An Integrate and Fire neural network model with chemicaland electric synapses


R.Cofr�e,B. Cessac: "Dynamics and spike trains statistics in conductance-based Integrate-and-Fire neural networks

with chemical and electric synapses", Chaos, Solitons and Fractals, 2013.


Sub-threshold dynamics:

CkdVk

dt= �gL;k(Vk � EL)

�Xj

gkj(t; !)(Vk � Ej)

�Xj

�gkj (Vk � Vj)

+i(ext)k (t) + �B�k(t)



CkdVk


�Xj


�Xj

�gkj (Vk � Vj)


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.5 1 1.5 2 2.5 3 3.5 4

PSP

t

t=1t=1.2t=1.6

t=3g(x)



CkdVk


�Xj


�Xj

�gkj (Vk � Vj)


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.5 1 1.5 2 2.5 3 3.5 4

PSP

t

g(x)



CkdVk


�Xj


�Xj

�gkj (Vk � Vj)


Sub-threshold regime

CdV

dt+�G (t; !)� G

�V = I (t; !);

Gkl(t; !) =

24 gL;k +

NXj=1

gkj(t; !)

35 �kl

def= gk(t; !)�kl :

I (t; !) = I (cs)(t; !) + I (ext)(t) + I (B)(t)

I(cs)k (t; !) =

Xj

Wkj�kj(t; !); Wkjdef= GkjEj :

Sub-threshold regime

8<:

dV = (�(t; !)V + f (t; !))dt + �BcINdW (t);

V (t0) = v ;

�(t; !) = C�1�G � G (t; !)

�

f (t; !) = C�1I (cs)(t; !) + C�1I (ext)(t)

Homogeneous Cauchy problem

�dV (t;!)

dt= �(t; !)V (t; !);

V (t0) = v ;

Theorem

�(t; !) square matrix with bounded elements.

M0(t0; t; !) = IN

Mk(t0; t; !) = IN +

Z t

t0

�(s; !)Mk�1(s; t)ds; t � t1;

converges uniformly in [t0; t1].Brockett, R. W., "Finite Dimensional Linear Systems",John Wiley and Sons, 1970.

Flow

�(t0; t; !)def= lim

k!1Mk(t0; t; !)


If �(t; !) and �(s; !) commute

�(t0; t; !) =1Xk=0

1

k!(

Z t

t0

�(s; !)ds)k = eR tt0�(s;!)ds



�(t0; t; !) =1Xk=0

1

k!(

Z t

t0

�(s; !)ds)k = eR tt0�(s;!)ds

This holds only in two cases :

G = 0;

�(t0; t; !) = e� 1

c

R tt0G(s;!)ds

B. Cessac, J. Math. Neuroscience, 2011.



�(t0; t; !) =1Xk=0

1

k!(

Z t

t0

�(s; !)ds)k = eR tt0�(s;!)ds

This holds only in two cases :

G = 0;

�(t0; t; !) = e� 1

c

R tt0G(s;!)ds

B. Cessac, J. Math. Neuroscience, 2011.

G (t; !) = �(t; !)IN


In general:

�(t0; t; !) = IN++1Xn=1

XX1 = ( B; A(s1; !) )X2 = ( B; A(s2; !) )

: : :

Xn = ( B; A(sn; !) )

Z t

t0

� � �Z sn�1

t0

nYk=1

Xk ds1 � � � dsn :

B = C�1G ; A(t; !) = �C�1G (t; !)

Exponentially bounded ow

De�nition: An exponentially bounded ow is a two parameter(t0; t) family f�(t0; t; !)gt�t0 of ows such that, 8! 2 :

1 �(t0; t0; !) = IN and �(t0; t; !)�(t; s; !) = �(t0; s; !)whenever t0 � t � s;

2 For each v 2 RN and ! 2 , (t0; t)! �(t0; t; !)v iscontinuous for t0 � t;

3 There is M > 0 and m > 0 such that :

jj�(s; t; !)jj � Me�m(t�s); s � t: (1)


Proposition

Let �1 be the largest eigenvalue of �G. If:

�1 < gL;

then the ow � in our model has the exponentially bounded ow

property.

Remark The typical electrical conductance values are of order 1nano-Siemens, while the leak conductance of retinal ganglion cellsis of order 50 micro-Siemens. Therefore, this condition is

compatible with the biophysical values of conductances in the

retina.


Theorem

If �(t0; t; !) is an exponentially bounded ow , there is a unique

strong solution for t � t0 given by:

V (t0; t; !) = �(t0; t; !)v+

Z t

t0

�(s; t; !)f (s; !)ds+�B

c

Z t

t0

�(s; t; !)dW (s):

R. Wooster, "Evolution systems of measures for non-autonomous stochastic di�erential equations with Levy noise",

Communications on Stochastic Analysis, vol 5, 353-370, 2011

Membrane potential decomposition

V (t; !) = V (d)(t; !) + V (noise)(t; !);

V (d)(t; !) = V (cs)(t; !) + V (ext)(t; !);

V (cs)(t; !) =1

c

Z t

�1�(s; t; !)�(s; !) I (cs)(s; !)ds;

V (ext)(t; !) =1

c

Z t

�1�(s; t; !)�(s; !) I (ext)(s; !)ds;

V (noise)(t; !) =�B

c

Z t

�k (t;!)�(s; t; !) dW (s):

Transition probabilities

Pb: to determine P�!(n)

��!n�1�1

�

Fix !, n and t < n. Set:

b�k(t; !) = � � V(d)k (t; !); (1)

Neuron k emits a spike at integer time n (!k(n) = 1) if:

9t 2 [n � 1; n]; V(noise)k (t; !) = b�k(t; !):

"First passage" problem, in N dimension, with a time

dependent boundary b�k(t; !). (general form unknown).

