Background Neuroscience models General class of models McKean-Vlasov limit for interacting systems with simultaneous jumps Luisa Andreis Prof. Paolo Dai Pra, Markus Fischer Università degli studi di Padova Berlin - Padova Young researchers Meeting in Probability Oct. 23-25, 2014
Background Neuroscience models General class of models Outline of the talk Brief overview on interacting particle systems, propagation of chaos and McKean-Vlasov limits Interacting particle systems for neuroscience A general class of models
Background Neuroscience models General class of models Interacting particle systems Mathematical models for different areas as Neuroscience, Genetics, Biology, Economics, etc., rely on interacting particle systems. These models start from a finite number N of particles interacting with each other. Particle system
Background Neuroscience models General class of models Interacting particle systems Mathematical models for different areas as Neuroscience, Genetics, Biology, Economics, etc., rely on interacting particle systems. These models start from a finite number N of particles interacting with each other. Particle system ւ ց microscopic macroscopic behaviour behaviour
Background Neuroscience models General class of models Propagation of chaos Definition ( p -chaotic sequence) Let E be a separable metric space and p N a sequence of symmetric probabilities on E N . p N is p -chaotic, with p probability on E , if for any sequence φ 1 , . . . , φ k of test functions and for all k ≥ 1, k � N →∞ � p N , φ 1 ⊗ · · · ⊗ φ k ⊗ 1 ⊗ · · · ⊗ 1 � = lim � p , φ i � . (1) i = 1
Background Neuroscience models General class of models Propagation of chaos Definition ( p -chaotic sequence) Let E be a separable metric space and p N a sequence of symmetric probabilities on E N . p N is p -chaotic, with p probability on E , if for any sequence φ 1 , . . . , φ k of test functions and for all k ≥ 1, k � N →∞ � p N , φ 1 ⊗ · · · ⊗ φ k ⊗ 1 ⊗ · · · ⊗ 1 � = lim � p , φ i � . (1) i = 1 Definition (Propagation of chaos) Consider a Markovian system of particles with symmetric law P N on C ( R + , E ) N (or D ( R + , E ) N ). Propagation of chaos means that when the initial conditions are p 0 -chaotic, for a certain p 0 probability on E , then there exists a suitable p probability on C ( R + , E ) (or D ( R + , E ) ), with initial condition p 0 , such that the sequence P N is P -chaotic.
Background Neuroscience models General class of models Propagation of chaos as law of large numbers Let us define the empirical measure N ρ N = 1 � δ X i , (2) N i = 1 where ( X 1 , . . . , X N ) is a random variable distributed according p N and therefore ρ N is a sequence of r.v. with values on M ( E ) . Theorem (Tanaka, Sznitman) Let E be a separable metric space, p a probability measure on E and p N a sequence of symmetric probability measures on E N , the fact that p N is p-chaotic is equivalent to law ( ρ N ) − → δ p , (3) where the convergence is in law.
Background Neuroscience models General class of models Proving propagation of chaos Classical approach to prove propagation of chaos: to prove tightness of the sequence; to identify the possible limits of the sequence; to prove uniqueness of this limit.
Background Neuroscience models General class of models Proving propagation of chaos Classical approach to prove propagation of chaos: to prove tightness of the sequence; to identify the possible limits of the sequence; to prove uniqueness of this limit. Example: McKean-Vlasov SDE of the particle system: � � � � dX i X i X i dB i t = b t , ρ N ( t ) dt + σ t , ρ N ( t ) for i = 1 , . . . , N , (4) t SDE of the limiting process: dX t = b ( X t , p t ) dt + σ ( X t , p t ) dB t . (5) Here X 1 , . . . , X N , X are stochastic processes with values on R d , b ( · , · ) : R d × M ( R d ) → R d , σ ( · , · ) : R d × M ( R d ) → R d × k , B 1 , . . . , B N , B are k -dimensional Brownian motion and p t is the law of the process X t itself.
