On Stability of Parametrized Families of Polynomials and Matrices

The Schur and Hurwitz stability problems for a parametric polynomial family as well as the Schur stability problem for a compact set of real matrix family are considered. It is established that the Schur stability of a family of real matrices is equivalent to the nonsingularity of the family if has at least one stable member. Based on the Bernstein expansion of a multivariable polynomial and extremal properties of a multilinear function, fast algorithms are suggested.

Cerebellar Flocculus and Paraflocculus Purkinje Cell Activity During Circular Pursuit in Monkey

Responses from 69 Purkinje cells in the flocculus and paraflocculus of two rhesus monkeys were studied during smooth pursuit of targets moving along circular trajectories and compared with responses during sinusoidal pursuit and fixation. A variety of interesting responses was observed during circular pursuit. Although some neurons fired most strongly in a single preferred direction during clockwise (CW) and counterclockwise (CCW) pursuit, others had directional preferences that changed with rotation direction. Some of these neurons showed similar modulation amplitudes during CW and CCW pursuit, whereas other neurons showed a preference for a particular rotation direction. Response specificity also was observed during sinusoidal pursuit. Some neurons showed responses that were much stronger during centrifugal pursuit, others showed a preference for centripetal pursuit, and still others showed responses during both centripetal and centrifugal motion. Differences in preferred response direction were sometimes observed for centripetal versus centrifugal pursuit. CW/CCW and centrifugal/centripetal preferences were not explained by a breakdown in component additivity. That is, modulations in firing rate during pursuit along a circular trajectory equaled the sum of modulations during horizontal and vertical sinusoidal components as well as for diagonal components. Instead all responses were well fit by a model that expressed the instantaneous firing rate of each neuron as a multilinear function of the two-dimensional position and velocity of the eye. This model generalized well to performance at different sinusoidal frequencies. It did somewhat less well for responses during fixation, suggesting some separation in the neural mechanisms of dynamic and static positioning. The model indicates that position sensitivity accounted for ∼36% of the modulation during circular pursuit, and velocity sensitivity accounted for ∼64%. When position and velocity sensitivity vectors were aligned, responses were simpler and modulations were similar during CW versus CCW pursuit. In contrast, when these vectors pointed in different directions, response complexity increased. Nonaligned position and velocity influences tended to reinforce during circular pursuit in one direction and to cancel each other during pursuit in the opposite direction. They also tended to produce response differences during centripetal versus centrifugal sinusoidal pursuit. The distinct roles played by position and velocity in shaping Purkinje cell responses are compatible with the control signals required to generate smooth pursuit along circular and other two-dimensional trajectories.