On the analytical solutions of the Hindmarsh-Rose neuronal model

In this article we analytically solve the Hindmarsh-Rose model (Proc R Soc Lond B221:87-102, 1984) by means of a technique developed for strongly nonlinear problems-the step homotopy analysis method. This analytical algorithm, based on a modification of the standard homotopy analysis method, allows us to obtain a one-parameter family of explicit series solutions for the studied neuronal model. The Hindmarsh-Rose system represents a paradigmatic example of models developed to qualitatively reproduce the electrical activity of cell membranes. By using the homotopy solutions, we investigate the dynamical effect of two chosen biologically meaningful bifurcation parameters: the injected current I and the parameter r, representing the ratio of time scales between spiking (fast dynamics) and resting (slow dynamics). The auxiliary parameter involved in the analytical method provides us with an elegant way to ensure convergent series solutions of the neuronal model. Our analytical results are found to be in excellent agreement with the numerical simulations.

Arylhydrazones of barbituric acid: synthesis, coordination ability and catalytic activity of their CoII, CoII/IIIand CuIIcomplexes toward peroxidative oxidation of alkanes

New ortho-substituted arylhydrazones of barbituric acid, 5-(2-(2-hydroxyphenyl)hydrazono) pyrimidine-2,4,6(1H,3H,5H)-trione (H4L1) and the sodium salt of 2-(2-(2,4,6-trioxotetra-hydropyrimidin-5(2H)-ylidene)hydrazinyl) benzenesulfonic acid (H4L2), [Na(H3L2)(mu-H2O)(H2O)(2)](2) (1), were used in the synthesis of Cu-II, Co-II and Co-II/III complexes, [Cu(H2L1)(H2O)(im)]center dot 3H(2)O (im = imidazole) (2), [Co(H2O)(6)] [Co(H2L1)(2)](2)center dot 8H(2)O (3), [Co(H2L2)(im)(3)] (4), [Cu(H2L2)(im)(2)]center dot H2O (5) and [Co(H2O)(6)][H3L2](2)center dot 8H(2)O (6). The complexes are water soluble and the mono-or di-deprotonated ligands display different coordination modes, depending on the synthetic conditions. The electrochemical behaviour of all the compounds was investigated by cyclic voltammetry and controlled potential electrolysis, revealing that the ligands are also redox active. All the compounds were evaluated as catalysts for the peroxidative (with H2O2) oxidation of cyclohexane at room temperature. The compounds 2 and 3 are the most active ones (yields up to 21% and TON up to 213 are achieved, in the presence of 3).