AbstractVertebrates are mostly large, agile creatures that can move rapidly and efficiently despite the significant inertial and interaction forces that are generated when they move. Vertebrates appear in the fossil record in the lower Cambrian, when predation began to apply selection pressure for increased size and, consequently, increased inertia. Agility depends critically upon the cerebellum, a brain structure that appears to have evolved as an elaboration of the vestibular and lateral‐line sensory processing regions in brains of aquatic ancestors. There is compelling evidence to suggest that the specific role of the cerebellum in motor control (etc) is state estimation. Analysis of responses of vestibular neurons in bullfrogs shows that these neurons have fractional‐order dynamics. Because of this property, the vestibular nerve forms a map of the state space of the animal's head, and individual action potentials can be regarded as assertions about the location of the head in that space. This view of vestibular coding suggests an analogy with particle filters, a new simulation‐based method for state estimation. In a particle filter the distribution of possible states consistent with observations is represented by a cloud of “particles” in a map of the state space. By regarding action potentials as particles in neural state estimators, we may develop a realistic model of neural computation in the cerebellum. At the same time—and independently of the biological verisimilitude of such a cerebellar model—we may develop useful new algorithms for state estimation. © 2005 Wiley Periodicals, Inc. Complexity 10: 56–65, 2005