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Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Estimation of local independence graphs via
Hawkes processes to unravel functional neuronalconnectivity
C. Tuleau-Malot (Nice), V. Rivoirard (Dauphine), N.R. Hansen(Copenhagen),P. Reynaud-Bouret(Nice),
F. Grammont (Nice), T. Bessaih, R. Lambert, N. Leresche(Paris 6)
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Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Contents
1 Biological motivation
2/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Contents
1 Biological motivation
2 Hawkes processes
2/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Contents
1 Biological motivation
2 Hawkes processes
3 Adaptive estimation
2/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Contents
1 Biological motivation
2 Hawkes processes
3 Adaptive estimation
4 Simulations
2/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Contents
1 Biological motivation
2 Hawkes processes
3 Adaptive estimation
4 Simulations
5 Real data analysis
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Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Description of the data (Equipe RNRP de Paris 6)
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Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Data = spike trainsmonkey trained to touch the correct target when illuminated
0.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 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1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.0
time in second
N3
N4
spike = point
4/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Data = spike trainsmonkey trained to touch the correct target when illuminated
0.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 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0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.0
time in second
N3
N4
spike = point→ point processes
4/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation Avec synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation Avec synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
potentiel d’action
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
potentiel d’action
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
potentiel d’action
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Synaptic integration
without synchronisation with synchronisation
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
potentiel d’action
différentes
dendrites
signaux d’entrées
seuil d’excitation
du neurone
potentiel de repos
du neurone
potentiel d’actionpotentiel d’action
différentes
5/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Point processes and conditional intensity
intensity
dNt︸ ︷︷ ︸
=︷ ︸︸ ︷
λ(t) dt︸ ︷︷ ︸
+ noise︸ ︷︷ ︸
Nbr observed points Expected Number Martingalesin [t, t + dt] given the past before t differences
λ(t) = instantaneous frequency= random, depends on previous points
6/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
λ(1)(t) = ν︸︷︷︸
spontaneous
7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
x
h2->1
x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
+∑
m 6= 1,T < t,
pt de Nm
hm→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction interaction 7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
x
h2->1
x
x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
+∑
m 6= 1,T < t,
pt de Nm
hm→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction interaction 7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
x
h2->1
x
x
h3->1 x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
+∑
m 6= 1,T < t,
pt de Nm
hm→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction interaction 7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
x
h2->1
x
x
h3->1 x x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
+∑
m 6= 1,T < t,
pt de Nm
hm→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction interaction 7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Multivariate Hawkes processes
t
N1
N2
N3
1->1h
x
x
h2->1
x
x
h3->1 x x
x
λ(1)(t) = ν︸︷︷︸
+∑
T < tpt de N1
h1→1(t − T )
︸ ︷︷ ︸
+∑
m 6= 1,T < t,
pt de Nm
hm→1(t − T )
︸ ︷︷ ︸
spontaneous self-interaction interaction 7/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Link Graphical model of local independence
(Didelez (2008))
1
23
1
23
1
23
8/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Link Graphical model of local independence
(Didelez (2008))
1
23
1
23
1
23
→ (?) functional connectivity graphs in Neurosciences,but also useful in sismology, genomics, marketing, finance,epidemiology ....
8/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
More formallyOnly excitation (all the h
(r)ℓ are positive): for all r ,
λ(r)(t) = νr +
M∑
ℓ=1
∫ t−
−∞
h(r)ℓ (t − u)dN
(ℓ)u .
Branching / Cluster representation, stationary process if the
spectral radius of(∫
h(r)ℓ (t)dt
)
is < 1.
9/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
More formallyOnly excitation (all the h
(r)ℓ are positive): for all r ,
λ(r)(t) = νr +
M∑
ℓ=1
∫ t−
−∞
h(r)ℓ (t − u)dN
(ℓ)u .
Branching / Cluster representation, stationary process if the
spectral radius of(∫
h(r)ℓ (t)dt
)
is < 1.
