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Many definitions of chaos have appeared in the last decades and there is a concerned question if they are equivalent in some more specific spaces. Our focus will be on distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. At the end of the paper we will also briefly look at their correlation with other types of chaos.

The development of neuronal branching patterns mainly proceeds by branching events at terminal segments. The randomness in the occurrences of these events is the major cause of the variability that is observed in the final topological structures of neuronal trees. A general observation in neuronal trees is the existence of a branch power relation between the diameters of the segments at branch points. Because of this correlation, total area and volume will become dependent on branching pattern topology. A quantitative assessment of this finding is given. How topological variation propagates into variation in total area and volume, using topological growth models for producing random tree topologies and a simple metrical parametrization is discussed.

Succinonitrile (SCN)–2 wt% Camphor (CAMP) alloy was unidirectionally solidified with a constant temperature gradient (G = 3.01 K/mm) at different growth rates (V = 6.5–113 μm/s) and with a constant growth rate (V = 6.5 μm/s) at different temperature gradients (G = 1.93 - 3.01 K/mm). Microstructural parameters (primary dendrite arm spacings, λ₁, secondary dendrite arm spacings, λ₂, dendrite tip radius, R, and mushy zone depth, d) were measured as a function of growth rate and temperature gradient. The experimental results have been compared with theoretical models and previous experimental works. The stability constant (σ) for this alloy system was calculated and compared with similar experimental results.

Pb-9.3wt.%Sb alloy was directionally solidified upwards under argon atmosphere under the two conditions; with different temperature gradients, (G = 0.93–3.67 K/mm) at a constant growth rate (V = 17.50 μm/s) and with different growth rates (V = 8.30–497.00 μm/s) at a constant (G = 3.67 K/mm) in a Bridgman furnace. The dependence of characteristic microstructure parameters such as primary dendrite arm spacing (λ₁), secondary dendrite arm spacing (λ₂) and dendrite tip radius (R) on the growth rate (V) and the temperature gradient (G) were determined by using a linear regression analysis. A detailed analysis of microstructure were also made and compared with the theoretical models and similar experimental works on dendritic solidification in the literature.

Hodgkin and Huxley's ionic theory of the nerve impulse embodies principles, applicable also to the impulses in vertebrate nerve fibers, as demonstrated by Bernhard Frankenhaeuser and Andrew Huxley 40 years ago. Frankenhaeuser and Huxley reformulated the classical Hodgkin-Huxley equations, in terms of electrodiffusion theory, and computed action potentials specifically for saltatory conduction in myelinated axons. In this paper, we obtain analytical solutions to the most difficult nonlinear partial differential equations in classical neurophysiology. We solve analytically the Frankenhaeuser-Huxley equations pertaining to a model of sparsely excitable, nonlinear dendrites with clusters of transiently activating, TTX-sensitive Na⁺ channels, discretely distributed as point sources of inward current along a continuous (non-segmented) leaky cable structure. Each cluster or hot-spot, corresponding to a mesoscopic level description of Na⁺ ion channels, includes known cumulative inactivation kinetics observed at the microscopic level. In such a third-order system, the 'recovery' variable is an electrogenic sodium-pump imbedded in the passive membrane, and the system is stabilized by the presence of a large leak conductance mediated by a composite number of ligand-gated channels permeable to monovalent cations Na⁺ and K⁺. In order to reproduce antidromic propagation and attenuation of action potentials, a nonlinear integral equation must be solved (in the presence of suprathreshold input, and a constant-field equation of electrodiffusion at each hot-spot with membrane gates controlling the flow of current). A perturbative expansion of the non-dimensional membrane potential (Φ) is used to obtain time-dependent analytical solutions, involving a voltage-dependent Na⁺ activation (μ) and a state-dependent inactivation (η) gating variables. It is shown that action potentials attenuate in amplitude in accordance with experimental findings, and that the spatial density distribution of transient Na⁺ channels along a long dendrite contributes significantly to their discharge patterns. A major significance of the analytical modeling, in contrast to the computational modeling of backpropagating action potentials, is the provision of a continuous description of the voltage as a function of position, allowing for greater feasibility in developing large-scale biophysical neural networks, without the need for ad hoc computational modeling.

Retinal amacrine cells regulate activities of retinal ganglion cells, the output neurons to higher visual centers, through cellular mechanism of lateral inhibition in the inner plexiform layer (IPL). Electrical properties of gap junction networks between amacrine cells in the IPL were investigated using combined techniques of intracellular recordings, Lucifer yellow and Neurobiotin injection, dual patch-clamp recordings and high voltage electron microscopy in isolated retinas of cyprinid fish. Six types of gap-junctionally connected amacrine cells were classified after their light-evoked responses to light flashes were recorded. Among them, gap junction networks of three types of amacrine cells were studied with structure-function correlation analysis. Cellular morphology of intercellular connections between three homologous cell classes was characterized. The interconnections between laterally extending dendrites in the IPL were localized at dendritic tip terminals. Three types of cells presented the dendrodendritic connections of tip-contact manner in the homologous cell population. High voltage as well as conventional electron microscopy revealed gap junctions between the dendritic tips of Neurobiotin-coupled cells. Receptive field properties of these amacrine cells were examined, displacing a slit of light along the distance from recording sites in the dorsal intermediate region of the retina. Receptive field size, space length constant, response latency and conduction velocity were measured. Spatial and temporal properties of receptive fields were symmetric along horizontally expanding dendrites in the dorsal retina. Simultaneous dual patch-clamp recordings revealed that the lateral gap junction connections between homologous amacrine cells expressed bidirectional electrical synapses passing Na⁺ spikes. These results demonstrate that bidirectional electrical transmission in gap junction networks of these amacrine cells is symmetric along the lateral gap junction connections between horizontally extending dendrites. Lateral inhibition regulated by amacrine cells in the IPL appears to be associated with the directional extension of the dendrites and the orientation of dendrodendritic gap junctions.

Progress towards detailed characterization of structural and biophysical properties of dendrites emphasizes the importance of finding analytical solutions for more realistic dendrite models with circular cross-section and varying diameter. In this regard, we employ symmetry methods and the passive cable theory to deduce a generalized analytical solution for electric propagation in a family of tapering dendrites. In particular, we study the effect of such tapering geometries on the obtained electric voltage. Simulations using the deduced analytical solution indicate that for a subfamily of tapering profiles neural integration is better than in the stereotypical profile given by a cylinder.

Alkaline zinc-based rechargeable batteries (AZRBs) are competitive candidates for future electrical energy storage because of their low-cost, eco-friendliness and high energy density. However, plagued by dendrites, the AZRBs suffer from drastic decay in electrochemical properties and safety. This review elucidates fundamentals of zinc dendritic formation and summarizes the strategies, including electrode design and modification, electrolyte optimization and separator improvement, for suppressing zinc dendritic growth.

Pb(${\text{In}}_{1∕2}$${\text{Nb}}_{1∕2}$)O₃-Pb(${\text{Mg}}_{1∕3}$${\text{Nb}}_{2∕3})$O₃-PbTiO₃ (PIN-PMN-PT) relaxor ferroelectric crystals in 5^″ diameter by 5^″ length were grown by the Bridgman (BR) method for the first time. Typical issues in the crystal growth concerning large volume of the melt, high density of the crystal and corrosion of the melt on the wall of Pt crucible are discussed.