Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Olivier Faugeras

Description

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Inter Spike Intervals probability distribution and Double Integral Processes Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris Workshop on Stochastic Models in Neuroscience 18-22 January 2010 Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

• ?s ?


• inter spike


• dip hits boundary


• ?m


• iwp iwp


• motivations dip


• ?s dws?


• s?t ?m

Informations

• Publié par: pefav
• Nombre de lectures: 29
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
Inter Spike Intervals probability distribution and
Double Integral Processes
Olivier Faugeras
NeuroMathComp project team - INRIA/ENS Paris
Workshop on Stochastic Models in Neuroscience
18-22 January 2010
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPI Synaptic currents:
dI (t) = I (t)dt +dWs s s t
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
LIF models of neurons
I Membrane potential:

dV (t) = (V (t) V ) + I (t) dt + dI (t)m rest e s
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPMotivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
LIF models of neurons
I Membrane potential:

dV (t) = (V (t) V ) + I (t) dt + dI (t)m rest e s
I Synaptic currents:
dI (t) = I (t)dt +dWs s s t
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPMotivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Reaching the threshold
Integrate the linear SDE:
t s tRt1 m mV (t) = V (1 e ) + e I (s) ds+rest e 0m Z Z 0t st t t ssI (0) s s m m s 0(e e ) + e e e dW dssm1 m s 0 0s
1 1with = .
m s
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPI Same as rst hitting time of
Z Z 0t s ss
sX = e e dW 0 dst s
0 0
I to the deterministic boundary a(t)

t t s tR t1 m m me a(t) =(t) V 1 e + e I (s) ds+rest e 0m m s
!
t tI (0)s s me em1
s
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Reaching the threshold
I A spike is emitted when V (t) reaches the threshold (t)
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPI to the deterministic boundary a(t)

t t s tR t1 m m me a(t) =(t) V 1 e + e I (s) ds+rest e 0m m s
!
t tI (0)s s me em1
s
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Reaching the threshold
I A spike is emitted when V (t) reaches the threshold (t)
I Same as rst hitting time of
Z Z 0t s ss
sX = e e dW 0 dst s
0 0
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPMotivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Reaching the threshold
I A spike is emitted when V (t) reaches the threshold (t)
I Same as rst hitting time of
Z Z 0t s ss
sX = e e dW 0 dst s
0 0
I to the deterministic boundary a(t)

t t s tR t1 m m me a(t) =(t) V 1 e + e I (s) ds+rest e 0m m s
!
t tI (0)s s me em1
s
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPMotivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Stopping times
De nition
A positive real random variable is called a stopping time with
respect to the ltration F provided thatf tg2F for allt t
t 0.
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPnI Let E be a non-empty open or closed set of , then
f = inf j X (t)2 Eg
t0
is a stopping time.
I Connection between SDEs and PDEs through the
Feynman-Kac formulae.
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Stopping times and di usion equations
I SDE:
dX (t) = b(X; t)dt + B(X; t)dWt
X (0) = X0
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPRI Connection between SDEs and PDEs through the
Feynman-Kac formulae.
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion
LIF models
Stopping times and di usion equations
I SDE:
dX (t) = b(X; t)dt + B(X; t)dWt
X (0) = X0
nI Let E be a non-empty open or closed set of , then
f = inf j X (t)2 Eg
t0
is a stopping time.
Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris
ISI and DIPR