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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

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Synchronization on complex networks

A model for neural networks Janusz Meylahn

Mathematical Institute - Leiden University

18 April 2018

Janusz Meylahn Synchronization on complex networks

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§ What will I tell you today?

a short introduction

Janusz Meylahn Synchronization on complex networks

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§ What will I tell you today?

a short introduction what a stochastic process is

Janusz Meylahn Synchronization on complex networks

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§ 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

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§ 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

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§ 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

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§ Introduction

Processes on networks

Janusz Meylahn Synchronization on complex networks

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§ Introduction

Processes on networks

1 Spreading of rumour Janusz Meylahn Synchronization on complex networks

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§ Introduction

Processes on networks

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

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§ 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

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§ 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

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§ 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

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§ 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

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§ 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

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§ 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

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§ Synchronization

Examples of synchronization

Janusz Meylahn Synchronization on complex networks

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§ Synchronization

Examples of synchronization

1 Fireflies flashing in the jungle Janusz Meylahn Synchronization on complex networks

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§ Synchronization

Examples of synchronization

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

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§ 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

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§ 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

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§ 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

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§ 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

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§ 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

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§ YouTube Video

Janusz Meylahn Synchronization on complex networks

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§ Stochastic processes

Question: What do you think?

Janusz Meylahn Synchronization on complex networks

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§ Stochastic processes

Question: What do you think? Ingredients some randomness

Janusz Meylahn Synchronization on complex networks

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§ Stochastic processes

Question: What do you think? Ingredients some randomness a recipe describing situation as function of randomness

Janusz Meylahn Synchronization on complex networks

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§ 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

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§ 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

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§ (Noisy) Kuramoto model

Janusz Meylahn Synchronization on complex networks

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§ (Noisy) Kuramoto model

Achtung! Mathematics ahead!

Janusz Meylahn Synchronization on complex networks

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§ (Noisy) Kuramoto model

Achtung! Mathematics ahead! Consider: N – oscillators θi(t) – phase of ith oscillator

Janusz Meylahn Synchronization on complex networks

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§ (Noisy) Kuramoto model

Achtung! Mathematics ahead! Consider: N – oscillators θi(t) – phase of ith oscillator Oscillators evolve according to a system of coupled stochastic differential equations dθi(t) = K N

N

  • j=1

sin

  • θj(t) − θi(t)
  • dt + D dWi(t).

(1) Here, K ∈ (0, ∞) is the interaction strength, D ∈ (0, ∞) is the noise strength, and (Wi(t))t≥0 are noise processes.

Janusz Meylahn Synchronization on complex networks

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§ (Noisy) Kuramoto model

Achtung! Mathematics ahead! Consider: N – oscillators θi(t) – phase of ith oscillator Oscillators evolve according to a system of coupled stochastic differential equations dθi(t) = K N

N

  • j=1

sin

  • θj(t) − θi(t)
  • dt + D dWi(t).

(1) Here, K ∈ (0, ∞) is the interaction strength, D ∈ (0, ∞) is the noise strength, and (Wi(t))t≥0 are noise processes. Question: Can you spot the network here?

Janusz Meylahn Synchronization on complex networks

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Cartoon of the Kuramoto model for N = 6

𝜕1 𝜕2 𝜕3 𝜕5 𝜕6 𝜄6 𝜄5 𝜕4 𝜄4 𝜄3 𝜄2 𝜄1

Janusz Meylahn Synchronization on complex networks

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Keeping track of the order

Janusz Meylahn Synchronization on complex networks

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Keeping track of the order

Order parameter rN(t) eiψN(t) = 1 N

N

  • j=1

eiθj(t). (2)

Janusz Meylahn Synchronization on complex networks

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Keeping track of the order

Order parameter rN(t) eiψN(t) = 1 N

N

  • j=1

eiθj(t). (2) rN(t) – synchronization level ψN(t) – average phase

Janusz Meylahn Synchronization on complex networks

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Keeping track of the order

Order parameter rN(t) eiψN(t) = 1 N

N

  • j=1

eiθj(t). (2) rN(t) – synchronization level ψN(t) – average phase

Phase distributions with r = 0.095 and r = 0.929.

