Biological Cybernetics

, Volume 107, Issue 2, pp 179–200

Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches

  • Shai Revzen
  • Samuel A. Burden
  • Talia Y. Moore
  • Jean-Michel Mongeau
  • Robert J. Full
Original Paper

DOI: 10.1007/s00422-012-0545-z

Cite this article as:
Revzen, S., Burden, S.A., Moore, T.Y. et al. Biol Cybern (2013) 107: 179. doi:10.1007/s00422-012-0545-z


Instantaneous kinematic phase calculation allows the development of reduced-order oscillator models useful in generating hypotheses of neuromechanical control. When perturbed, changes in instantaneous kinematic phase and frequency of rhythmic movements can provide details of movement and evidence for neural feedback to a system-level neural oscillator with a time resolution not possible with traditional approaches. We elicited an escape response in cockroaches (Blaberus discoidalis) that ran onto a movable cart accelerated laterally with respect to the animals’ motion causing a perturbation. The specific impulse imposed on animals (0.50 \(\pm \) 0.04 m s\(^{-1}\); mean, SD) was nearly twice their forward speed (0.25 \(\pm \) 0.06 m s\(^{-1})\). Instantaneous residual phase computed from kinematic phase remained constant for 110 ms after the onset of perturbation, but then decreased representing a decrease in stride frequency. Results from direct muscle action potential recordings supported kinematic phase results in showing that recovery begins with self-stabilizing mechanical feedback followed by neural feedback to an abstracted neural oscillator or central pattern generator. Trials fell into two classes of forward velocity changes, while exhibiting statistically indistinguishable frequency changes. Animals pulled away from the side with front and hind legs of the tripod in stance recovered heading within 300 ms, whereas animals that only had a middle leg of the tripod resisting the pull did not recover within this period. Animals with eight or more legs might be more robust to lateral perturbations than hexapods.


BiomechanicsPhaseNeuromechanical controlNeural clockLocomotionCockroachPerturbation



Y-axis is positive along the line of platform translation. Also called lateral axis. Z-axis is perpendicular to the Y-axis, positive vertical of the platform. X-axis is perpendicular to both the Y- and Z-axes, and positive in the direction of cockroach locomotion. Also called the forward axis


Component of cockroach velocity in trackway direction


Component of cockroach velocity across trackway


Center of mass


Central pattern generator




Inter-burst interval. The time between two bursts of muscle action potentials in an electromyography


Inter-spike interval


Lateral leg spring model


Muscle action potential


Principal component analysis


Lateral leg spring model


Anterior extreme position. The transition from swing to stance.


Posterior extreme position. The transition from stance to swing.


Spring loaded inverted pendulum model

List of symbols

\(\varPhi \)

Phase threshold between classes (one class has \(\varPhi -\pi <\phi _{0} < \varPhi \), the other \(\varPhi < \phi _{0} < \varPhi + \pi \))

\(\phi _{0}\)

Predictor phase

\(\phi , \theta \)


\(\omega \)

Derivative of phase with respect to time, i.e. instantaneous frequency

\(\Delta \phi \)

Residual phase

\(x, v\)

Position, velocity time series used to create complex phase time series

\(z \)

Complex phase time series \(\langle .\rangle \) mean value; \(\langle w(t)\rangle \) is the expectation of the variable \(w(t)\)

\(t_{1\text{ pre}}\)

Starting time window pre-perturbation

\(t_{2\text{ pre}}\)

Ending time window pre-perturbation

\(t_{\text{ step}}\)

Step duration

\(t_{\text{ on}}\)

Onset of perturbation

\(t_{1\text{ post}}\)

Starting time window post-perturbation

\(t_{2\text{ post}}\)

Ending time window post-perturbation


Standard deviation operator; std[\(w(t)\)] is the standard deviation of the variable \(w(t)\)


(Complex) exponential function


Complex argument (i.e., polar angle) function


Class 0, one of the two phase classes (in red)


Class 1, one of the two phase classes (in blue)


Mean of cockroach velocity in trackway direction for C\(_{0}\)


Mean of cockroach velocity in trackway direction for C\(_{1}\)

L\(_{1}\) norm

Sum of absolute differences

L\(_{2}\) norm

Square root of sum of squared differences, same as root mean square (RMS) up to a scale


Parameter governing the number of bootstrap trials used for testing classification significance; \(N^{2}\) trials for H\(_{1}\) and H\(_{0(\mathrm{a})}\) are compared with a nested bootstrap of \(N\) trials of \(N\) nested trials each.


Number of trials provided by an individual animal


Statistical hypothesis that classes the C\(_{0}\) and C\(_{1}\) obtained from \(\phi _{ 0}\) and \(\varPhi \) describe animals that behave differently.


Statistical hypothesis that trial classes C\(_{0}\) and C\(_{1}\) are selected at random from the same distribution of animal motions.

H\(_{0(\mathrm{b}) }\)

Statistical hypothesis that trial classes C\(_{0}\) and C\(_{1}\) are selected to be most dissimilar classes that can be obtained based on a choice of \(\varPhi \), while still being selected at random from the same distribution of animal motions.

\(\chi ^{2}\)

Statistical distribution and associated test

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shai Revzen
    • 1
  • Samuel A. Burden
    • 2
  • Talia Y. Moore
    • 1
  • Jean-Michel Mongeau
    • 3
  • Robert J. Full
    • 1
  1. 1.Department of Integrative BiologyUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  3. 3.Biophysics Graduate GroupUniversity of CaliforniaBerkeleyUSA