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
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models LIF models of neurons ◮ Membrane potential: � � τ m dV ( t ) = − ( V ( t ) − V rest ) + I e ( t ) dt + dI s ( t ) Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models LIF models of neurons ◮ Membrane potential: � � τ m dV ( t ) = − ( V ( t ) − V rest ) + I e ( t ) dt + dI s ( t ) ◮ Synaptic currents: τ s dI s ( t ) = − I s ( t ) dt + σ dW t Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Reaching the threshold Integrate the linear SDE: s − t V ( t ) = V rest (1 − e − t � t τ m ) + 1 τ m I e ( s ) ds + 0 e τ m � t �� s s ′ I s (0) ( e − t τ s − e − t e − t � σ s τ m ) + τ s dW s ′ τ m e e ds α 1 − τ m τ m τ s 0 0 τ s τ m − 1 1 with α = τ s . Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Reaching the threshold ◮ A spike is emitted when V ( t ) reaches the threshold θ ( t ) Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Reaching the threshold ◮ A spike is emitted when V ( t ) reaches the threshold θ ( t ) ◮ Same as first hitting time of � t �� s s ′ s � τ s dW s ′ X t = e e ds α 0 0 Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Reaching the threshold ◮ A spike is emitted when V ( t ) reaches the threshold θ ( t ) ◮ Same as first hitting time of � t �� s s ′ s � τ s dW s ′ X t = e e ds α 0 0 ◮ to the deterministic boundary a ( t ) � e − t 1 − e − t s − t σ � t � � + 1 τ m a ( t ) = θ ( t ) − τ m I e ( s ) ds + V rest τ m 0 e τ m τ m τ s � � I s (0) � e − t τ s − e − t τ m 1 − τ m τ s Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Stopping times Definition A positive real random variable is called a stopping time with respect to the filtration F t provided that { τ ≤ t } ∈ F t for all t ≥ 0. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
❘ Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Stopping times and diffusion equations ◮ SDE: dX ( t ) = b ( X , t ) dt + B ( X , t ) dW t X (0) = X 0 Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Stopping times and diffusion equations ◮ SDE: dX ( t ) = b ( X , t ) dt + B ( X , t ) dW t X (0) = X 0 ◮ Let E be a non-empty open or closed set of ❘ n , then { τ = inf t ≥ 0 | X ( t ) ∈ E } is a stopping time. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Stopping times and diffusion equations ◮ SDE: dX ( t ) = b ( X , t ) dt + B ( X , t ) dW t X (0) = X 0 ◮ Let E be a non-empty open or closed set of ❘ n , then { τ = inf t ≥ 0 | X ( t ) ∈ E } is a stopping time. ◮ Connection between SDEs and PDEs through the Feynman-Kac formulae. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models Stopping times and diffusion equations Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks A neural network Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable ◮ For each neuron i, define X ( i ) ( t ) ≥ 0 to be the remaining time until the next emission of a spike by neuron i if it does not receive any spike meanwhile. ◮ This process has a very simple dynamics: d X ( i ) ( t ) = − 1 d t Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable ◮ At time t , the next spike will occur in neuron i = Arg Min j ∈{ 1 ... N } X ( j ) ( t ) at time t + X ( i ) ( t ). Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable ◮ At time t , the next spike will occur in neuron i = Arg Min j ∈{ 1 ... N } X ( j ) ( t ) at time t + X ( i ) ( t ). ◮ At spike time, the membrane potential of the neuron that just spiked is reset. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable ◮ At time t , the next spike will occur in neuron i = Arg Min j ∈{ 1 ... N } X ( j ) ( t ) at time t + X ( i ) ( t ). ◮ At spike time, the membrane potential of the neuron that just spiked is reset. ◮ The countdown value is also reset to a value Y i corresponding to the next spike time of this neuron if nothing occurs meanwhile. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable ◮ At time t , the next spike will occur in neuron i = Arg Min j ∈{ 1 ... N } X ( j ) ( t ) at time t + X ( i ) ( t ). ◮ At spike time, the membrane potential of the neuron that just spiked is reset. ◮ The countdown value is also reset to a value Y i corresponding to the next spike time of this neuron if nothing occurs meanwhile. ◮ This value is a random variable, the reset random variable . Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable ◮ At time t , the next spike will occur in neuron i = Arg Min j ∈{ 1 ... N } X ( j ) ( t ) at time t + X ( i ) ( t ). ◮ At spike time, the membrane potential of the neuron that just spiked is reset. ◮ The countdown value is also reset to a value Y i corresponding to the next spike time of this neuron if nothing occurs meanwhile. ◮ This value is a random variable, the reset random variable . ◮ Depending upon the neurone model, its law is that of the first hitting time of a Brownian, an IWP or a DIP to a deterministic boundary. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable The interaction random variable η ij between neurons i and j is the modification of the time to the next spike of neuron j caused by its receiving a spike from neuron i . Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Countdown process and reset variable Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Markov description of the network For a large variety of IF and LIF models, the state of the network can be described by a Markov chain (or process) (Touboul, Faugeras, in preparation), e.g. ( X ( t ) , I s ( t ) , t ). Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks Neural networks and queuing theory ◮ A lot can probably be gained in the study of neural networks by looking at the work in queuing theory. ◮ The countdown process is called an hourglass model (introduced by Marie Cottrell 1992). ◮ Later studied in (Turova 1996, Asmussen and Turova 1998, Cottrell and Turova 2000, Turova 2000). ◮ In order to apply this modeling we need to define in each case the reset and the interaction random variables. Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP
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