Detection of dependence patterns with delay J. Chevallier T. Lalo - PowerPoint PPT Presentation

Detection of dependence patterns with delay J. Chevallier T. Laloë LJAD University of Nice Journées de la SFdS 4 Juin 2015

Introduction Our method Simulations Multiple testing Overview Biological context Structure of a typical neuron Connected neurons Neural network: Interacting cells. Information transport via electric pulses: action potentials.

Introduction Our method Simulations Multiple testing Overview Biological context Structure of a typical neuron Connected neurons � ��������� � � � � Neural network: Interacting cells. � � � � � � � � � � � Information transport via electric pulses: � � � � � � � � � � action potentials. � � ������������������ ����������������� � � � � � � � � � � � � � � � � � � After preprocessing, we dispose of M trials of � � � � � � � � simultaneously recorded spike trains. � �� ��� �����������������������������������

Introduction Our method Simulations Multiple testing Overview Synchronization phenomenon V V potentiel d’action potentiel d’action potentiel d’action seuil d’excitation seuil d’excitation du neurone du neurone potentiel de repos potentiel de repos du neurone du neurone plusieurs plusieurs t t dendrites dendrites signaux d’entrées signaux d’entrées Without synchronization With synchronization The synchronization phenomenon can occur during sensory-motor tasks. The repetition of a given task may give birth to neuronal assemblies. Goal Detection of synchronizations.

Introduction Our method Simulations Multiple testing Overview Statistical analysis Cross-correlogram (Perkel et al., ’67). Peristimulus time histogram (PSTH, (Aertsen et al., ’89)).

Introduction Our method Simulations Multiple testing Overview Statistical analysis Cross-correlogram (Perkel et al., ’67). Peristimulus time histogram (PSTH, (Aertsen et al., ’89)). Unitary events (Grün, ’96).

Introduction Our method Simulations Multiple testing Overview Statistical analysis Cross-correlogram (Perkel et al., ’67). Peristimulus time histogram (PSTH, (Aertsen et al., ’89)). Unitary events (Grün, ’96). UE method Unitary event: spike synchrony that recurs more often than expected. The test statistic is based on the number of coincidences . Introduced in the PhD thesis of S. Grün (’96). Applied to time discrete data.

Introduction Our method Simulations Multiple testing Overview Statistical analysis Cross-correlogram (Perkel et al., ’67). Peristimulus time histogram (PSTH, (Aertsen et al., ’89)). Unitary events (Grün, ’96). UE method Unitary event: spike synchrony that recurs more often than expected. The test statistic is based on the number of coincidences . Introduced in the PhD thesis of S. Grün (’96). Applied to time discrete data. GAUE method for two neurons (Tuleau-Malot et al., 2014) Notion of coincidence transposed to the continuous time framework. Independence test between Poisson processes based on this new notion.

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices.

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The delayed coincidence count of delay δ < ( b − a ) / 2 is � X J := � ≤ δ N i 1 ( dx 1 ) ... N i J ( dx J ) . 1 � � � � � max i ∈{ 1 ,..., J } x i − min i ∈{ 1 ,..., J } x i � � [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The general coincidence count is � X J := c ( x 1 ,..., x J ) N i 1 ( dx 1 ) ... N i J ( dx J ) . [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b

Introduction Our method Simulations Multiple testing Overview Notion of delayed coincidences N 1 ,..., N n are point processes on [ a , b ] . J ⊂ { 1 ,..., n } is a set of indices. Definition The general coincidence count is � X J := c ( x 1 ,..., x J ) N i 1 ( dx 1 ) ... N i J ( dx J ) . [ a , b ] J Neuron 1 Neuron 2 Neuron 3 a b Goal: Test H 0 against H 1 � H 0 : The processes N j , j ∈ J are independent; H 1 : The processes N j , j ∈ J are not independent.

Introduction Our method Simulations Multiple testing Overview Asymptotic properties Let ( N ( k ) ,..., N ( k ) ) 1 ≤ k ≤ M denote a M -sample. We compare two estimates. n 1 √ M m − E [ X J ] M → ∞ k = 1 X ( k ) → N ( 0 , 1 ) , where m = 1 M ∑ M CLT ⇒ − − − − J . � V ar ( X J )