Primary roots of adzuki bean were incubated under an external electric field in the transverse direction across the root. A root was laid horizontally in a aqueous solution, and the electric field was generated by flowing the electric current in the solution. Roots bent to the side of the positive electrode. This bending tendency increased with increasing the electric field. The surface-electric potential and pH distributions were measured under the electric field and the normal condition. The application of electric field caused the asymmetries of the electric potential pattern and the acidification. At the activated side in the growth, corresponding to the electrically negative side of the root, the amplitudes of the potential and the acidification were found to increase whereas the opposite tendency was observed at the opposite side. The results with a theoretical consideration suggest that the symmetry of the growth was broken by an asymmetrical H+ re-distribution made by the external electrical force.

Theoretical description with a potential is made for inhomogeneous structures of high field domain and current filament in semiconductors with a negative differential conductivity (NDC) appearing under voltage- and current-controlled conditions, respectively. The potential proposed here can describe systematically a route from homogeneous state to the patterned state through the instability of homogeneous state, whereas previously proposed potentials can describe only the patterned state. The potential is constructed from two internal variables: one is the variable dependent on the spatial coordinate which exhibits the spatial pattern in the NDC region, while another remains constant spatially but changes discontinuously its value when the patterned state bifurcates from a thermodynamic branch of the homogeneous state. The bifurcation to spatial pattern is examined in a similar way to the first-order phase transition in equilibrium systems. At the same time, the property of the resulting pattern is discussed from analogy with the phase separation.

A nonperturbative method is presented for describing approximately the behavior of a self-oscillation of electric voltage in the Van der Pol equation over a wide range of the value of external parameter µ. To express an appreciably distorted wave form for the steady self-oscillation at µ1, a phase F of the voltage x, defined by x2A cos F (ωt), is approximated by a combination of several straight lines as a function of ωt from 0 to 2π with several numerical coefficients determined mainly from asymptotic behaviors of x for µ1 and µ1. It is shown that the resultant expression for x can describe well the numerical result over the wide range of µ. A bursting phenomenon induced by an oscillation of µ with a long period is also discussed on the basis of the present method, and the analytical results are in good agreement with the numerical ones.

Two-dimensional pattern of an electric potential around a horizontally-placed root of adzuki bean was detected by a multi-electrode measuring system. The vectors of electric current calculated from the electric potential pattern show a long route from the basal part of the root to the apical part.

The growth of a primary root was studied for adzuki beans incubated under an external electric field. An electric current was applied to the solution around the root, which was laid horizontally. The electric potential and pH patterns were measured along the root. The change of the growth speed was observed when the electric field, corresponding to a current of 1-10 µA/cm2, was applied for a period of 15 to 60 min. The application of the electric field in the direction from the base to the tip of a root activated the growth of the root, whereas the growth was inhibited by applying the electric field of the reverse polarity. The changes of the growth rates were both about 20% for the currents of 10 µA/cm2. The root was generating an electric current pattern in the aqueous solution around its surface with the maximum current density of the order of 1 µA/cm2. In the activation of the growth, the amplitude of the electric potential pattern and the acidification around the root were found to be increased. The opposite tendency was obtained in the growth inhibition. The degree of acidification corresponds to the change of the growth rate by a well-known acid growth mechanism.