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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

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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

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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