Mechanisms underlying rhythmic locomotion: interactions between activation, tension and body curvature waves.

Undulatory animal locomotion arises from three closely related propagating waves that sweep rostrocaudally along the body: activation of segmental muscles by motoneurons (MNs), strain of the body wall, and muscle tension induced by activation and strain. Neuromechanical models that predict the relative propagation speeds of neural/muscle activation, muscle tension and body curvature can reveal crucial underlying control features of the central nervous system and the power-generating mechanisms of the muscle. We provide an analytical explanation of the relative speeds of these three waves based on a model of neuromuscular activation and a model of the body–fluid interactions for leech anguilliform-like swimming. First, we deduced the motoneuron spike frequencies that activate the muscle and the resulting muscle tension during swimming in intact leeches from muscle bending moments. Muscle bending moments were derived from our video-recorded kinematic motion data by our body–fluid interaction model. The phase relationships of neural activation and muscle tension in the strain cycle were then calculated. Our study predicts that the MN activation and body curvature waves have roughly the same speed (the ratio of curvature to MN activation speed ≈0.84), whereas the tension wave travels about twice as fast. The high speed of the tension wave resulting from slow MN activation is explained by the multiplicative effects of MN activation and muscle strain on tension development. That is, the product of two slower waves (activation and strain) with appropriate amplitude, bias and phase can generate a tension wave with twice the propagation speed of the factors. Our study predicts that (1) the bending moment required for swimming is achieved by minimal MN spike frequency, rather than by minimal muscle tension; (2) MN activity is greater in the mid-body than in the head and tail regions; (3) inhibitory MNs not only accelerate the muscle relaxation but also reduce the intrinsic tonus tension during one sector of the swim cycle; and (4) movements of the caudal end are passive during swimming. These predictions await verification or rejection through further experiments on swimming animals. * a(f) : activation factor for leech muscle a d( t ) : time course of activation factor of dorsal muscle ai(t) : activation factor at the i th body segment a v( t ) : time course of activation factor of ventral muscle a d : column vector of discretized a d( t ) a v : column vector of discretized a v( t ) a * : column vector of discretized a d( t ) or a v( t ) CPG : central pattern generator DE-3 : dorsal excitatory motoneuron 3 DE-5 : dorsal excitatory motoneuron 5 EMG : electromyographic f : motoneuron spike frequency f d : net spike frequency of MN activating the dorsal muscle f v : net spike frequency of MN activating the ventral muscle h(x) : strain factor in tension model h d( t ) : time course of strain factor of dorsal muscle tension hi(t) : strain factor at the i th body segment h v( t ) : time course of strain factor of ventral muscle tension h d : column vector of discretized h d( t ) h v : column vector of discretized h v( t ) h * : column vector of discretized h d( t ) or h v( t ) H * : diagonal matrix with diagonal being h * LMI : linear matrix inequality MN : motoneuron P(s) : transfer function in muscle activation model r : half of body thickness ri, θi : amplitude and phase of the fundamental harmonic component of the product of hi ( t ) ai ( t ) s : Laplace variable T : tension of leech body wall T d( t ) : time course of tension of dorsal muscle T v( t ) : time course of tension of ventral muscle T d : column vector of discretized T d( t ) T v : column vector of discretized T v( t ) T * : column vector of discretized T d( t ) or T v( t ) u(t) : time course of muscle torque u : column vector of discretized u ( t ) x : muscle strain x d : strain of dorsal muscle x v : strain of ventral muscle α : static gain of the transfer function for muscle activation αi, βi, γi : amplitude, phase and bias of activation factor in the tension model at the i th joint ![Graphic][1] : amplitude, phase and bias of strain factor in the tension model at the i th joint δ : phase difference between ai ( t ) and hi ( t ) e : small perturbation parameter η : length constant of tension model μ : passive (tonus) tension at nominal swim length (i.e. x =0) ρ : ratio of normalized amplitudes of ai ( t ) and hi ( t ) τ1 : time constant in muscle activation τ2 : time delay in muscle activation ω : oscillation frequency * : representing dorsal or ventral muscle [1]: /embed/inline-graphic-2.gif