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

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Abstract

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.

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Notes

  1. When the series all have the same amplitude this is precisely the circular average (Fisher 1993) of their phase angles.

Abbreviations

Axes:

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

Vx:

Component of cockroach velocity in trackway direction

Vy:

Component of cockroach velocity across trackway

COM:

Center of mass

CPG:

Central pattern generator

EMG:

Electromyography

IBI:

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

ISI:

Inter-spike interval

LLS:

Lateral leg spring model

MAP:

Muscle action potential

PCA:

Principal component analysis

LLS:

Lateral leg spring model

AEP:

Anterior extreme position. The transition from swing to stance.

PEP:

Posterior extreme position. The transition from stance to swing.

SLIP:

Spring loaded inverted pendulum model

\(\varPhi \) :

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

\(\phi _{0}\) :

Predictor phase

\(\phi , \theta \) :

Phases

\(\omega \) :

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

std:

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

exp:

(Complex) exponential function

arg:

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

C\(_{0}\) :

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

C\(_{1}\) :

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

Vx\(_{0}\) :

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

Vx\(_{1}\) :

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

\(N\) :

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.

\(n\) :

Number of trials provided by an individual animal

H\(_{1}\) :

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

H\(_{0(\mathrm{a})}\) :

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

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Acknowledgments

We would like to thank Teressa Alexander for laboratory assistance in collecting EMG data. This work was funded by NSF Frontiers for Integrative Biology Research (FIBR) Grant No. 0425878-Neuromechanical Systems Biology to RJF. SB and JMM were partially supported by NSF Graduate Research Fellowships and an NSF IGERT Traineeship to JMM.

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Authors and Affiliations

  1. Department of Integrative Biology, University of California, Berkeley, CA, 94720, USA

    Shai Revzen, Talia Y. Moore & Robert J. Full

  2. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720, USA

    Samuel A. Burden

  3. Biophysics Graduate Group, University of California, Berkeley, CA, 94720, USA

    Jean-Michel Mongeau

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  1. Shai Revzen
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Revzen, S., Burden, S.A., Moore, T.Y. et al. Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches. Biol Cybern 107, 179–200 (2013). https://doi.org/10.1007/s00422-012-0545-z

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