Interaction, for instance, (in general any 1-Lipschitz function,Brémaud Massoulié 1996)
λ(r)(t) =
(
νr +
M∑
ℓ=1
∫ t−
−∞
h(r)ℓ (t − u)dN
(ℓ)u
)
+
.
9/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
More formallyOnly excitation (all the h
(r)ℓ are positive): for all r ,
λ(r)(t) = νr +
M∑
ℓ=1
∫ t−
−∞
h(r)ℓ (t − u)dN
(ℓ)u .
Branching / Cluster representation, stationary process if the
spectral radius of(∫
h(r)ℓ (t)dt
)
is < 1.
Interaction, for instance, (in general any 1-Lipschitz function,Brémaud Massoulié 1996)
λ(r)(t) =
(
νr +
M∑
ℓ=1
∫ t−
−∞
h(r)ℓ (t − u)dN
(ℓ)u
)
+
.
Exponential (Multiplicative shape but no guarantee of astationary version ...)
λ(r)(t) = exp
(
νr +M∑
ℓ=1
∫ t−
−∞
h(r)ℓ (t − u)dN
(ℓ)u
)
.9/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Previous works
Maximum likelihood estimates eventually + AIC (Ogata,Vere-Jones etc mainly for sismology, Chornoboy et al., forneuroscience, Gusto and Schbath for genomics)
Parametric tests for the detection of edge + Maximumlikelihood + exponential formula + spline estimation(Carstensen et al., in genomics)
Univariate processes + ℓ0 penalty, oracle inequalities (RB andSchbath)
Maximum likelihood + exponential formula + ℓ1 ”groupLasso” penalty (Pillow et al. in neuroscience)
Thresholding + tests for very particular bivariate models,oracle inequality (Sansonnet)
10/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another parametric method : Least-squares
A good estimate of the parameters should correspond to anintensity candidate close to the true one.
11/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another parametric method : Least-squares
A good estimate of the parameters should correspond to anintensity candidate close to the true one.
Hence one would like to minimize‖η − λ‖2 =
∫ T
0 [η(t)− λ(t)]2dt in η
11/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another parametric method : Least-squares
A good estimate of the parameters should correspond to anintensity candidate close to the true one.
Hence one would like to minimize‖η − λ‖2 =
∫ T
0 [η(t)− λ(t)]2dt in η
or equivalently −2∫ T
0 η(t)λ(t)dt +∫ T
0 η(t)2dt.
11/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another parametric method : Least-squares
A good estimate of the parameters should correspond to anintensity candidate close to the true one.
Hence one would like to minimize‖η − λ‖2 =
∫ T
0 [η(t)− λ(t)]2dt in η
or equivalently −2∫ T
0 η(t)λ(t)dt +∫ T
0 η(t)2dt.
But dNt randomly fluctuates around λ(t)dt and is observable.
11/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another parametric method : Least-squares
A good estimate of the parameters should correspond to anintensity candidate close to the true one.
Hence one would like to minimize‖η − λ‖2 =
∫ T
0 [η(t)− λ(t)]2dt in η
or equivalently −2∫ T
0 η(t)λ(t)dt +∫ T
0 η(t)2dt.
But dNt randomly fluctuates around λ(t)dt and is observable.
Hence one minimizes
−2
∫ T
0η(t)dNt +
∫ T
0η(t)2dt,
for a model η = λa(t)
11/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On Hawkes processesRecall that for the process N(r)
λ(r)(t) = ν(r) +
M∑
ℓ=1
∑
T
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On Hawkes processesRecall that for the process N(r)
λ(r)(t) = ν(r) +
M∑
ℓ=1
∑
T
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On Hawkes processesRecall that for the process N(r)
λ(r)(t) = ν(r) +
M∑
ℓ=1
∑
T
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On Hawkes processesRecall that for the process N(r)
λ(r)(t) = ν(r) +
M∑
ℓ=1
∑
T
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On Hawkes processesRecall that for the process N(r)
λ(r)(t) = ν(r) +
M∑
ℓ=1
∑
T
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
An heuristic for the least-square estimatorInformally, the link between the point process and its intensity canbe written as
dN(r)(t) ≃ (Rct)′a
(r)∗ dt + noise.
