Estimationoflocalindependencegraphsvia ...20140319... · ℓ (t)dt is < 1. Interaction, for...

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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Contents

1 Biological motivation

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Contents


2 Hawkes processes

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Contents


2 Hawkes processes

3 Adaptive estimation

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Contents


2 Hawkes processes


4 Simulations

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Contents


2 Hawkes processes


4 Simulations

5 Real data analysis

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Description of the data (Equipe RNRP de Paris 6)

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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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2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.00.0 0.5 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

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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


Synaptic integration

without synchronisation with synchronisation

dendrites

signaux d’entrées

seuil d’excitation

du neurone

potentiel de repos

du neurone

différentes

dendrites



du neurone

potentiel de repos

du neurone

différentes

5/30



without synchronisation Avec synchronisation

dendrites



du neurone

potentiel de repos

du neurone

différentes

dendrites



du neurone

potentiel de repos

du neurone

différentes

5/30




dendrites



du neurone

potentiel de repos

du neurone

différentes

dendrites



du neurone

potentiel de repos

du neurone

différentes

5/30



without synchronisation Avec synchronisation

dendrites



du neurone

potentiel de repos

du neurone

différentes

dendrites



du neurone

potentiel de repos

du neurone

différentes

5/30




dendrites



du neurone

potentiel de repos

du neurone

différentes

dendrites



du neurone

potentiel de repos

du neurone

différentes

5/30




dendrites



du neurone

potentiel de repos

du neurone

potentiel d’action

différentes

dendrites



du neurone

potentiel de repos

du neurone

différentes

5/30




dendrites



du neurone

potentiel de repos

du neurone


différentes

dendrites



du neurone

potentiel de repos

du neurone


différentes

5/30




dendrites



du neurone

potentiel de repos

du neurone


différentes

dendrites



du neurone

potentiel de repos

du neurone

potentiel d’actionpotentiel d’action

différentes

5/30


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


Multivariate Hawkes processes

t

N1

N2

N3

7/30



t

N1

N2

N3

λ(1)(t) = ν︸︷︷︸

spontaneous

7/30



t

N1

N2

N3

1->1h

λ(1)(t) = ν︸︷︷︸

+∑

T < tpt de N1

h1→1(t − T )

︸︷︷︸

spontaneous self-interaction7/30



t

N1

N2

N3

1->1h

x

λ(1)(t) = ν︸︷︷︸

+∑

T < tpt de N1

h1→1(t − T )

︸︷︷︸




t

N1

N2

N3

1->1h

x

x

λ(1)(t) = ν︸︷︷︸

+∑

T < tpt de N1

h1→1(t − T )

︸︷︷︸




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



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 )

︸︷︷︸




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 )

︸︷︷︸




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 )

︸︷︷︸




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 )

︸︷︷︸



Link Graphical model of local independence

(Didelez (2008))

1

23

1

23

1

23

8/30


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


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




λ(r)(t) = νr +

M∑

ℓ=1

∫ t−

−∞

h(r)ℓ (t − u)dN

(ℓ)u .



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




λ(r)(t) = νr +

M∑

ℓ=1

∫ t−

−∞

h(r)ℓ (t − u)dN

(ℓ)u .



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


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


Another parametric method : Least-squares

A good estimate of the parameters should correspond to anintensity candidate close to the true one.

11/30




Hence one would like to minimize‖η − λ‖2 =

∫ T

0 [η(t)− λ(t)]2dt in η

11/30





∫ T


or equivalently −2∫ T

0 η(t)λ(t)dt +∫ T

0 η(t)2dt.

11/30





∫ T




0 η(t)2dt.

But dNt randomly fluctuates around λ(t)dt and is observable.

11/30





∫ 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


On Hawkes processesRecall that for the process N(r)

λ(r)(t) = ν(r) +

M∑

ℓ=1

∑

T


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




(r)∗ dt + noise.

Let

G =

∫ T

0Rct(Rct)

′dt,

the integrated covariation of the renormalized instantaneous count.

13/30




(r)∗ dt + noise.

Let

G =

∫ T

0Rct(Rct)

′dt,


b(r) :=

∫ T

0RctdN

(r)(t) ≃ Ga(r)∗ + noise.

13/30




(r)∗ dt + noise.

Let

G =

∫ T

0Rct(Rct)

′dt,


b(r) :=

∫ T

0RctdN


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




(r)∗ dt + noise.

Let

G =

∫ T

0Rct(Rct)

′dt,


b(r) :=

∫ T

0RctdN


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


What gain wrt cross correlogramm ?

1

23

5ms

5ms

only 2 non zero interaction functions over 9

14/30



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



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


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


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


ExplanationsMain Point: d controls the random fluctuations / noise →should be data-driven and sharp !

17/30



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



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



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



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



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


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 .

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∫ t


V̂ µτ =µ

µ−φ(µ)

∫ τ

0H2s dNs +

b2x



P

(

Mτ ≥

√


)


NB : V̂ µτ is an estimate of the bracket of the martingale, i.e. themartingale equivalent of the variance → Gaussian behavior for themain term.

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∫ t


V̂ µτ =µ

µ−φ(µ)

∫ τ

0H2s dNs +

b2x



P

(

Mτ ≥

√


)


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.

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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


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


Another example with inhibition

T = 20s 21/30


Another (more realistic?) neuronal network: Integrate andFire

1 2 3 4 5

6 7 8 9 10

22/30



T = 60s

22/30



1 2 3 4 5

6 7 8 9 10

22/30



Number of times that a given

interaction is detected Mean energy (� h ) when detected2

22/30


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


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


Application on real dataData:


Simulation:Neuron TT102 TT201 TT202 TT204 TT301 TT303 TT401 TT402 TT403 TT404Spikes 9327 92 585 148 13 23 133 271 8 3

25/30


Application on real dataData:


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


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


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


Thank you !

28/30


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


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

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Biological motivationHawkes processesAdaptive estimationSimulationsReal data analysis

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