The influence of the disorder in the Kuramoto model Eric Luçon Université René Descartes - Paris 5 Partial joint works with Giambattista Giacomin and Christophe Poquet and with Wilhelm Stannat Random Dynamical Systems, Bielefeld, Nov. 2, 2013 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 1 / 33
Mean-field interacting diffusions 1 N → ∞ : Law of Large Numbers 2 The example of the Kuramoto model 3 The symmetric case: fluctuations around the McKean-Vlasov equation 4 Spatially extended neurons 5 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 2 / 33
Synchronization of individuals Emergence of synchrony is widely encountered in complex systems of individuals in interaction (networks of neurons, collective behavior of social insects, chemical interactions between cells, planets orbiting, . . . ). Three main ingredients for interacting individuals: a dynamics for each individual (e.g. FitzHugh-Nagumo or Hodgkin-Huxley 1 for neurons) a network of interactions (possibly heterogeneous and delayed) 2 absence or presence of a thermal noise (possibly correlated). 3 Under these conditions, for a sufficiently strong interaction between individuals and for a sufficiently large population, synchronization should occur (individuals exhibit similar simultaneous behavior). Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 3 / 33
Outline Mean-field interacting diffusions 1 N → ∞ : Law of Large Numbers 2 The example of the Kuramoto model 3 The symmetric case: fluctuations around the McKean-Vlasov equation 4 Spatially extended neurons 5 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 4 / 33
General framework: mean-field interacting diffusions in R m Here, each individual θ is a diffusion in R p . For T > 0, N � 1, consider t ∈ [ 0 , T ] �→ ( θ 1 ( t ) ,..., θ N ( t )) solution to N d θ i ( t ) = c ( θ i ) d t + 1 ∑ Γ( θ i , θ j ) d t + σ d B i ( t ) , i = 1 , ··· , N , N j = 1 c ( · ) : local dynamics of one individual Γ( · , · ) : interaction kernel B i : i.i.d. Brownian motions (thermal noise). Exchangeability If at t = 0, the vector ( θ 1 ( 0 ) ,..., θ N ( 0 )) is exchangeable, then, at all time t > 0, the law of the vector ( θ 1 ( t ) ,..., θ N ( t )) is also exchangeable. Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 5 / 33
An example Interesting examples include the granular media system: N d θ i ( t ) = − ∇ V ( θ i ) d t − 1 ∑ ∇ W ( θ i − θ j ) d t + σ d B i ( t ) , N j = 1 for V and W having convexity properties. V ( θ ) θ [ ✎ Carillo, McCann, Villani, Malrieu, Guillin, Cattiaux, Berglund, Gentz, Tugaut, etc.] Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 6 / 33
Interacting diffusions in a random environment Exchangeability may not be a suitable property: one needs to encode the fact that the dynamics may not be the same for each individual (e.g., inhibition or excitation for a neuron). Idea: set a sequence of i.i.d. random variables ( ω i ) encoding the intrinsic behavior of the individual θ i . For this choice of disorder ( ω 1 , ω 2 ,..., ω N ) , we modify the dynamics and the interaction in the following way: N d θ i ( t ) = c ( θ i , ω i ) d t + 1 ∑ Γ( θ i , θ j , ω i , ω j ) d t + σ d B i ( t ) , i = 1 , ··· , N . N j = 1 Question What is the (quenched) influence of the disorder on the long-time/large N behavior of the system in comparison with the case without disorder? Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 7 / 33
Disordered mean-field models in neuroscience Consider FitzHugh-Nagumo dynamics for the spiking activity of one neuron θ = ( v , W ) ∈ R 2 i.e. � ε ˙ = V − V 3 / 3 + W + I V ˙ = aW + bV , W where V is the membrane potential and W is a recovery variable. The disorder ω = ( a , b ) encodes the state (inhibited/excited) of one neuron. N d θ i ( t ) = c ( θ i , ω i ) d t + 1 ∑ Γ( θ i , θ j , ω i , ω j ) d t + σ d B i ( t ) , i = 1 , ··· , N , N j = 1 Here Γ models synaptic connections between neurons. [ ✎ O. Faugeras, J. Touboul et al.: similar systems with delay, two-scale of population, disorder, etc.] Difficulty: absence of reversibility. Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 8 / 33
