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...

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

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

other neurons...

Neural network activity.

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

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

other neurons...

Neural network activity.

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

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 ?

i(t)=1 if i fires at t

=0 otherwise.

~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.

● Definite succession of spikes during a definite time period.

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

● 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.

● Definite succession of spikes during a definite time period.

●Statistical “coding”.

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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

Cambridge, 1997).

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

● Definite succession of spikes during a definite time period.

●Statistical “coding”.

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

Cambridge, 1997).

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

● Definite succession of spikes during a definite time period.

●Statistical “coding”.

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

Cambridge, 1997).

● 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.

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.

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

● Definite succession of spikes during a definite time period.

●Statistical “coding”.

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

Cambridge, 1997).

● 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.

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

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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.

● 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.

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

Sample averaging.Sample averaging.

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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.

● 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.

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

Time averaging.Time averaging.

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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.

● 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.

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

Time averaging.Time averaging.

Neural network activity. Spike generation.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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.

Spike generation.Neural network activity.

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

Spike generation.Neural network activity.

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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

distribution .... ?

Spike generation.Neural network activity.

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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

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

distribution .... ?

Spike generation.Neural network activity.

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

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

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).

Neural network activity.

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

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

Spike generation.

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

Neural network activity.

● 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.

i(t)=1 if i fires at t

=0 otherwise.

~A raster plot is a sequence

={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

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

Neural network activity.

● 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.

Spike generation.

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

Neural network activity.

● 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.

Spike generation.

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

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

Conductances depend on past spikes over a finite finite time.

Neural network activity.

● 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.

Spike generation.

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

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

Conductances depend on past spikes over a finite finite time.

Neural network activity.

● 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.

Discrete time version.Discrete time version.

Spike generation.

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

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

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

Conductances depend on past spikes over a finite finite time.

Neural network activity.

● 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.

Discrete time version.Discrete time version.

Spike generation.

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

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

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).

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

Conductances depend on past spikes over a finite finite time.

Neural network activity.

● 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.

Discrete time version.Discrete time version.

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

Spike generation.

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

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

Spikes trains provide a symbolic coding.

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

on the initial condition).

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

Conductances depend on past spikes over a finite finite time.

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

Neural network activity.

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).

Discrete time version.Discrete time version.

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

Spike generation.

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

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

Spikes trains provide a symbolic coding.

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

on the initial condition).

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

Conductances depend on past spikes over a finite finite time.

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

Neural network activity.

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).

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.

Discrete time version.Discrete time version.

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

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

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.

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

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.

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

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 :

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

quantities whose average C is known).

Statistical model

Examples

Firing rate of neuron i :

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

Spike coincidence

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

quantities whose average C is known).

Statistical model

Examples

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

Firing rate of neuron i :

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

Spike coincidence

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

quantities whose average C is known).

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

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

quantities whose average C is known).

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

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

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

=C=C

, , αα = 1...K. = 1...K.

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

quantities whose average C is known).

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

Statistical model

Examples

Firing rate of neuron i :

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

Spike coincidence

Gibbs measures.

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

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 Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

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

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

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.

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

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

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.

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

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

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

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.

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

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

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

Statistical model Gibbs measures.

Examples

Firing rate of neuron i :

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

Spike coincidence

Bernoulli distributionBernoulli distribution

« Ising like » distribution« Ising like » distribution

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

(2006)

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

quantities whose average C is known).

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

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

=C=C

, , αα = 1...K. = 1...K.

Gibbs distribution with potentialGibbs distribution with potential

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

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.

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

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)

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

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).

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 ?

Neural network activity.

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

other neurons...

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

Synaptic weight evolution.Synaptic weight evolution.

Synaptic weight evolution.Synaptic weight evolution.

Synaptic weight evolution.Synaptic weight evolution.

Synaptic weight evolution.Synaptic weight evolution.

Example

Synaptic weight evolution.Synaptic weight evolution.

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

Synaptic weight evolution.Synaptic weight evolution.

Convergence

Dynamics and statistics evolutionDynamics and statistics evolution

Changing synaptic weights

changing membrane potential dynamics

changing raster plots dynamicsand statistics

Synaptic weight evolution.Synaptic weight evolution.

Convergence

Dynamics and statistics evolutionDynamics and statistics evolution

Changing synaptic weights

changing membrane potential dynamics

changing raster plots dynamicsand statistics

Variational principle.Variational principle.

Synaptic weight evolution.Synaptic weight evolution.

Convergence

Dynamics and statistics evolutionDynamics and statistics evolution

Changing synaptic weights

changing membrane potential dynamics

changing raster plots dynamicsand statistics

Variational principle.Variational principle.

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

Synaptic weight evolution.Synaptic weight evolution.

Convergence

Dynamics and statistics evolutionDynamics and statistics evolution

Changing synaptic weights

changing membrane potential dynamics

changing raster plots dynamicsand statistics

Variational principle.Variational principle.

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

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

Synaptic weight evolution.Synaptic weight evolution.

Convergence

Dynamics and statistics evolutionDynamics and statistics evolution

Changing synaptic weights

changing membrane potential dynamics

changing raster plots dynamicsand statistics

Variational principle.Variational principle.

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

● 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:

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 ?

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

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).

How to extract the statistical model from empirical data ?

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

Application to real data

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