Synchronization on complex networks A model for neural networks - PowerPoint PPT Presentation

Synchronization on complex networks A model for neural networks Janusz Meylahn Mathematical Institute - Leiden University 18 April 2018 Janusz Meylahn Synchronization on complex networks

§ What will I tell you today? a short introduction Janusz Meylahn Synchronization on complex networks

§ What will I tell you today? a short introduction what a stochastic process is Janusz Meylahn Synchronization on complex networks

§ What will I tell you today? a short introduction what a stochastic process is synchronization: what, how and why? Janusz Meylahn Synchronization on complex networks

§ What will I tell you today? a short introduction what a stochastic process is synchronization: what, how and why? Kuramoto: a mathematical model Janusz Meylahn Synchronization on complex networks

§ What will I tell you today? a short introduction what a stochastic process is synchronization: what, how and why? Kuramoto: a mathematical model synchronization on networks Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet 3 Formation of polymers Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet 3 Formation of polymers 4 Synchronization of neurons firing in brain What are some differences here? Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet 3 Formation of polymers 4 Synchronization of neurons firing in brain What are some differences here? 1 Network as interactions or as paths Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet 3 Formation of polymers 4 Synchronization of neurons firing in brain What are some differences here? 1 Network as interactions or as paths 2 Process on each site or moving on network Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet 3 Formation of polymers 4 Synchronization of neurons firing in brain What are some differences here? 1 Network as interactions or as paths 2 Process on each site or moving on network 3 Continuous space or discrete space Janusz Meylahn Synchronization on complex networks

§ Introduction Processes on networks 1 Spreading of rumour 2 Searching for information on the internet 3 Formation of polymers 4 Synchronization of neurons firing in brain What are some differences here? 1 Network as interactions or as paths 2 Process on each site or moving on network 3 Continuous space or discrete space 4 Dynamic or static network Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle 2 Electricity generators on power grid Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle 2 Electricity generators on power grid 3 Audience clapping after concert Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle 2 Electricity generators on power grid 3 Audience clapping after concert 4 Neurons firing in the brain Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle 2 Electricity generators on power grid 3 Audience clapping after concert 4 Neurons firing in the brain 5 Gravitational synchronization of meteors Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle 2 Electricity generators on power grid 3 Audience clapping after concert 4 Neurons firing in the brain 5 Gravitational synchronization of meteors 6 and many more... Janusz Meylahn Synchronization on complex networks

§ Synchronization Examples of synchronization 1 Fireflies flashing in the jungle 2 Electricity generators on power grid 3 Audience clapping after concert 4 Neurons firing in the brain 5 Gravitational synchronization of meteors 6 and many more... If you are looking for your next popular science book to read try: ‘Sync: The emerging science of spontaneous order’ - Steven Strogatz Janusz Meylahn Synchronization on complex networks

§ YouTube Video Janusz Meylahn Synchronization on complex networks

§ Stochastic processes Question: What do you think? Janusz Meylahn Synchronization on complex networks

§ Stochastic processes Question: What do you think? Ingredients some randomness Janusz Meylahn Synchronization on complex networks

§ Stochastic processes Question: What do you think? Ingredients some randomness a recipe describing situation as function of randomness Janusz Meylahn Synchronization on complex networks

§ Stochastic processes Question: What do you think? Ingredients some randomness a recipe describing situation as function of randomness some idea of time Janusz Meylahn Synchronization on complex networks

§ Stochastic processes Question: What do you think? Ingredients some randomness a recipe describing situation as function of randomness some idea of time Example: Coin flipping win 1 e if heads lose 1 e if tails Exercise: ω = { H , T , T , T , T , H , T , H . . . } Janusz Meylahn Synchronization on complex networks

§ (Noisy) Kuramoto model Janusz Meylahn Synchronization on complex networks

§ (Noisy) Kuramoto model Achtung! Mathematics ahead! Janusz Meylahn Synchronization on complex networks

§ (Noisy) Kuramoto model Achtung! Mathematics ahead! Consider: N – oscillators θ i ( t ) – phase of i th oscillator Janusz Meylahn Synchronization on complex networks

§ (Noisy) Kuramoto model Achtung! Mathematics ahead! Consider: N – oscillators θ i ( t ) – phase of i th oscillator Oscillators evolve according to a system of coupled stochastic differential equations N d θ i ( t ) = K � � � sin θ j ( t ) − θ i ( t ) d t + D d W i ( t ) . (1) N j =1 Here, K ∈ (0 , ∞ ) is the interaction strength, D ∈ (0 , ∞ ) is the noise strength, and ( W i ( t )) t ≥ 0 are noise processes. Janusz Meylahn Synchronization on complex networks

§ (Noisy) Kuramoto model Achtung! Mathematics ahead! Consider: N – oscillators θ i ( t ) – phase of i th oscillator Oscillators evolve according to a system of coupled stochastic differential equations N d θ i ( t ) = K � � � sin θ j ( t ) − θ i ( t ) d t + D d W i ( t ) . (1) N j =1 Here, K ∈ (0 , ∞ ) is the interaction strength, D ∈ (0 , ∞ ) is the noise strength, and ( W i ( t )) t ≥ 0 are noise processes. Question: Can you spot the network here? Janusz Meylahn Synchronization on complex networks

Cartoon of the Kuramoto model for N = 6 𝜄 2 𝜕 1 𝜕 2 𝜄 1 𝜄 6 𝜕 6 𝜕 3 𝜄 3 𝜄 5 𝜄 4 𝜕 5 𝜕 4 Janusz Meylahn Synchronization on complex networks

Keeping track of the order Janusz Meylahn Synchronization on complex networks

Keeping track of the order Order parameter N r N ( t ) e i ψ N ( t ) = 1 � e i θ j ( t ) . (2) N j =1 Janusz Meylahn Synchronization on complex networks

Keeping track of the order Order parameter N r N ( t ) e i ψ N ( t ) = 1 � e i θ j ( t ) . (2) N j =1 r N ( t ) – synchronization level ψ N ( t ) – average phase Janusz Meylahn Synchronization on complex networks

Keeping track of the order Order parameter N r N ( t ) e i ψ N ( t ) = 1 � e i θ j ( t ) . (2) N j =1 r N ( t ) – synchronization level ψ N ( t ) – average phase Phase distributions with r = 0 . 095 and r = 0 . 929. Janusz Meylahn Synchronization on complex networks

Taking limits Rewriting using the order parameter (exercise) gives � � d θ i ( t ) = Kr N ( t ) sin ψ N ( t ) − θ i ( t ) d t + D d W i ( t ) , (3) Janusz Meylahn Synchronization on complex networks

Taking limits Rewriting using the order parameter (exercise) gives � � d θ i ( t ) = Kr N ( t ) sin ψ N ( t ) − θ i ( t ) d t + D d W i ( t ) , (3) The large N limit As N gets ever larger, you can describe the evolution of the oscillators as the evolution of a density. Janusz Meylahn Synchronization on complex networks