Relative phase and Traveling waves:
Building blocks of coordination patterns in free locomotor behavior

Neri Kafkafi    National Institute of Drug Abuse, Baltimore, USA

Ilan Golani     Tel-Aviv University, Tel Aviv, Israel

Phase is the cyclic variable of periodic movement

Phase is measured in parts of a cycle, from 0 to 1 (which is 0 again!). Since it is cyclic, we code it with the circle of colors

The simplest case: strictly periodic movement in one joint. We use the color to code the phase in both the time series and the animation of movement

To operate the animation  -  
        double click on the black and red rod below .  
            You can control the animation parameters by pressing the buttons on the toolbar at the lower left corner.

There are three parameters of periodic movement

Amplitude: You can make it smaller or larger

Period (the inverse of frequency): You can make it slower or faster

Center of oscillation: You can make it oscillate around another angle

A discrete movement can be treated as a part of a periodic movement

Each of these discrete movements can be trated as half a cycle of a periodic movement. Every instant of it can thus be assigned a phase (i.e., color). During immobility, phase has no meaning (grey)

While turning, each joint in the body axis of a quadruped or a fish bends to one side, then straightens again. This can be treated as a single cycle of a periodic movement, which is centered around a center of oscillation different than zero

Coordination patterns of several periodic movements usually involve stable relative phase

Relative Phase = 0 cycles (in-phase)

Relative Phase = 0.5 cycles (anti-phase)

Relative Phase = 0.2 cycles (a slight delay of one joint relative to the other)

A travelling wave is created when the phase in each consecutive joint delays after the former joint

For example, six joints with a relative phase of 0.2 cycles between them (the last one has the same phase as the first!)

A travelling  wave in a model animal which is built of six consecutive joints. The length of this travelling wave (i.e., the distance between same colores) is one body length.

Conclusion: Relative phase and travelling waves may be used as building blocks of complex coordination patterns, not only with strictly periodic behaviors, but also with free behavior   (continued in the "phase animation notebook)

Neri Kafkafi and Ilan Golani 

A traveling wave of lateral movement coordinates both turning and forward walking in the ferret 

Volume 78, Issue 6, pp 441-453 
Biological Cybernetics, July 20, 1998