Glossary - Robust Methods of Amorphous Synchronization

## Glossary

A note on terminology: very little of the terminology in the literature is standardized. I have tried to include the most often-used terminology, and in some cases have included multiple definitions. When necessary, it will be specified which definition I am using.

Action potential: See Potential.

Coupling: excitatory, inhibitory: Excitatory coupling is when the coupling strength is positive, and inhibitory when the coupling strength is negative.

Coupling strength: The amount by which the state is increased in the integrate-and-fire algorithms. When the coupling strength is only dependent on the transmitting processor, it is sometimes referred to as the firing strength.

Excitatory: neuron, coupling: An excitatory neuron returns to its rest potential rather than to its threshold in the absence of external inputs.
See Coupling.

Fire: The event we are trying to synchronize (e.g. taking a reading), and often coincides with transmitting a message.

Firing map: The map h(p) of the phase of one integrate-and-fire processor to the phase the other will be in after the first fires.

Firing strength: See Coupling strength.

Frequency-locking: The entire network running at the same average frequency, according to Mirollo & Strogatz (1990).
Being consistent with the definition of phase-locking leads to all nodes have fixed differences between their frequencies.

Graphs: all-to-all, circulant, connected, lattice: A graph is composed of nodes and vertices between nodes.
An all-to-all, or completely connected graph, is one in which every node is connected to every other.
A circulant graph is defined by a set of positive integers. The nodes are arranged (spatially, for the purposes of thinking about it) in a ring, and every node is connected to the one N to the right and N to the left of it, for each N in the set.
A connected graph is one in which there is a path from one node to every other node.
A lattice is a graph in which (effectively) the nodes are spatially arranged in a grid and every node is connected to its N,W,S,and E neighbors.

Inhibitory coupling: See Coupling.

Integrate-and-fire: The basic model of the neuron in which a neuron receives inputs which increase its voltage from its resting voltage. When it reaches its threshold voltage, it sends out an action potential.

Lag: See Latency.

Latency: processing, transmission: Processing latency is the amount of time it takes a processor to transmit a message once it realizes it has to, or the time it takes a processor to interpret and respond to a message once it has received one.
Transmission latency is the amount of time it takes a message to get from one processor to another. Sometimes processing latency on the transmitter's end is included in the transmission latency for simplicity.

Node: See Graph. Also used interchangably with Oscillator and Processor (in this document).

Oscillator: An object which oscillates at a certain frequency. Used interchangeably with Node and Processor (in this document).

Oscillatory neuron: As opposed to an Excitatory neuron, oscillatory neurons approach (and eventually reach) their threshold in the absence of external inputs.

PRC: See Phase response curve.

Period-locking: See Frequency-locking.

Phase-locking: When the phase difference between any two nodes in the network is fixed.

Phase: A percentage of the time to firing.

Phase response curve: A curve defining how much the phase is advanced (or retarded) when a node receives a spike at a given phase.

Potential: action, reset, resting, threshold: The action potential is the potential transmitted from a spiking neuron. Analogous to a processor firing.
The reset potential is the voltage to which a neuron resets after firing (not always, but usually the same, as the resting potential).
The resting potential is the reference potential. When nothing has affected the neuron at all, it is in this state.
The threshold potential is the potential at which a neuron will fire and then reset to its reset potential.

Processor: The basic element of amorphous computing. Used interchangably with Node and Oscillator (in this document).

Refractory period: A period after firing in which a neuron is resetting to its reset potential. In our integrate-and-fire algorithm, this is also a period in which the node does not respond to inputs.

Reset potential: See Potential.

Resting potential: See Potential.

Return map: In a two-oscillator system in which node A has just fired, the return map takes in the initial phase of B and returns the phase of B after the next time A fires.

Spike: See Fire.

State function: Equivalent to the voltage function of a neuron. Lets you know the state given phase. Also determines the PRC. Often it is easier to skip this and go straight to the PRC.

Symmetry-breaking: When two oscillators are effectively identical, how we decide which one to synchronize on. Basic symmetry-breaking does this by having each choose a random number and whoever has the higher wins.

Synchronization: alpha, full, in-phase, noisy, out-of-phase: Alpha-synchronization is when a network all fires within alpha*d were d is the diameter of the network.
Full synchronization is when the entire network will fire at the same time (note that this does not require that the network be phase-locked throughout its phase, only when it fires).
In-phase synchronization is another name for full synchronization.
Noisy synchronization is when an entire network fires within a time alpha (a more restrained version of alpha-synchronization, not dependent on the size of the network).
Out-of-phase synchronization is just phase-locking.

Tetany: A state of rapid firing, usually induced in laggy situations with large lag or large coupling strength.

Threshold: See Potential.

Transmit: When one processor sends a message to another. This is used to distinguish sending a message from, for example, taking a reading, though the two often coincide.

Voltage: action, reset, resting, threshold: See Potential.