A simpler model: the Kuramoto model ✎ Kuramoto [’75], Strogatz, Giacomin, Pakdaman, Pellegrin, Poquet, Bertini, L. The state space here is the one-dimensional circle: S := R / 2 π : N d θ i ( t ) = ω i d t + K ∑ sin ( θ j − θ i ) d t + σ d B i ( t ) , i = 1 ,..., N , N j = 1 Intuition: competition between ω i dt : random intrinsic speed of rotation for each rotator θ i , K sin ( · ) dt : synchronizing kernel between rotators. Absence of disorder = reversibility If ∀ i = 1 ,..., N , ω i = 0, the dynamics is reversible. Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 9 / 33
Example: The disorder is chosen by a toss of coins µ = 1 2 ( δ − 1 + δ 1 ) . Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
Example: The disorder is chosen by a toss of coins µ = 1 2 ( δ − 1 + δ 1 ) . − 1 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
Example: The disorder is chosen by a toss of coins µ = 1 2 ( δ − 1 + δ 1 ) . − 1 +1 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
Example: The disorder is chosen by a toss of coins µ = 1 2 ( δ − 1 + δ 1 ) . − 1 +1 Questions: what is the influence of the disorder on the system? Does it depend only on its law µ (centered, symmetric or not) or on a typical realization ( ω 1 ,..., ω N ) ? Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
Example: The disorder is chosen by a toss of coins µ = p δ − 1 +( 1 − p ) δ 1 . − 1 Questions: what is the influence of the disorder on the system? Does it depend only on its law µ (centered, symmetric or not) or on a typical realization ( ω 1 ,..., ω N ) ? Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 10 / 33
Simulation I: N = 500, K = 3, σ = 1, no disorder N d θ i ( t ) = K ∑ sin ( θ j − θ i ) d t + σ d B i ( t ) , i = 1 ,..., N , N j = 1 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 11 / 33
Simulation II: N = 600, K = 6, σ = 1, µ = 1 2 ( δ − 1 + δ 1 ) N d θ i ( t ) = ω i d t + K ∑ sin ( θ j − θ i ) d t + σ d B i ( t ) , i = 1 ,..., N , N j = 1 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 12 / 33
Outline Mean-field interacting diffusions 1 N → ∞ : Law of Large Numbers 2 The example of the Kuramoto model 3 The symmetric case: fluctuations around the McKean-Vlasov equation 4 Spatially extended neurons 5 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 13 / 33
The empirical measure N d θ i ( t ) = c ( θ i , ω i ) d t + 1 ∑ Γ( θ i , θ j , ω i , ω j ) d t + σ d B i ( t ) , i = 1 , ··· , N . N j = 1 We want to understand the (quenched vs annealed) behavior as N → ∞ of the empirical measure N ν N , t := 1 ∑ δ ( θ j ( t ) , ω j ) . N j = 1 Law of Large Numbers? Does the continuous limit say anything on the particle system? Central Limit Theorem? Large Deviations? Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 14 / 33
Quenched convergence of the empirical measure ν N Under Lipschitz regularity on c and Γ , moment condition on µ and N → ∞ convergence of the initial condition ν ( ω ) → ν 0 , − N , 0 Proposition (Quenched LLN - L. 2011) For a.e. ( ω i ) i � 1 , the empirical measure ν ( ω ) converges in law, as a process to N t �→ ν t ( d θ , d ω ) that is the unique weak solution to the following McKean-Vlasov equation: � � �� ∂ t ν t = 1 � � σσ T ∇ θ ν t � − div θ ν t c ( θ , ω )+ Γ( θ , ω , · , · ) d ν t . 2 div θ Self-averaging phenomenon At the level of the LLN, the dependence in the disorder lies in its law, not a typical realization. Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 15 / 33
Outline Mean-field interacting diffusions 1 N → ∞ : Law of Large Numbers 2 The example of the Kuramoto model 3 The symmetric case: fluctuations around the McKean-Vlasov equation 4 Spatially extended neurons 5 Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 16 / 33
McKean-Vlasov equation in the Kuramoto model In the limit of an infinite population: q t ( θ , ω ) , density at time t of oscillators with phase θ and frequency ω solves ∂ t q t ( θ , ω ) = σ 2 � � �� 2 ∆ q t ( θ , ω ) − K ∂ θ q t ( θ , ω ) � sin ∗ q t � µ ( θ )+ ω . What makes the Kuramoto tractable is that the nonlinearity is nice (it only concerns the first Fourier coefficients of q ). In particular, if there is no disorder, one can show that the microscopic system is reversible, there exists a Lyapounov functional for the continuous model. From this, one can derive many things in the case of small disorder, by perturbation arguments. Eric Luçon (Paris 5) The disordered Kuramoto model Nov. 2, 2013 17 / 33
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