On a mean-field model of interacting neurons Q. Cormier 1 , E Tanr 1 - PowerPoint PPT Presentation

On a mean-field model of interacting neurons Q. Cormier 1 , E Tanré 1 , R Veltz 2 1 Inria TOSCA 2 Inria MathNeuro July 5, 2019 (Inria) On a mean-field model of interacting neurons July 5, 2019 1 / 25

Introduction Model of coupled noisy Integrate and Fire neurons. Mean-Field description through a McKean-Vlasov SDE . From a Dynamical System point of view: What are the invariant measures (equilibrium points for ODEs), what can we say about their stability (local, global) ? What happens if the invariant measure is not locally stable ( bifurcations ) ? (Inria) On a mean-field model of interacting neurons July 5, 2019 2 / 25

The model: Interspikes dynamics N neurons characterized by their membrane potential: V i t ∈ R + Between the spikes, ( V i t ) t ≥ 0 solves a simple deterministic ODE: dV i dt = b ( V i t t ) . (Example: b ≡ constant: the potential of each neuron grows linearly between its spikes). (Inria) On a mean-field model of interacting neurons July 5, 2019 3 / 25

The model: Interspikes dynamics N neurons characterized by their membrane potential: V i t ∈ R + Between the spikes, ( V i t ) t ≥ 0 solves a simple deterministic ODE: dV i dt = b ( V i t t ) . (Example: b ≡ constant: the potential of each neuron grows linearly between its spikes). (Inria) On a mean-field model of interacting neurons July 5, 2019 3 / 25

The model: Spiking dynamic Each neuron i spikes randomly at a rate f ( V i t ) . When such a spike occurs (say at time τ ): 1. The potential of the neuron i is reset to 0 : V i τ = 0 2. The potentials of the other neurons are increased by J i → j : j � = i, V j τ = V j τ − + J i → j . (Inria) On a mean-field model of interacting neurons July 5, 2019 4 / 25

Illustration with N = 2 neurons (Inria) On a mean-field model of interacting neurons July 5, 2019 5 / 25

The parameters of the problem The 4 parameters of the model are: 1. the drift b : R + → R , with b (0) > 0 : it gives the dynamic of the neurons between the spikes 2. the rate function f : R + → R + : it encodes the probability for a neuron of a given potential to spike between t and t + dt . 3. The connectivity parameters ( J i → j ) i,j . 4. the law of the initial potentials: we assume the neurons are initially i.i.d. with probability law ν . (Inria) On a mean-field model of interacting neurons July 5, 2019 6 / 25

The particle systems Let ( N i ( du, dz )) i =1 , ... ,N N independent Poisson measures on R + × R + with intensity measure dudz . Let ( V i 0 ) i =1 , ... ,N a family of N random variables on R + , i.i.d. of law ν Then ( V i t ) is a càdlàg process solution of the SDE:  � t � t � �   J j → i V i t = V i b ( V i u − ) } N j ( du, dz )  0 + u ) du + ✶ { z ≤ f ( V j   0 0 R + j � = i � t �    V i u − ) } N i ( du, dz ) .  − u − ✶ { z ≤ f ( V i  0 R + (Inria) On a mean-field model of interacting neurons July 5, 2019 7 / 25

The limit equation Simplification: J i → j = J N for some constant J ≥ 0 � t � t � t J � � V i t = V i b ( V i u − ) } N j ( du, dz ) − V i u − ) } N i ( du, dz ) . � 0 + u ) du + ✶ { z ≤ f ( V j u − ✶ { z ≤ f ( V i 0 N 0 0 R + R + j � = i N → ∞ : the Mean-Field equation � t � t � t � V t = V 0 + b ( V u ) du + J E f ( V u ) du − V u − ✶ { z ≤ f ( V u − ) } N ( du, dz ) 0 0 0 R + (M-F) or equivalently:  d   dtV t = b ( V t ) + J E f ( V t )     + ( V t ) t ≥ 0 jumps to 0 with rate f ( V t ) (Inria) On a mean-field model of interacting neurons July 5, 2019 8 / 25

The Fokker-Planck PDE The law of V t solves (weakly) the Fokker-Planck equation:  ∂tν ( t, x ) = − ∂ ∂   ∂x [( b ( x ) + Jr t ) ν ( t, x )] − f ( x ) ν ( t, x )  � ∞ r t   ν ( t, 0) = r t = f ( x ) ν ( t, x ) dx.  , b (0) + Jr t 0 N-L transport equation with a (N-L) boundary condition. (Inria) On a mean-field model of interacting neurons July 5, 2019 9 / 25

A brief tour of previous results 1. Many earlier considerations by physicists ( Keywords: hazard rate model, generalized I & F ) 2. A. De Masi, A. Galves, E. Löcherbach, E. Presutti, “Hydrodynamic limit for interacting neuron” 3. N. Fournier, E. Löcherbach, “On a toy model of interacting neurons” results on the long time behavior for b ≡ 0 4. A. Drogoul, R. Veltz “Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics” numerical evidence of an Hopf bifurcation in a closed setting. 5. A. Drogoul, R. Veltz “Exponential stability of the stationary distribution of a mean field of spiking neural network” results on the long time behavior for b ≡ 0 . (Inria) On a mean-field model of interacting neurons July 5, 2019 10 / 25

