Spiking neural models: from point processes to partial differential - PowerPoint PPT Presentation

Spiking neural models: from point processes to partial differential equations. J. Chevallier Advisors: P. Reynaud Bouret (Nice) and F. Delarue (Nice) Colloque jeunes probabilistes et statisticiens Les Houches 2016/04/18

Context Two scales Mean field approximation Summary Biological context Action potential +40 Voltage (mV) Depolarization R 0 e p o l a r i z a t i o Failed n Threshold -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period.

Context Two scales Mean field approximation Summary Biological context microscopic scale Action potential +40 Voltage (mV) Depolarization Repolarization 0 Failed Threshold -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period.

Context Two scales Mean field approximation Summary Biological context microscopic scale Action potential +40 Voltage (mV) Depolarization Repolarization 0 Failed Threshold -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period.

Context Two scales Mean field approximation Summary Biological context microscopic scale Action . potential +40 Voltage (mV) Depolarization . Repolarization 0 Failed Threshold . -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period.

Context Two scales Mean field approximation Summary Biological context microscopic scale macroscopic scale Action . potential +40 Voltage (mV) Depolarization . Repolarization 0 Failed Threshold . -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period.

Context Two scales Mean field approximation Summary Biological context microscopic scale macroscopic scale Action . potential +40 Voltage (mV) Depolarization . Repolarization 0 Failed Threshold . -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period.

Context Two scales Mean field approximation Summary Biological context microscopic scale macroscopic scale Action . potential +40 Voltage (mV) Depolarization . Repolarization 0 Failed Threshold . -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period. Model interacting spiking neurons.

Context Two scales Mean field approximation Summary Biological context microscopic scale macroscopic scale Action . potential +40 Voltage (mV) Depolarization . Repolarization 0 Failed Threshold . -55 initiations Resting state -70 Stimulus Refractory period 0 1 2 3 4 5 Time (ms) Neurons = electrically excitable cells. Action potential = spike of the electrical potential. Physiological constraint: refractory period. Model interacting spiking neurons.

Context Two scales Mean field approximation Summary Microscopic modelling Microscopic modelling of spike trains Time point processes = random countable sets of times (points of R or R + ). Point process: N = { T i , i ∈ Z } s.t. ··· < T 0 ≤ 0 < T 1 < ··· . � f ( t ) N ( dt ) = ∑ i ∈ Z f ( T i ) . Point measure: N ( dt ) = ∑ i ∈ Z δ T i ( dt ) . Hence,

Context Two scales Mean field approximation Summary Microscopic modelling Microscopic modelling of spike trains Time point processes = random countable sets of times (points of R or R + ). Point process: N = { T i , i ∈ Z } s.t. ··· < T 0 ≤ 0 < T 1 < ··· . � f ( t ) N ( dt ) = ∑ i ∈ Z f ( T i ) . Point measure: N ( dt ) = ∑ i ∈ Z δ T i ( dt ) . Hence, Age process: ( S t − ) t ≥ 0 . Age = delay since last spike.

Context Two scales Mean field approximation Summary Microscopic modelling Microscopic modelling of spike trains Time point processes = random countable sets of times (points of R or R + ). Point process: N = { T i , i ∈ Z } s.t. ··· < T 0 ≤ 0 < T 1 < ··· . � f ( t ) N ( dt ) = ∑ i ∈ Z f ( T i ) . Point measure: N ( dt ) = ∑ i ∈ Z δ T i ( dt ) . Hence, Age process: ( S t − ) t ≥ 0 . Stochastic intensity Heuristically, � � 1 N ([ t , t +∆ t ]) = 1 | F N λ t = lim ∆ t P , t − ∆ t → 0 where F N t − denotes the history of N before time t . Local behaviour: probability to find a new spike. May depend on the past (e.g. refractory period, excitation, inhibition).

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period).

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period)

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period) � � t − � Hawkes process: λ t = Φ h ( t − x ) N ( dx ) . 0

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period) � � t − � Hawkes process: λ t = Φ h ( t − x ) N ( dx ) . 0 � �� � ∑ h ( t − T ) T ∈ N T < t

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period) � � t − � Hawkes process: λ t = Φ h ( t − x ) N ( dx ) . 0 � �� � ∑ h ( t − T ) T ∈ N T < t Model Poisson Renewal Hawkes Goodness-of-fit × � �

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period) � � t − h i → i ( t − x ) N i ( dx ) Multivariate HP: λ i t = Φ � 0 � t − h j → i ( t − x ) N j ( dx ) ( i = 1 ,..., n ) + ∑ j � = i . 0

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period) � � t − h i → i ( t − x ) N i ( dx ) Multivariate HP: λ i t = Φ � 0 � t − h j → i ( t − x ) N j ( dx ) ( i = 1 ,..., n ) + ∑ j � = i . 0 i → i

Context Two scales Mean field approximation Summary Some classical point processes Poisson process: λ t = λ ( t ) (no refractory period). Renewal process: λ t = f ( S t − ) ⇔ i.i.d. ISIs. (refractory period) � � t − h i → i ( t − x ) N i ( dx ) Multivariate HP: λ i t = Φ � 0 � t − h j → i ( t − x ) N j ( dx ) ( i = 1 ,..., n ) + ∑ j � = i . 0 j � = i

Context Two scales Mean field approximation Summary Age structured equations (K. Pakdaman, B. Perthame, D. Salort, 2010) Age = delay since last spike. � probability density of finding a neuron with age s at time t . u ( t , s ) = ratio of the neural population with age s at time t .  ∂ u ( t , s ) + ∂ u ( t , s )  +Ψ( s , X ( t )) u ( t , s ) = 0   ∂ t ∂ s (PPS) � + ∞   u ( t , 0 ) = Ψ( s , X ( t )) u ( t , s ) ds .  0 Key Parameter � t X ( t ) = 0 h ( t − x ) u ( x , 0 ) dx (global neural activity)

Context Two scales Mean field approximation Summary Age structured equations (K. Pakdaman, B. Perthame, D. Salort, 2010) Age = delay since last spike. � probability density of finding a neuron with age s at time t . u ( t , s ) = ratio of the neural population with age s at time t .  ∂ u ( t , s ) + ∂ u ( t , s )  +Ψ( s , X ( t )) u ( t , s ) = 0   ∂ t ∂ s (PPS) � + ∞   u ( t , 0 ) = Ψ( s , X ( t )) u ( t , s ) ds .  0 Key Parameter � t X ( t ) = 0 h ( t − x ) u ( x , 0 ) dx (global neural activity) � t − X ( t ) ← → h ( t − x ) N ( dx ) . 0