13/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
An heuristic for the least-square estimatorInformally, the link between the point process and its intensity canbe written as
dN(r)(t) ≃ (Rct)′a
(r)∗ dt + noise.
Let
G =
∫ T
0Rct(Rct)
′dt,
the integrated covariation of the renormalized instantaneous count.
13/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
An heuristic for the least-square estimatorInformally, the link between the point process and its intensity canbe written as
dN(r)(t) ≃ (Rct)′a
(r)∗ dt + noise.
Let
G =
∫ T
0Rct(Rct)
′dt,
the integrated covariation of the renormalized instantaneous count.
b(r) :=
∫ T
0RctdN
(r)(t) ≃ Ga(r)∗ + noise.
13/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
An heuristic for the least-square estimatorInformally, the link between the point process and its intensity canbe written as
dN(r)(t) ≃ (Rct)′a
(r)∗ dt + noise.
Let
G =
∫ T
0Rct(Rct)
′dt,
the integrated covariation of the renormalized instantaneous count.
b(r) :=
∫ T
0RctdN
(r)(t) ≃ Ga(r)∗ + noise.
The least-square estimate is
â(r) = G−1b(r),
where in b lies again the number of couples with a certain delay.
13/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
An heuristic for the least-square estimatorInformally, the link between the point process and its intensity canbe written as
dN(r)(t) ≃ (Rct)′a
(r)∗ dt + noise.
Let
G =
∫ T
0Rct(Rct)
′dt,
the integrated covariation of the renormalized instantaneous count.
b(r) :=
∫ T
0RctdN
(r)(t) ≃ Ga(r)∗ + noise.
The least-square estimate is
â(r) = G−1b(r),
where in b lies again the number of couples with a certain delay.→ simpler formula than Maximum Likelihood Estimators forsimilar properties
13/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
What gain wrt cross correlogramm ?
1
23
5ms
5ms
only 2 non zero interaction functions over 9
14/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
What gain wrt cross correlogramm ?
Histogram of dist2
1−>2
Fre
quen
cy
−0.03 −0.02 −0.01 0.00 0.01 0.02 0.03
020
4060
8010
0Histogram of dist2
2−>3
Fre
quen
cy
−0.03 −0.02 −0.01 0.00 0.01 0.02 0.03
050
100
150
200
Histogram of dist2
1−>3
Fre
quen
cy
−0.03 −0.02 −0.01 0.00 0.01 0.02 0.03
020
4060
80
14/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
What gain wrt cross correlogramm ?
0.000 0.005 0.010 0.015 0.020 0.025 0.030
01
23
4
1−>2
inte
ract
ion
0.000 0.005 0.010 0.015 0.020 0.025 0.030
01
23
4
2−>3
inte
ract
ion
0.000 0.005 0.010 0.015 0.020 0.025 0.030
01
23
4
1−>3
inte
ract
ion
14/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Full Multivariate Hawkes process and ℓ1 penaltywith N.R. Hansen (Copenhagen), and V. Rivoirard (Dauphine) (2012)
The Lasso criterion can be expressed independently for eachsub-process by :
Lasso criterion
â(r) = argmina{γT (λ(r)a ) + pen(a)}
= argmina{−2a′br + a
′Ga+ 2(d(r))′|a|}
The crucial choice is the d(r), should be data-driven !
The theoretical validation : be able to state that our choice isthe best possible choice.
The practical validation : on simulated Hawkes processes(done), on simulated neuronal networks, on real data (RNRPParis 6 work in progress)...
15/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Theoretical Validation = Oracle inequalityRecall that
b(r) =
∫ T
0RctdN
(r)(t)
and
G =
∫ T
0Rct(Rct)
′dt.