Assumptions Given ( a t ) t ≥ 0 ∈ C ( R + , R ) “any external current”, let ϕ ( a. ) t,s ( x ) be the flow solution of: d dtϕ ( a. ) t,s ( x ) = b ( ϕ ( a. ) ϕ ( a. ) t,s ( x )) + a t , s,s ( x ) = x. A1 b : R + → R is continuous, b (0) > 0 , bounded from above. A2 There exists a constant C : for all ( a t ) t ≥ 0 , ( d t ) t ≥ 0 : � t ∀ x ≥ 0 , ∀ t ≥ s, | ϕ ( a. ) t,s ( x ) − ϕ ( d. ) t,s ( x ) | ≤ C | a u − d u | du. s A3 f : R + → R + is C 1 convex increasing, f (0) = 0 + some technical assumptions on the grow of f . A4 The initial condition ν = L ( V 0 ) satisfies: � ν ( f 2 ) := f 2 ( x ) ν ( dx ) < ∞ . R + (Inria) On a mean-field model of interacting neurons July 5, 2019 11 / 25

What are the invariant measures of this N-L process? � t � t � t � V t = V 0 + b ( V u ) du + J E f ( V u ) du − V u − ✶ { z ≤ f ( V u − ) } N ( du, dz ) 0 0 0 R + In (M-F) replace the interactions J E f ( X t ) by the constant α ≥ 0 : � t � t � Y α t = Y α b ( Y α Y α 0 + u ) du + αt − u − ✶ { z ≤ f ( Y α u − ) } N ( du, dz ) . 0 0 R + This process has an unique invariant measure given by: � � � x γ ( α ) f ( y ) ν ∞ α ( dx ) = b ( x ) + α exp − b ( y ) + αdy ✶ { x ∈ [0 ,σ α ] } dx, 0 σ α = lim t →∞ ϕ α t, 0 (0) ∈ R ∗ + ∪ { + ∞} . � ν ∞ γ ( α ) is the normalizing factor (such that α ( dx ) = 1 .) It holds that γ ( α ) = ν ∞ α ( f ) . The invariant measures of (M-F) are exactly: { ν ∞ α : α = Jγ ( α ) , α ≥ 0 } . (Inria) On a mean-field model of interacting neurons July 5, 2019 12 / 25

The case of small interactions Theorem (C., Tanré, Veltz 2018) Under A1, A2, A3, A4 : 1. the N-L SDE (M-F) has a path-wise unique solution with sup t ≥ 0 E f ( V t ) < ∞ . 2. if the interaction parameter J is small enough, then ( V t ) has an unique invariant measure which is globally stable : starting from any initial condition, V t converges in law to the unique invariant measure. The convergence is exponentially fast. (Inria) On a mean-field model of interacting neurons July 5, 2019 13 / 25

Examples Consider for all x ≥ 0 : f ( x ) = x ξ . b ( x ) = b 0 − b 1 x, For b 0 > 0 , b 1 ≥ 0 and ξ ≥ 1 , its satisfies all the assumptions. (Inria) On a mean-field model of interacting neurons July 5, 2019 14 / 25

Examples For b ( x ) = 0 . 1 − x , f ( x ) = x 2 . J < J 1 : one unique invariant measure. J 1 < J < J 2 : three invariant measures, 2 are stable: bi-stability . J > J 2 : one unique invariant measure. (Inria) On a mean-field model of interacting neurons July 5, 2019 15 / 25

Examples For b ( x ) = 2 − 2 x , f ( x ) = x 10 . Always exactly one invariant measure. But if J ∈ [0 . 73 , 1 . 04] spontaneous oscillation of t → E f ( V t ) appears! The law of V t asymptotically oscillates (Video !). The invariant measure looses its stability. There is a Hopf bifurcation for J ≈ 0 . 73 . (Inria) On a mean-field model of interacting neurons July 5, 2019 16 / 25

Sketch of the Proof 1) Introduce a linearized version of the N-L equation (M-F). Given ( a t ) t ≥ 0 , consider � t � t � t � Y ν, ( a. ) = Y ν, ( a. ) Y ν, ( a. ) b ( Y ν, ( a. ) + ) du + a u du − ✶ { z ≤ f ( Y ν, ( a. ) ) } N ( du, dz ) . t 0 u u − u − 0 0 0 R + The interactions J E f ( V t ) have been replaced by a t . Then ( Y ν, ( a. ) ) is a solution of (M-F) if and only if: t ∀ t ≥ 0 : a t = J E f ( Y ν, ( a. ) ) . t (Inria) On a mean-field model of interacting neurons July 5, 2019 17 / 25

Sketch of the Proof 2) The jump rate of this linearized process solves a Volterra equation : ( a. ) ( t ) := E f ( Y ν, ( a. ) let r ν ) . Then t � t ∀ t ≥ 0 , r ν ( a. ) ( t ) = K ν K δ 0 ( a. ) ( t, u ) r ν ( a. ) ( t, 0) + ( a. ) ( u ) du, 0 with for all x ≥ 0 , t ≥ s � � � t ( a. ) ( t, s ) := f ( ϕ ( a. ) K δ x f ( ϕ ( a. ) t,s ( x )) exp − u,s ( x )) du , s � ∞ K ν K δ x ( a. ) ( t, s ) := ( a. ) ( t, s ) ν ( dx ) . 0 (Inria) On a mean-field model of interacting neurons July 5, 2019 18 / 25

Sketch of the Proof 3) We first study the case ( a. ) constant and equal to α . In that case the Volterra equation become a convolution Volterra equation . We prove that for all 0 ≤ λ < λ ∗ α t ) − γ ( α ) | e λt < ∞ . | E f ( Y α sup t ≥ 0 The number λ ∗ α > 0 is the largest real part of the Complex zeros of the � � � t 0 f ( ϕ α Laplace transform of H α ( t ) := exp − u ) du . (Inria) On a mean-field model of interacting neurons July 5, 2019 19 / 25