Numerical Implementation of Gradient Algorithms

A numerical method for computational implementation of gradient dynamical systems is presented. The method is based upon the development of geometric integration numerical methods, which aim at preserving the dynamical properties of the original ordinary differential equation under discretization. In particular, the proposed method belongs to the class of discrete gradients methods, which substitute the gradient of the continuous equation with a discrete gradient, leading to a map that possesses the same Lyapunov function of the dynamical system, thus preserving the qualitative properties regardless of the step size. In this work, we apply a discrete gradient method to the implementation of Hopfield neural networks. Contrary to most geometric integration methods, the proposed algorithm can be rewritten in explicit form, which considerably improves its performance and stability. Simulation results show that the preservation of the Lyapunov function leads to an improved performance, compared to the conventional discretization.Spanish Government project no. TIN2010-16556 Junta de Andalucía project no. P08-TIC-04026 Agencia Española de Cooperación Internacional para el Desarrollo project no. A2/038418/1

Markers of cartilage matrix metabolism in human joint fluid and serum: the effect of exercise.

The concentrations of cartilage proteoglycan (aggrecan), stromelysin-1, tissue inhibitor of metalloproteinases-1 (TIMP-1) and procollagen II C-propeptide in knee joint fluid and the levels of aggrecan, hyaluronan and keratan sulfate in serum were measured before and after exercise in 33 healthy athletes. The samples before exercise were obtained after 24 h rest from running or soccer and the samples after exercise were obtained 30-60 min after the exercise. Nine athletes ran on a treadmill for 60 min, 16 ran on road for 80 min and 8 played one soccer game (90 min). A reference group of 28 patients with knee pain but not evidence of joint pathology or injury was used for comparison. In joint fluid no single marker from the degradative processes in cartilage matrix changed significantly with exercise but all showed a rising trend. All markers except stromelysin showed lower concentrations in athletes at rest compared to the reference group. In serum from runners before exercise the concentration of keratan sulfate was significantly higher than in both the soccer and reference groups and further increased after exercise. The increase in markers after exercise may reflect an effect of mechanical loading in combination with a possible high turnover rate of body cartilage matrix in these individuals