Background Neuroscience models General class of models Interacting particle systems in neuroscience Modelling membrane potential ւ ց neurons spikes neurons spikes occur when their occur randomly potentials reach according some a treshold point processes
Background Neuroscience models General class of models Toy model for interacting neurons Presented by De Masi, Galves, Löcherbach, Presutti (2014) and deeper investigated by Fournier, Löcherbach (2014). � � U N ( t ) = U N 1 ( t ) , . . . , U N ∈ R N N ( t ) + , configuration of membrane potentials for N interacting neurons. Chemical synapses: each neuron randomly spikes, it sets its energy at 0 and it gives all the other neurons a small and deterministic quantity of engergy, i.e. 1 N . Electrical synapses: each neuron’s potential tends to reach the value of the center of mass. Infinitesimal generator of the N dimensional particle systems For all φ : R N + → R smooth test function, N N � � x ] ∂φ L N φ ( x ) = � � [ x i − ¯ f ( x i )[ φ ( x + ∆ i ( x )) − φ ( x )] − λ ( x ) . (6) ∂ x i i = 1 i = 1 Here f ∈ C ( R + , R + ) is strictly positive for x > 0 and non-decreasing, λ ≥ 0 and � 1 for j � = i ∆ i ( x ) j = N − x i for j = i
Background Neuroscience models General class of models Approach to prove propagation of chaos The peculiarity of this model are simultaneous jumps . In the framework of propagation of chaos, non-simultaneous jumps in particle systems and in non-linear limits have been treated before, Graham(1992), Méléard (1996). Authors approach for simultaneous jumps - They build an ad hoc Markov process that is a simplification and a discretization of the model. - They consider an elaborate coupling algorithm to prove that the discrete time approximation of the initial Markov process and the simpler discrete process are closer with N → ∞ . - They obtain a limiting equation for the density of the simpler discrete process when N → ∞ and the time interval goes to 0.
Background Neuroscience models General class of models Toy model for interacting neurons Infinitesimal generator of the limit process, for all ϕ : R + → R smooth test function, ρ ) + p ] ∂ϕ L ϕ ( x ) = f ( x )[ ϕ ( 0 ) − ϕ ( x )] + [ − λ ( x − ¯ ∂ x ( x ) . (7) Here f ∈ C ( R + , R + ) is strictly positive for x > 0 and non-decreasing, λ ≥ 0 and � ∞ ¯ ρ = x ρ ( dx ) , (8) 0 � ∞ p = f ( x ) ρ ( dx ) . (9) 0
Background Neuroscience models General class of models Motivation Do particle systems with simultaneous jumps need ad hoc treatment to prove propagation of chaos?
Background Neuroscience models General class of models Motivation Do particle systems with simultaneous jumps need ad hoc treatment to prove propagation of chaos? ↓ Our aim is to build a sufficiently general framework to include the previous model and to obtain limiting results with the classical tools of propagation of chaos.
Background Neuroscience models General class of models Particle system We start from a system of N interacting particles, each of them associated to a random process X N i ( t ) with values on R , i.e. � � R d × · · · × R d . X N ( t ) = X N 1 ( t ) , . . . , X N N ( t ) ∈ Infinitesimal generator For every φ : R d × · · · × R d → R smooth test function, L N φ ( x ) = � N � F ( x i , ρ N ) ▽ i φ ( x ) i = 1 � (10) � + λ ( x i , ρ N ) [ 0 , 1 ] N [ φ ( x + ∆ i ( x , h )) − φ ( x )] ν N ( dh ) , with � Θ( x i , x j ,ρ N , h i , h j ) for j � = i , ∆ i ( x , h ) j = (11) N ψ ( x i , ρ N , h i ) for j = i . Here λ ( · , · ) is bounded and ν N is the projection on N coordinates of a symmetric probability measure ν on [ 0 , 1 ] N and gives the randomness of the amplitude of the jumps.
Background Neuroscience models General class of models Candidate to be the limit process Through heuristic computation we arrive at a possible limit process, X ( t ) ∈ R d . Infinitesimal generator For every ϕ : R d → R smooth test function, L ϕ ( x ) = F ( x , ρ ) ▽ ϕ ( x ) �� � � + R d λ ( y , ρ ) [ 0 , 1 ] 2 Θ( y , x , ρ, h 1 , h 2 ) ν 2 ( dh 1 , dh 2 ) ρ ( dy ) ▽ ϕ ( x ) � � + λ ( x , ρ ) [ ϕ ( x + ψ ( x , ρ, h 1 )) − ϕ ( x )] ν 1 ( dh 1 ) , [ 0 , 1 ] (12) where ρ is the law of the process itself.
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