UC Berkeley

We define a theoretical framework for the experimental study of neuromechanical control in animals, based on mathematical concepts from dynamical systems theory. This approach allows experiments, results and theoretical models to be shared among biologists, engineers and mathematicians, and is applicable to the study of control in any system, biological, artificial or simulated, provided the system exhibits stable rhythmic solutions. The basis of this framework is the notion that rhythmic systems are best expressed as periodic functions of their phase. Using phase as a predictor, an extrapolated prediction of future animal motions can be compared with the motions that occur when a perturbation is applied. Phase also serves as a succinct summary of the kinematic state, allowing the difference between the expected state as summarized by phase and the phase found in the perturbed animal -- a “residual phase”. In the first chapter we introduce the key concepts and describe how the residual phase may be used to identify the neuromechanical control architecture of an animal. In the following two chapters we use residual phase to analyze running arthropods subjected to perturbations. In the final chapter, we extend the kinematic phase based models to the construction of a linearized approximation of animal dynamics based on Floquet theory. The Floquet model allows us to directly test the “Templates and Anchors Hypothesis” of motor control and to characterize a “template” -- a low dimensional model of the dynamics of the animal.

In chapter 2, our residual phase results from running cockroaches over a hurdle show that kinematic phase was reset, while running frequency was closely maintained to within ±5%. Kinematic phase changes were distributed bi-modally with modes one step (half a cycle) apart, which corresponds to a left-right reflection of the kinematic state of the body. The results decrease the plausibility of feedforward control and support the use of neural feedback for this task. Based on the results, we propose a controller that expresses the timing of the two leg tripods of the animals as two coupled phase oscillators, which in turn, may also be coupled to a master clock.

In chapter 3, we analyze cockroaches which ran onto a movable cart that translated laterally. The specific impulse imposed on animals was 50±4 cm/s (mean,SD), nearly twice their forward speed 25±6 cm/s. Animals corrected for these perturbations by decreasing stride frequency, thereby demonstrating neural feedback. Trials fell into two classes, one class slowing down after a step (50 ms), the other after nearly three steps (130 ms). Classes were predicted by the kinematic phase of the animal at onset of perturbation. We hypothesize that the differences in response time is a consequence of the mechanical posture of the animal during perturbation, as expressed by the phase, and the coupling of neural and mechanical control.

In chapter 4 we attempted to use kinematic phase methods to reconstruct the linearized (Floquet) structure of running cockroaches when viewed as nonlinear oscillators. The development of this approach required several innovations in applied mathematics and statistics. We analyzed foot and body positions of 34 animals running on a treadmill. Results showed that cockroaches running at preferred speed possess a six dimensional template with each dimension recovering by less than 50% in a stride (P<0.05, 11 animals, 24 trials, 532 strides).