Hansen,Rivoirard,RB
If G ≥ cI with c > 0 and if
∣∣
∫ T
0Rct
(
dN(r)(t)− λ(r)(t)dt) ∣∣ ≤ d(r), ∀r
then
∑
r
‖λ(r)−Rct â(r)‖2 ≤ � inf
a
∑
r
‖λ(r) − Rcta‖2 +
1
c
∑
i∈supp(a)
(d(r)i )
2
.
16/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !
17/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !
The loss (1/T )∑
i∈supp(a)(d(r)i )
2 is unavoidable even for onlyone choice of set of non-zeros coefficients. Should be read as”capacity of approximation” + unavoidable loss due to thenoise.
17/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !
The loss (1/T )∑
i∈supp(a)(d(r)i )
2 is unavoidable even for onlyone choice of set of non-zeros coefficients. Should be read as”capacity of approximation” + unavoidable loss due to thenoise.Factor 1/c gives a ”quality indicator” since G observable inpractice. Maybe not the best thing, but with stationnaryHawkes process, one can prove that c ≃ T .
17/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !
The loss (1/T )∑
i∈supp(a)(d(r)i )
2 is unavoidable even for onlyone choice of set of non-zeros coefficients. Should be read as”capacity of approximation” + unavoidable loss due to thenoise.Factor 1/c gives a ”quality indicator” since G observable inpractice. Maybe not the best thing, but with stationnaryHawkes process, one can prove that c ≃ T .Gives the ”best” linear approximation and in this sense valideven if λ not of the prescribed shape, in particular in case ofinhibition.
17/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !
The loss (1/T )∑
i∈supp(a)(d(r)i )
2 is unavoidable even for onlyone choice of set of non-zeros coefficients. Should be read as”capacity of approximation” + unavoidable loss due to thenoise.Factor 1/c gives a ”quality indicator” since G observable inpractice. Maybe not the best thing, but with stationnaryHawkes process, one can prove that c ≃ T .Gives the ”best” linear approximation and in this sense valideven if λ not of the prescribed shape, in particular in case ofinhibition.In practice small unavoidable bias, possible to correct by twostep procedures (OLS on the support of the Lasso estimate).
17/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !
The loss (1/T )∑
i∈supp(a)(d(r)i )
2 is unavoidable even for onlyone choice of set of non-zeros coefficients. Should be read as”capacity of approximation” + unavoidable loss due to thenoise.Factor 1/c gives a ”quality indicator” since G observable inpractice. Maybe not the best thing, but with stationnaryHawkes process, one can prove that c ≃ T .Gives the ”best” linear approximation and in this sense valideven if λ not of the prescribed shape, in particular in case ofinhibition.In practice small unavoidable bias, possible to correct by twostep procedures (OLS on the support of the Lasso estimate).Possible to do it with other basis (Fourier) or other linearmodel (Aalen)
17/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
One of the main probabilistic ingredients
Bernstein type inequality for counting processes (H., R.B., R.)
Let (Hs)s≥0 be a predictable process andMt =
∫ t
0 Hs(dNs − λ(s)ds). Let b > 0 and v > w > 0.For all x , µ > 0 such that µ > φ(µ), let
V̂ µτ =µ
µ−φ(µ)
∫ τ
0H2s dNs +
b2x
µ−φ(µ) , where φ(u) = exp(u)− u − 1.
Then for every stopping time τ and every ε > 0
P
(
Mτ ≥
√
2(1 + ε)V̂ µτ x + bx/3, w ≤ V̂ µτ ≤ v and sups∈[0,τ ] |Hs | ≤ b
)
≤ 2 log(v/w)log(1+ε) e−x .
18/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
One of the main probabilistic ingredients
Bernstein type inequality for counting processes (H., R.B., R.)
Let (Hs)s≥0 be a predictable process andMt =
∫ t
0 Hs(dNs − λ(s)ds). Let b > 0 and v > w > 0.For all x , µ > 0 such that µ > φ(µ), let
V̂ µτ =µ
µ−φ(µ)
∫ τ
0H2s dNs +
b2x
µ−φ(µ) , where φ(u) = exp(u)− u − 1.
