Statistics of spike trains: A dynamical systems perspective. · Spikes: Exploring the Neural Code. F Rieke, D Warland, R de Ruyter van Steveninck & W Bialek (MIT Press, Cambridge,

Bruno Cessac, Horacio Rostro,

JuanCarlos Vasquez, Thierry Viéville

Statistics of spike trains: A dynamical systems Statistics of spike trains: A dynamical systems perspective.perspective.

●Multiples scales. ●Non linear and collective dynamics.● Adaptation.● Interwoven evolution.

Neural network activity.

● Spontaneous activity;● Response to external stimuli ;● Response to excitations from

other neurons...



other neurons...




other neurons...



i(t)=1 if i fires at t

=0 otherwise.

Spike generation.

~A raster plot is a sequence

={i(t)}, i=1...N, t=1...

● Spontaneous activitySpontaneous activity;● Response to external stimuliResponse to external stimuli ;● Response to excitations from Response to excitations from other neuronsother neurons...

●Multiples scales. ●Non linear and collective dynamics.Non linear and collective dynamics.● Adaptation.● Interwoven evolution.

Neural network activity. Spike generation.

Neural response to some stimulus ?Neural response to some stimulus ?


=0 otherwise.


={i(t)}, i=1...N, t=1...



● Definite succession of spikes during a definite time period.



=0 otherwise.


={i(t)}, i=1...N, t=1...





●Statistical “coding”.



=0 otherwise.


={i(t)}, i=1...N, t=1...


Spikes: Exploring the Neural Code. F Rieke, D Warland, R de Ruyter van Steveninck & W Bialek (MIT Press,

Cambridge, 1997).

http://mitpress.mit.edu/catalog/item/default.asp?sid=3CA2950F-5938-4E41-BCAE-450652BF2660&ttype=2&tid=3977





=0 otherwise.


={i(t)}, i=1...N, t=1...





Cambridge, 1997).






=0 otherwise.


={i(t)}, i=1...N, t=1...





Cambridge, 1997).




The probability of The probability of occurrence of R depends on occurrence of R depends on the stimulus and on system the stimulus and on system

parameters.parameters.



=0 otherwise.


={i(t)}, i=1...N, t=1...





Cambridge, 1997).




How to compute P(R|S) ?How to compute P(R|S) ?



=0 otherwise.


={i(t)}, i=1...N, t=1...







Sample averaging.Sample averaging.



=0 otherwise.


={i(t)}, i=1...N, t=1...







Time averaging.Time averaging.



=0 otherwise.


={i(t)}, i=1...N, t=1...







Time averaging.Time averaging.



=0 otherwise.


={i(t)}, i=1...N, t=1...




Spike generation.Neural network activity.




=0 otherwise.


={i(t)}, i=1...N, t=1...

Which type of probability distribution can we expect ?Which type of probability distribution can we expect ?





=0 otherwise.


={i(t)}, i=1...N, t=1...


(Inhomogeneous) Poisson, Cox process, “Ising”-like

distribution .... ?





=0 otherwise.


={i(t)}, i=1...N, t=1...


This certainly depends on This certainly depends on what you measure.what you measure.

(Inhomogeneous) Poisson, Cox process, “Ising”-like

distribution .... ?





=0 otherwise.


={i(t)}, i=1...N, t=1...


Can we answer this question in simple Can we answer this question in simple models ?models ?

Spike generation.

gIF Models M. Rudolph, A. Destexhe, Neural Comput., 18, 2146–2210 (2006).





=0 otherwise.


={i(t)}, i=1...N, t=1...

Spike generation.






=0 otherwise.


={i(t)}, i=1...N, t=1...

Spike generation.

I-F models are I-F models are (maybe) good enough.(maybe) good enough.

Approximating real raster plots from orbits

of IF models with suitable parameters.

R. Jolivet, T. J. Lewis, W. Gerstner (2004)J. Neurophysiology 92: 959-976





Spike generation.





Spike generation.


There is a minimal time scale t below which spikes are indistinguishable.

Conductances depend on past spikes over a finite finite time.




Spike generation.







Discrete time version.Discrete time version.

Spike generation.


Generic dynamics. B. Cessac, T. Viéville, Front. Comput. Neurosci. 2:2 (2008).







Spike generation.



There is a weak form of initial condition sensitivity.

Attractors are generically stable period orbits.

The number of stable periodic orbit diverges exponentially with the number of neurons.

Depending on parameters (synaptic weights, input current), periods can be quite large (well beyond any accessible computational time).








Spike generation.



Spikes trains provide a symbolic coding.

To a given “input” one can associate a finite number of periodic orbits (depending

on the initial condition).











Spike generation.



Spikes trains provide a symbolic coding.

To a given “input” one can associate a finite number of periodic orbits (depending

on the initial condition).









Adding noise :Adding noise :

- renders dynamics “ergodic”;- renders dynamics “ergodic”;

- provides a rich variety of spike train statistics.- provides a rich variety of spike train statistics.

Adding noise :Adding noise :

- renders dynamics “ergodic”;- renders dynamics “ergodic”;

- provides a rich variety of spike train statistics.- provides a rich variety of spike train statistics.


The MACACC Project The MACACC Project

Modelling Cortical Activity and Analysing the Modelling Cortical Activity and Analysing the Brain Neural CodeBrain Neural Code

http://www-sop.inria.fr/nhttp://www-sop.inria.fr/neuromathcomp/contractseuromathcomp/contracts

http://www-sop.inria.fr/n




Extract canonical forms of probability distribution on raster plots from the study of Extract canonical forms of probability distribution on raster plots from the study of gIF models with noisegIF models with noise

=>=>

Statistical models.Statistical models.