Janusz Meylahn Synchronization on complex networks

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Taking limits

Rewriting using the order parameter (exercise) gives dθi(t) = KrN(t) sin

  • ψN(t) − θi(t)
  • dt + D dWi(t),

(3)

Janusz Meylahn Synchronization on complex networks

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Taking limits

Rewriting using the order parameter (exercise) gives dθi(t) = KrN(t) sin

  • ψN(t) − θi(t)
  • dt + D dWi(t),

(3) The large N limit As N gets ever larger, you can describe the evolution of the

  • scillators as the evolution of a density.

Janusz Meylahn Synchronization on complex networks

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Taking limits

Rewriting using the order parameter (exercise) gives dθi(t) = KrN(t) sin

  • ψN(t) − θi(t)
  • dt + D dWi(t),

(3) The large N limit As N gets ever larger, you can describe the evolution of the

  • scillators as the evolution of a density.

But what is a density??

Janusz Meylahn Synchronization on complex networks

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Taking limits

Rewriting using the order parameter (exercise) gives dθi(t) = KrN(t) sin

  • ψN(t) − θi(t)
  • dt + D dWi(t),

(3) The large N limit As N gets ever larger, you can describe the evolution of the

  • scillators as the evolution of a density.

But what is a density?? The large time limit (steady-state) Question: Does the density of the system stop evolving at some point?

Janusz Meylahn Synchronization on complex networks

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Critical threshold

There exists a critical threshold Kc such that: (I) For K < Kc the system relaxes to an unsynchronized state (r = 0). (II) For K > Kc the system relaxes to a partially synchronized state (r > 0).

Janusz Meylahn Synchronization on complex networks

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Critical threshold

There exists a critical threshold Kc such that: (I) For K < Kc the system relaxes to an unsynchronized state (r = 0). (II) For K > Kc the system relaxes to a partially synchronized state (r > 0). Theorem Kc = 2 (4)

2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 K rK

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus SCN has a strong community structure

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus SCN has a strong community structure interaction between the communities is negative

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus SCN has a strong community structure interaction between the communities is negative this is the case in all mammals

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus SCN has a strong community structure interaction between the communities is negative this is the case in all mammals structure might play a role in richness and robustness of SCN

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus SCN has a strong community structure interaction between the communities is negative this is the case in all mammals structure might play a role in richness and robustness of SCN malfunctioning can cause health problems ranging from epilepsy to narcolepsy

Janusz Meylahn Synchronization on complex networks

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§ ‘Complex’ Network

Suprachiasmatic nucleus SCN has a strong community structure interaction between the communities is negative this is the case in all mammals structure might play a role in richness and robustness of SCN malfunctioning can cause health problems ranging from epilepsy to narcolepsy

Janusz Meylahn Synchronization on complex networks

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Synchronization on complex network

4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 K L2 rK

Janusz Meylahn Synchronization on complex networks

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§ Conclusion

What should you remember?

Janusz Meylahn Synchronization on complex networks

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§ Conclusion

What should you remember? There are many different types of processes to study on networks

Janusz Meylahn Synchronization on complex networks

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§ Conclusion

What should you remember? There are many different types of processes to study on networks Networks really play an important role almost everywhere

Janusz Meylahn Synchronization on complex networks

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§ Conclusion

What should you remember? There are many different types of processes to study on networks Networks really play an important role almost everywhere Synchronization is an example that is particularly interesting

Janusz Meylahn Synchronization on complex networks

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§ Conclusion

What should you remember? There are many different types of processes to study on networks Networks really play an important role almost everywhere Synchronization is an example that is particularly interesting Using mathematics we can tell neuroscientists something of value

Janusz Meylahn Synchronization on complex networks

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§ Conclusion

What should you remember? There are many different types of processes to study on networks Networks really play an important role almost everywhere Synchronization is an example that is particularly interesting Using mathematics we can tell neuroscientists something of value Inter-disciplinary research is becoming more and more important

Janusz Meylahn Synchronization on complex networks