Then for every stopping time τ and every ε > 0
P
(
Mτ ≥
√
2(1 + ε)V̂ µτ x + bx/3, w ≤ V̂ µτ ≤ v and sups∈[0,τ ] |Hs | ≤ b
)
≤ 2 log(v/w)log(1+ε) e−x .
NB : V̂ µτ is an estimate of the bracket of the martingale, i.e. themartingale equivalent of the variance → Gaussian behavior for themain term.
18/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
One of the main probabilistic ingredients
Bernstein type inequality for counting processes (H., R.B., R.)
Let (Hs)s≥0 be a predictable process andMt =
∫ t
0 Hs(dNs − λ(s)ds). Let b > 0 and v > w > 0.For all x , µ > 0 such that µ > φ(µ), let
V̂ µτ =µ
µ−φ(µ)
∫ τ
0H2s dNs +
b2x
µ−φ(µ) , where φ(u) = exp(u)− u − 1.
Then for every stopping time τ and every ε > 0
P
(
Mτ ≥
√
2(1 + ε)V̂ µτ x + bx/3, w ≤ V̂ µτ ≤ v and sups∈[0,τ ] |Hs | ≤ b
)
≤ 2 log(v/w)log(1+ε) e−x .
NB : V̂ µτ is an estimate of the bracket of the martingale, i.e. themartingale equivalent of the variance → Gaussian behavior for themain term.Applied to
∫ T
0 Rct(dN(r)(t)− λ(r)(t)dt
): d is given by the right
hand-side (x ≃ γ ln(T )) → Bernstein Lasso.
18/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Choices of the weights dλ
spont1 spont2
00.02
30
10
0.02
1->1 2->1 1->2 2->2
00.02 0.02
10
30
1->1 2->1 1->2 2->2spont1 spont2
o b
o b
o |b-b|
dgamma=1o
dgamma=1.5o
10 4.2 0 1.4 0 10 0 0 0 0 10 4.2 0 1.4 0 10 0 0 0 0Coeff: Coeff:
NB : Adaptive Lasso (Zou) dλ = γ/|âλ|
19/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Simulation study - Estimation (T = 20, M = 8, K = 8)
Interactions reconstructed with ’Adaptive Lasso’, ’Bernstein Lasso’ and’Bernstein Lasso+OLS’. Above graphs, estimation of spontaneous rates
20/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another example with inhibition
T = 20s 21/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another (more realistic?) neuronal network: Integrate andFire
1 2 3 4 5
6 7 8 9 10
22/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another (more realistic?) neuronal network: Integrate andFire
T = 60s
22/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another (more realistic?) neuronal network: Integrate andFire
1 2 3 4 5
6 7 8 9 10
22/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Another (more realistic?) neuronal network: Integrate andFire
Number of times that a given
interaction is detected Mean energy (� h ) when detected2
22/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On neuronal data (sensorimotor task)
0.00 0.01 0.02 0.03 0.04
−60
−40
−20
0
SB= 22.2 SBO= 37.2
seconds
Hz
0.00 0.01 0.02 0.03 0.04
−1.
00.
00.
51.
0
SB= 22.2 SBO= 37.2
seconds
Hz
0.00 0.01 0.02 0.03 0.04
−1.
00.
00.
51.
