Statistics of spike trains: A dynamical systems
perspective.
Bruno Cessac, Horacio Rostro, Juan­Carlos Vasquez, Thierry Viéville
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.
●
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.
●
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...
Spontaneous activity;
● Response to external stimuli ;
● Response to excitations from
other neurons...
●
Multiples scales.
●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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Neural response to some stimulus ?
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Neural response to some stimulus ?
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
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).
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
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).
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
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).
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
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).
The probability of
occurrence of R depends on
the stimulus and on system
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Neural response to some stimulus ?
How to compute P(R|S) ?
The probability of
occurrence of R depends on
the stimulus and on system
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Neural response to some stimulus ?
How to compute P(R|S) ?
Sample averaging.
The probability of
occurrence of R depends on
the stimulus and on system
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Neural response to some stimulus ?
How to compute P(R|S) ?
Time averaging.
The probability of
occurrence of R depends on
the stimulus and on system
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Neural response to some stimulus ?
How to compute P(R|S) ?
Time averaging.
The probability of
occurrence of R depends on
the stimulus and on system
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Which type of probability distribution can we expect ?
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Which type of probability distribution can we expect ?
(Inhomogeneous) Poisson, Cox
process, “Ising”-like
distribution .... ?
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Which type of probability distribution can we expect ?
(Inhomogeneous) Poisson, Cox
process, “Ising”-like
distribution .... ?
This certainly depends on
what you measure.
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Which type of probability distribution can we expect ?
Can we answer this question in simple
models ?
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
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...
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
Neural network activity.
Spike generation.
I-F models are
(maybe) good enough.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
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
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
Neural network activity.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Spike generation.
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
Neural network activity.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Spike generation.
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
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 time.
Neural network activity.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Spike generation.
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
Discrete time version.
There is a minimal time scale  t below
which spikes are indistinguishable.
Conductances depend on past spikes
over a finite time.
Neural network activity.
Spike generation.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Generic dynamics. B. Cessac, T. Viéville, Front. Comput. Neurosci. 2:2
(2008).
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
Discrete time version.
There is a minimal time scale  t below
which spikes are indistinguishable.
Conductances depend on past spikes
over a finite time.
Neural network activity.
Spike generation.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Generic dynamics. B. Cessac, T. Viéville, Front. Comput. Neurosci. 2:2
(2008).
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
There is a weak form of initial
condition sensitivity.
Attractors are generically stable
period orbits.
Discrete time version.
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 time.
Neural network activity.
Spike generation.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
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).
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
There is a weak form of initial
condition sensitivity.
Attractors are generically stable
period orbits.
Discrete time version.
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 time.
Neural network activity.
Spike generation.
Spontaneous activity
activity;
● Response to external stimuli ;
● Response to excitations from
other neurons
neurons...
●
Multiples scales.
●Non linear and collective dynamics.
● Adaptation.
● Interwoven evolution.
●
Generic dynamics. B. Cessac, T. Viéville, Front. Comput. Neurosci. 2:2
(2008).
Spikes trains provide a
symbolic coding.
gIF Models
M. Rudolph, A. Destexhe,
Neural Comput., 18, 2146–2210 (2006).
There is a weak form of initial
condition sensitivity.
Attractors are
Adding
noise
::
orbits.
Addingperiod
noise
generically stable
Discrete time version.
To a given “input” one can
associate a finite number of
The number
of stable periodic orbit
- -renders
dynamics
“ergodic”;
periodic orbits
(depending
renders dynamics
diverges“ergodic”;
exponentially with the
on the initial condition).
number of neurons.
- -provides
providesaarich
richvariety
varietyof
ofspike
spiketrain
trainstatistics.
statistics.
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 time.
The MACACC Project
Modelling Cortical Activity and Analysing the
Brain Neural Code
http://www-sop.inria.fr/neuromathcomp/contracts
Extract canonical forms of probability distribution on raster plots from the study of gIF models with noise
=>
Statistical models.
The MACACC Project
Modelling Cortical Activity and Analysing the
Brain Neural Code
http://www-sop.inria.fr/neuromathcomp/contracts
Extract canonical forms of probability distribution on raster plots from the study of gIF models with noise
=>
Statistical models.
The MACACC Project
Infer algorithmic methods to obtain a statistical model from empirical data,
Modelling Cortical Activity and Analysing the
evaluate the finite sampling effects, Brain Neural Code
http://www-sop.inria.fr/neuromathcomp/contracts
find a quantitative and tractable way to discriminate between several statistical models.
Extract canonical forms of probability distribution on raster plots from the study of gIF models with noise
=>
Statistical models.