=>=>


Infer algorithmic methods to obtain a statistical model from empirical data,Infer algorithmic methods to obtain a statistical model from empirical data,

evaluate the finite sampling effects, evaluate the finite sampling effects,

find a quantitative and tractable way to discriminate between several statistical find a quantitative and tractable way to discriminate between several statistical models.models.






=>=>





Apply these methods to biological data.Apply these methods to biological data.


Statistical model

Statistical model

Fix , = 1 ...K, a set of observables (prescribed

quantities whose average C is known).

Statistical model

Examples

Firing rate of neuron i :



Statistical model

Examples


Prob[j fires at some time t and i fires at t+]

Spike coincidence



Statistical model

Examples

An ergodic probability measure on the set of admissible raster plots is called a statistical model if, for all … :



Spike coincidence



Statistical model Gibbs measures.

Examples



Spike coincidence





Examples



Spike coincidence

Maximising the statistical entropy under Maximising the statistical entropy under the constraints the constraints

=C=C

, , αα = 1...K. = 1...K.




Statistical model

Examples



Spike coincidence

Gibbs measures.





=C=C

, , αα = 1...K. = 1...K.


Examples



Spike coincidence





=C=C

, , αα = 1...K. = 1...K.


Examples



Spike coincidence





=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential


Examples



Spike coincidence





=C=C

, , αα = 1...K. = 1...K.



Examples



Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

E. Schneidman, M.J. Berry, R. Segev, W. Bialek, Nature,440,

(2006)





=C=C

, , αα = 1...K. = 1...K.






=>=>





Apply these methods to biological data.Apply these methods to biological data.



Examples



Spike coincidence




(2006)





=C=C

, , αα = 1...K. = 1...K.


Discrete Time IF models with noise have Gibbs measures.Discrete Time IF models with noise have Gibbs measures.(Cessac, in preparation)(Cessac, in preparation)


Examples



Spike coincidence




(2006)





=C=C

, , αα = 1...K. = 1...K.


Parametric estimation of the Gibbs potential.Parametric estimation of the Gibbs potential.(Vasquez, Cessac, Viéville, submitted).(Vasquez, Cessac, Viéville, submitted).



Examples



Spike coincidence




(2006)





=C=C

, , αα = 1...K. = 1...K.



Explicit computation of the topological pressure from spectral Explicit computation of the topological pressure from spectral methods.methods.

Computation of the K-L divergence between empirical measure and Computation of the K-L divergence between empirical measure and Gibbs distribution => Comparison between statistical models.Gibbs distribution => Comparison between statistical models.



Examples



Spike coincidence




(2006)





=C=C

, , αα = 1...K. = 1...K.





Control of finite sample corrections.Control of finite sample corrections.



Examples



Spike coincidence




(2006)





=C=C

, , αα = 1...K. = 1...K.






http://enas.gforge.inria.fr/http://enas.gforge.inria.fr/


http://enas.gforge.inria.fr/


Examples



Spike coincidence




(2006)





=C=C

, , αα = 1...K. = 1...K.






http://enas.gforge.inria.fr/http://enas.gforge.inria.fr/

Application to the characterization of ganglion cells(with A. Palacios, C.Neurociencia, Valparaiso)

Application to spike train analysis in motor cortical neurons(F. Grammont LJAD, A. Riehle, INCM)



The knowledge of prescribed The knowledge of prescribed observables average fixes the observables average fixes the

statistical model.statistical model.

B. Cessac, H. Rostro, J.C. Vasquez, T. Viéville , “How Gibbs distributions may naturally arise from synaptic adaptation

mechanisms” , J. Stat. Phys,136, (3), 565-602 (2009).





Which observables ?Which observables ?Which observables ?Which observables ?



other neurons...

●Multiples scales. ●Non linear and collective dynamics.● Adaptation.Adaptation.● Interwoven evolution.

Synaptic weight evolution.Synaptic weight evolution.




Example


Example

Convergence

Synaptic weight evolution.Synaptic weight evolution. Dynamics and statistics evolutionDynamics and statistics evolution

Changing synaptic weights

changing membrane potential dynamics

changing raster plots dynamicsand statistics

Example

Convergence


Convergence

Dynamics and statistics evolutionDynamics and statistics evolution





Convergence





Variational principle.Variational principle.


Convergence






● There is a functional F that decreases whenever synaptic weights change smoothly (regular periods).


Convergence







● Regular periods are separated by sharp variations of synaptic weights (phase transitions).


Convergence







● Regular periods are separated by sharp variations of synaptic weights (phase transitions).

● If the synaptic adaptation rule “converges” then the corresponding statistical model is a Gibbs measure whose potential contains the term:





Which observables ?Which observables ?Which observables ?Which observables ?

The synaptic adaptation mechanism The synaptic adaptation mechanism fixes the form of the potential.fixes the form of the potential.



How to extract the statistical model from empirical data ?


Application to real data

http://www-sop.inria.fr/odyssee/contracts/MACACC/macacc.html

Documents

Statistics of spike trains: A dynamical systems perspective. · Spikes: Exploring the Neural Code. F Rieke, D Warland, R de Ruyter van Steveninck & W Bialek (MIT Press, Cambridge,