0
SB= 55.5 SBO= 88.9
seconds
Hz
0.00 0.01 0.02 0.03 0.04
−15
0−
500
SB= 55.5 SBO= 88.9
seconds
Hz
30 trials : monkey trained to touch the correct target whenilluminated. Accept the test of Hawkes hypothesis. Work with F.Grammont, V. Rivoirard and C. Tuleau-Malot 23/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
On neuronal data (vibrissa excitation)Joint work with RNRP (Paris 6). Behavior: vibrissa excitation atlow frequency. T = 90.5, M = 10
Neuron TT102 TT201 TT202 TT204 TT301 TT303 TT401 TT402 TT403 TT404Spikes 9191 99 544 149 15 18 136 282 8 6
0 1 2 3 4
01
23
4
Comportement 1 ; k 10 ; delta 0.005 ; gamma 1
TT1_02
TT2_01
TT2_02
TT2_04
TT3_01 TT3_03
TT4_01
TT4_02
TT4_03
TT4_04
0.00 0.01 0.02 0.03 0.04 0.05
010
2030
40
Hz
TT1_02 −> TT1_02
TT2_02 −> TT1_02
TT1_02 −> TT2_02
TT1_02 −> TT2_04
TT1_02 −> TT4_02
24/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Application on real dataData:
Neuron TT102 TT201 TT202 TT204 TT301 TT303 TT401 TT402 TT403 TT404Spikes 9191 99 544 149 15 18 136 282 8 6
Simulation:Neuron TT102 TT201 TT202 TT204 TT301 TT303 TT401 TT402 TT403 TT404Spikes 9327 92 585 148 13 23 133 271 8 3
25/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Application on real dataData:
Neuron TT102 TT201 TT202 TT204 TT301 TT303 TT401 TT402 TT403 TT404Spikes 9191 99 544 149 15 18 136 282 8 6
Simulation:Neuron TT102 TT201 TT202 TT204 TT301 TT303 TT401 TT402 TT403 TT404Spikes 9327 92 585 148 13 23 133 271 8 3
0 1 2 3 4
01
23
4
TT1_02
TT2_01
TT2_02
TT2_04
TT3_01 TT3_03
TT4_01
TT4_02
TT4_03
TT4_04
0.00 0.01 0.02 0.03 0.04 0.05
010
2030
40
Hz
TT1_02 −> TT1_02
TT2_02 −> TT1_02
TT1_02 −> TT2_02
25/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Evolution of the dependance graph as a fonction of thevibrissa excitation
10
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
25
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
50
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
75
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
100
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
125
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
150
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
175
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
200
TT1_01
TT1_02
TT2_0F
TT2_03
TT2_14
TT3_04
TT4_0A
TT4_05TT4_18
26/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Conclusions and Perspectives
It is possible to estimate interaction functions and even agraph of possible interactions, to have some mathematicalguarantee on it and this even if the Hawkes linear model isnot completely true (robustness).
It is even possible to (non parametrically) test that suchinteractions exists. However the power of such tests (notstudied yet) should be linked to a ”distance” to the Hawkesmodel.
One of the main issue is the lack of stationarity → a work inprogress with F. Picard (Lyon) and C. Tuleau-Malot (Nice)based on segmentation and clustering.
27/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Thank you !
28/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
References
Hansen, N.R., Reynaud-Bouret, P. and Rivoirard, V. Lasso andprobabilistic inequalities for multivariate point processes. to appearin Bernoulli (2012).
Tuleau-Malot, C., Rouis, A., Grammont, F., and Reynaud-Bouret,P. Multiple Tests based on a Gaussian Approximation of the UnitaryEvents method. in revision (2013)
Reynaud-Bouret, P., Tuleau-Malot, C., Rivoirard, V. andGrammont, F. Goodness-of-fit tests and nonparametric adaptiveestimation for spike train analysis to appear in Journal ofMathematical Neuroscience (2013)
29/30
Biological motivation Hawkes processes Adaptive estimation Simulations Real data analysis
Link with cross-correlogram and Joint PeriStimulusHistogram
N1 trial 1
N1 trial 2
N2
tri
al 2
N2
tri
al 1
B 0 Tmax
0
Tmax
N1
N2
0 Tmax
0
Tmax
N1
N2
0 TmaxN1
0
Tmax
N2
a b
a
b
A
delta
2
1
1 2
DC
A
see also Zhang et al (2007) in genomics and Aertsen et al (1989)in Neurosciences
30/30
Biological motivationHawkes processesAdaptive estimationSimulationsReal data analysis