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Locomotion is a complex process involving specific interactions between the central neural controller and the mechanical components of the system. The basic rhythmic activity generated by locomotor circuits in the spinal cord defines rhythmic limb movements and their central coordination. The operation of these circuits is modulated by sensory feedback from the limbs providing information about the state of the limbs and the body. However, the specific role and contribution of central interactions and sensory feedback in the control of locomotor gait and posture remain poorly understood. We use biomechanical data on quadrupedal locomotion in mice and recent findings on the organization of neural interactions within the spinal locomotor circuitry to create and analyse a tractable mathematical model of mouse locomotion. The model includes a simplified mechanical model of the mouse body with four limbs and a central controller composed of four rhythm generators, each operating as a state machine controlling the state of one limb. Feedback signals characterize the load and extension of each limb as well as postural stability (balance). We systematically investigate and compare several model versions and compare their behaviour to existing experimental data on mouse locomotion. Our results highlight the specific roles of sensory feedback and some central propriospinal interactions between circuits controlling fore and hind limbs for speed-dependent gait expression. Our models suggest that postural imbalance feedback may be critically involved in the control of swing-to-stance transitions in each limb and the stabilization of walking direction. 

Abstract Alinear forestis a disjoint union of path graphs. Thelinear arboricity of a graph, denoted by , is the least number of linear forests into which the graph can be partitioned. Clearly, for any graph of maximum degree . For the upper bound, the long‐standingLinear Arboricity Conjecture(LAC) due to Akiyama, Exoo, and Harary from 1981 asserts that . A graph is apseudoforestif each of its components contains at most one cycle. In this paper, we prove thatthe union of any two pseudoforests of maximum degree up to 3 can be decomposed into three linear forests. Combining it with a recent result of Wdowinski on the minimum number of pseudoforests into which a graph can be decomposed, we prove that the LAC holds for the following simple graph classes: ‐degenerate graphs with maximum degree , all graphs on nonnegative Euler characteristic surfaces provided the maximum degree , and graphs on negative Euler characteristic surfaces provided the maximum degree , as well as graphs with no ‐minor satisfying some conditions on maximum degrees.