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sync

by Yuanzhao Zhang (Submitted: 01/03/2014)

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## WHAT IS IT?

This model explores the synchronization of pulse-coupled oscillators on complex networks when the network itself is influenced by oscillator dynamics.

## HOW IT WORKS

The oscillators in this model agree with Strogatz’s integrate and fire model in spirit. But instead of continuous time I use discrete tick. At every tick the phase will advance by a positive value, until it reaches a threshold and fires immediately, at which time the phase is reset to 0. The value becomes smaller as the phase approaches the threshold, which makes the phase course monotonously increasing and concave (down). A pulse is sent by a firing node to all other connected nodes, which will induce a phase jump at next tick. But the firing will also strengthen a node's connected links, who comes with an initial strength that will be reduced by 1 at every tick. If the strength of a link reaches 0, it will die out, at which circumstance a new link will be generated randomly to keep the total number constant. This mechanism is inspired by neural network development, where seldom used links soon die out and heavily used links get strengthened.

In the scheme described above, All the nodes firing in the same tick can make their full impact, this I shall refer to as rule "0". I also investigated two other rules where things are not so strongly coupled. For rule "1", all the firing nodes can still strengthen its links, but will not introduce phase jump unless it is the first to fire among all the nodes connected with it. The firing order in one tick is random. With rule "2" this restriction is extended to both phase jump and link strengthen. In other words, a node will be completely muted if there is a connected node fires before it. These two new rules introduce a new time varying mechanism for network structure, since the effect of these restrictions can be roughly viewed as removing links connected to the muted nodes at every tick, thus make the "connection" more sparse and ever-changing even when there is no link breaking.

## HOW TO USE IT

You can choose among three different rules by selecting the chooser for "0", "1" or "2".

The "nonlinear" switch let you choose between linear and nonlinear phase course for nodes, and the "delay" switch let you decide whether there is a delay between firing and phase jump.

The "add-clustered" switch let you add the link randomly or cluster-favorablely.

## THINGS TO TRY

Try and see what's the difference induced by three different rules.

What's the difference between clustered network and random network on network stabillity, transition time, etc?

What's the difference made by "delay"?

Try this model for all-to-all connection.Is there any novel form of synchronization?

## EXTENDING THE MODEL

Make the network directed; Includes refactory period; Set up Assortative, Small-world, or Scale-free networks; Use non-indentical oscillators.

## CREDITS AND REFERENCES

ArXiv.org (2014) "Novel Synchronization of Pulse-coupled Oscillators on Time-varying Networks" Yuanzhao Zhang

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