Transition probabilities

Pb: to determine P�!(n)

��!n�1�1

�Fix !, n and t < n. Set:

b�k(t; !) = � � V(d)k (t; !); (1)

Neuron k emits a spike at integer time n (!k(n) = 1) if:

9t 2 [n � 1; n]; V(noise)k (t; !) = b�k(t; !):

"First passage" problem, in N dimension, with a time

dependent boundary b�k(t; !). (general form unknown).

Conditional probability

Without electric synapses the probability of !(n) conditionally to!n�1�1 can be approximated by:

P�!(n)

��!n�1�1

�=

NYk=1

P�!k(n)

��!n�1�1

�;

with P�!k(n)

��!n�1�1

�=

!k(n)� (Xk(n � 1; !)) + (1� !k(n)) (1� � (Xk(n � 1; !))) ;

where

Xk(n � 1; !) =� � V

(det)k (n � 1; !)

�k(n � 1; !);

and

�(x) =1p2�

Z +1

x

e�u2

2 du:

Conditional probability

�(!) = logP�!(n)

��!n�1�1

�de�nes a (in�nite range)

normalized potential de�ning a unique Gibbs distribution.

It depends explicitly on networks parameters and external

stimulus.

Its de�nition holds for a time-dependent stimulus (nonstationary).

It is similar to the so-called Generalized Linear Model used forretina analysis, although with a more complex structure.

The general form (with electric synapses) is yet unknown.

Back to our second "small" problem

Is there a Maximum Entropy potential corresponding to � (inthe stationary case) ?


One can make a Taylor expansion of �(!).


Using !i (n)k = !i (n); k � 1 one ends up with a potential of the

form:

�(!) =NXi=1

hi!i (0) +NX

i ;j=1

J(0)ij !i (0)!j(0) + : : :


The expansion is in�nite although one can approximate the in�niterange potential � by a �nite range approximation (�nite memory),

giving rise to a �nite expansion.


The coe�cients of the expansion are non linear functions of thenetwork parameters and stimulus.

They are therefore somewhat redundant.


Rodrigo Cofr�e, Bruno Cessac, "Exact computation of the maximum-entropy potential of spiking neural-network

models",Phys. Rev. E 89, 052117.

Given a set of stationary transition probabilities P�!(D)

��!D�1

0

�> 0

there is a unique (up to a constant) Maximum Entropy potential, written

as a linear combination of spike interactions terms with a minimal

number of terms (normal form). This potential can be explicitly (and

algorithmically) computed.

Hints: Using variable change one can eliminate terms in thepotential ("normal" form).

The construction is based on equivalence between Gibbs potentials(cohomology) and periodic orbits expansion.


However, there is still a number of terms growing exponentially

with the number of neurons and the memory depth.

These terms are generically non zero.

Back to the retina

Neuromimetic models have typically O(N2) parameters whereN is the number of neurons.

The equivalent MaxEnt potential has generically a number ofparameters growing exponentially with N, non linear andredundant functions of the network parameters (synapticweights, stimulus).

)Intractable determination of parameters;

Stimulus dependent parameters;

Over�tting.

BUT Real neural networks are not generic

Back to the retina





Over�tting.

BUT

Real neural networks are not generic

Back to the retina





Over�tting.

BUT Real neural networks are not generic

Back to the retina

MaxEnt approach might be useful if there is some hidden law ofnature/ symmetry which cancels most terms in the expansion.

Acknowledgment

Neuromathcomp team

Rodrigo Cofr�e (pHd, September 2014)

Dora Karvouniari (M2)

Pierre Kornprobst (CR1 INRIA)

Slim Kraria (IR)

Gaia Lombardi (M2! Paris)

Hassan Nasser (pHd! Startup)

Daniela Pamplona (PostDoc)

Geo�rey Portelli (Post Doc)

Vivien Robinet (Post Doc! MCF Kourou)

Horacio Rostro (pHd! Docent Mexico)

Wahiba Taouali (Post Doc! Post Doc INTMarseille)

Juan-Carlos Vasquez (pHd! Post Doc Bogota)

Princeton University

Michael J. Berry II

ANR KEOPS

Maria-Jos�e Escobar (CN Valparaiso)

Adrian Palacios (CN Valparaiso)

Cesar Ravelo (CN Valparaiso)

Thierry Vi�eville (INRIA Mnemosyne)

Renvision FP7 project

Luca Bernondini (IIT Genova)

Matthias Hennig (Edinburgh)

Alessandro Maccionne (IIT Genova)

Evelyne Sernagor (Newcastle)

Institut de la Vision

Olivier Marre

Serge Picaud


Can we hear the shape of a Maximum entropy potential

Two distinct potentials H(1);H(2) of range R = D + 1 correspondto the same Gibbs distribution (are \equivalent"), if and only ifthere exists a range D function f such that (Chazottes-Keller(2009)):

H(2)�!D0

�= H(1)

�!D0

�� f

�!D�10

�+ f

�!D1

�+�; (2)

where � = P(H(2))� P(H(1)).


Summing over periodic orbits we get rid of the function f

RXn=1

�(!�nl1) =RXn=1

H�(!�nl1)� RP(H�); (3)

We eliminate equivalent constraints.


Conclusion

Given a set of transition probabilities Ph!(D)

��!D�10

i> 0 there

is a unique, up to a constant, MaxEnt potential, written as a linearcombination of constraints (average of spike events) with aminimal number of terms. This potential can be explicitly (andalgorithmically) computed.

Statistical analysis of spike trains in neuronal networks · 2017. 1. 28. · Statistical analysis...

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