The MACACC Project
Infer algorithmic methods to obtain a statistical model from empirical data,
Modelling Cortical Activity and Analysing the
evaluate the finite sampling effects, Brain Neural Code
http://www-sop.inria.fr/neuromathcomp/contracts
find a quantitative and tractable way to discriminate between several statistical models.
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
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
Examples
Firing rate of neuron i :
Statistical model
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Statistical model
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 …:
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Statistical model
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 …:
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Gibbs measures.
Statistical model
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 …:
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Gibbs measures.
Maximising the statistical entropy under
the constraints =C, α = 1...K.
Statistical model
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 …:
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Gibbs measures.
Maximising the statistical entropy under
the constraints =C, α = 1...K.
Statistical model
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 …:
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Gibbs measures.
Maximising the statistical entropy under
the constraints =C, α = 1...K.
Statistical model
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
Gibbs measures.
Maximising the statistical entropy under
the constraints =C, α = 1...K.
An ergodic probability measure  on the set of
admissible raster plots is called a statistical
model if, for all …:
Gibbs distribution with potential
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Statistical model
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
Gibbs measures.
Maximising the statistical entropy under
the constraints =C, α = 1...K.
An ergodic probability measure  on the set of
admissible raster plots is called a statistical
model if, for all …:
Gibbs distribution with potential
Examples
Firing rate of neuron i :
Spike coincidence
Prob[j fires at some time t and i fires at t+]
Statistical model
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
Gibbs measures.
Maximising the statistical entropy under
the constraints =C, α = 1...K.
An ergodic probability measure  on the set of
admissible raster plots is called a statistical
model if, for all …:
Gibbs distribution with potential
Examples
Firing rate of neuron i :
Bernoulli distribution
Spike coincidence
« Ising like » distribution
Prob[j fires at some time t and i fires at t+]
E. Schneidman, M.J. Berry, R.
Segev, W. Bialek, Nature,440,
(2006)
Extract canonical forms of probability distribution on raster plots from the study of gIF models with noise
=>
Statistical models.
The MACACC Project
Infer algorithmic methods to obtain a statistical model from empirical data,
Modelling Cortical Activity and Analysing the
evaluate the finite sampling effects, Brain Neural Code
http://www-sop.inria.fr/neuromathcomp/contracts
find a quantitative and tractable way to discriminate between several statistical models.
Apply these methods to biological data.
Statistical model
Gibbs measures.
Discrete Time IF models with noiseMaximising
have Gibbs
measures.
the statistical entropy under
(Cessac, in preparation)
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
the constraints =C, α = 1...K.
An ergodic probability measure  on the set of
admissible raster plots is called a statistical
model if, for all …:
Gibbs distribution with potential
Examples
Firing rate of neuron i :
Bernoulli distribution
Spike coincidence
« Ising like » distribution
Prob[j fires at some time t and i fires at t+]
E. Schneidman, M.J. Berry, R.
Segev, W. Bialek, Nature,440,
(2006)
Statistical model
Gibbs measures.
Discrete Time IF models with noiseMaximising
have Gibbs
measures.
the statistical entropy under
(Cessac, in preparation)
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
the constraints =C, α = 1...K.
An ergodic probability
measure estimation
on the set of of the Gibbs potential.
Parametric
admissible raster plots is called
a statistical
(Vasquez,
Cessac, Viéville, submitted).
model if, for all …:
Gibbs distribution with potential
Examples
Firing rate of neuron i :
Bernoulli distribution
Spike coincidence
« Ising like » distribution
Prob[j fires at some time t and i fires at t+]
E. Schneidman, M.J. Berry, R.
Segev, W. Bialek, Nature,440,
(2006)
Statistical model
Gibbs measures.
Discrete Time IF models with noiseMaximising
have Gibbs
measures.
the statistical entropy under
(Cessac, in preparation)
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
the constraints =C, α = 1...K.
An ergodic probability
measure estimation
on the set of of the Gibbs potential.
Parametric
admissible raster plots is called
a statistical
(Vasquez,
Cessac, Viéville, submitted).
model if, for all …:
Explicit computation of the topological pressure from spectral
Gibbs distribution with potential
methods.
Computation of the K-L divergence between empirical measure and
Gibbs distribution => Comparison between statistical models.
Examples
Firing rate of neuron i :
Bernoulli distribution
Spike coincidence
« Ising like » distribution
Prob[j fires at some time t and i fires at t+]
E. Schneidman, M.J. Berry, R.
Segev, W. Bialek, Nature,440,
(2006)
Statistical model
Gibbs measures.
Discrete Time IF models with noiseMaximising
have Gibbs
measures.
the statistical entropy under
(Cessac, in preparation)
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
the constraints =C, α = 1...K.
An ergodic probability
measure estimation
on the set of of the Gibbs potential.
Parametric
admissible raster plots is called
a statistical
(Vasquez,
Cessac, Viéville, submitted).
model if, for all …:
Explicit computation of the topological pressure from spectral
Gibbs distribution with potential
methods.
Computation of the K-L divergence between empirical measure and
Gibbs distribution => Comparison between statistical models.
Examples
Control of finite sample corrections.
Firing rate of neuron i :
Bernoulli distribution
Spike coincidence
« Ising like » distribution
Prob[j fires at some time t and i fires at t+]
E. Schneidman, M.J. Berry, R.
Segev, W. Bialek, Nature,440,
(2006)
Statistical model
Gibbs measures.
Discrete Time IF models with noiseMaximising
have Gibbs
measures.
the statistical entropy under
(Cessac, in preparation)
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
the constraints =C, α = 1...K.
An ergodic probability
measure estimation
on the set of of the Gibbs potential.
Parametric
admissible raster plots is called
a statistical
(Vasquez,
Cessac, Viéville, submitted).
model if, for all …:
Explicit computation of the topological pressure from spectral
Gibbs distribution with potential
methods.
Computation of the K-L divergence between empirical measure and
Gibbs distribution => Comparison between statistical models.
Examples
Firing rate of neuron i :
Control of finite sample corrections.
http://enas.gforge.inria.fr/
Bernoulli distribution
Spike coincidence
« Ising like » distribution
Prob[j fires at some time t and i fires at t+]
E. Schneidman, M.J. Berry, R.
Segev, W. Bialek, Nature,440,
(2006)
Statistical model
Gibbs measures.
Discrete Time IF models with noiseMaximising
have Gibbs
measures.
the statistical entropy under
(Cessac, in preparation)
Fix ,  = 1 ...K, a set of observables (prescribed
quantities whose average C is known).
the constraints =C, α = 1...K.
An ergodic probability
measure estimation
on the set of of the Gibbs potential.
Parametric
admissible raster plots is called
a statistical
(Vasquez,
Cessac, Viéville, submitted).
model if, for all …:
Explicit computation of the topological pressure from spectral
Gibbs distribution with potential
methods.
Computation of the K-L divergence between empirical measure and
Gibbs distribution => Comparison between statistical models.
Examples
Control of finite sample corrections.
http://enas.gforge.inria.fr/
Firing rate of neuron i :
Application
Spike coincidence
Bernoulli distribution
to the characterization of ganglion cells
(with A. Palacios, C.Neurociencia, Valparaiso)
« Ising like » distribution
E. Schneidman,
M.J. Berry, R.
Application to spike train analysis in motor cortical
neurons
Prob[j fires at some time (F.
t and
i fires at t+]
Grammont
Segev, W. Bialek, Nature,440,
LJAD, A. Riehle, INCM)
(2006)
The knowledge of prescribed
observables average fixes the
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
observables average fixes the
statistical
model.?
Which
observables
Which observables ?
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).
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.
●
Synaptic weight evolution.
Synaptic weight evolution.
Synaptic weight evolution.
Synaptic weight evolution.
Example
Synaptic weight evolution.
Convergence
Example
Synaptic weight evolution.
Dynamics and statistics evolution
Changing synaptic weights
changing membrane potential dynamics
Convergence
Example
changing raster plots dynamics
and statistics
Synaptic weight evolution.
Dynamics and statistics evolution
Changing synaptic weights
changing membrane potential dynamics
Convergence
changing raster plots dynamics
and statistics
Synaptic weight evolution.
Dynamics and statistics evolution
Changing synaptic weights
changing membrane potential dynamics
Convergence
changing raster plots dynamics
and statistics
Variational principle.
Synaptic weight evolution.
Dynamics and statistics evolution
Changing synaptic weights
changing membrane potential dynamics
Convergence
changing raster plots dynamics
and statistics
Variational principle.
There is a functional F that decreases whenever
synaptic weights change smoothly (regular periods).
●
Synaptic weight evolution.
Dynamics and statistics evolution
Changing synaptic weights
changing membrane potential dynamics
Convergence
changing raster plots dynamics
and statistics
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.
Dynamics and statistics evolution
Changing synaptic weights
changing membrane potential dynamics
Convergence
changing raster plots dynamics
and statistics
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
observables average fixes the
statistical
model.?
Which
observables
Which observables ?
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 synaptic adaptation mechanism
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