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An alternative method of attributing the long-range order to neurons can be done by following the technique of Little except that such a long-range order will be referred to the spatial or orientational anisotropy instead of time correlations. To facilitate this approach, the free-point molecular dipole interactions can be considered in lieu of magnetic spin interactions [15]. The free-point molecular dipole interactions with partial anisotropy in spatial arrangement refer to the nematic phase in a liquid crystal. Hence, the relevant analysis equates the neural statistics to that of a nematic phase system consistent with the Inown dogma that “the living cell is actually a liquid crystal” [81]. That is, as Brown and Wolken [81] observed, the characteristics of molecular patterns, structural and behavioral properties of liquid crystals, make them unique model-systems to investigate a variety of biological phenomena. The general physioanatomical state of biological cells depicts neither real crystals nor real liquid phase (and constitutes what is popularly known as the mesomorphous state) much akin to several organic compounds which have become known as the “flüssige Kristalle” or liquid crystals; and both the liquid crystalline materials as well as the biological cells have a common, irregular pattern of side-by-side spatial arrangements in a series of layers (known as the nematic phase).

The microscopic structural studies of biological cells indicate that they are constituted by very complex systems of macromolecules which are organized into various bodies or “organelles” that perform specific functions for the cell. From the structural and functional point of view, Brown and Wolken have drawn an analogy of the description of the living cells to liquid crystals on the basis that a cell has a structural order. This in fact is a basic property of liquid crystals as well, for they have a structural order of a solid. Furthermore, in many respects it has been observed that the physical, chemical, structural, and optical properties of biological cells mimic closely those of liquid crystals.

Due to its liquid crystalline nature, a cell through its own structure forms a proto-organ facilitating electrical activity. Further, the anisotropically oriented structure of cellular assembly (analogous to liquid crystals) has been found responsible for the complex catalytic action needed to account for cellular regeneration. In other words, by nature the cells are inherently like liquid crystals with similar functional attributions.

On the basis of these considerations a neural cell can be modeled via liquid-crystal analogy, and the squashing action of the neural cells pertinent to the input-output relations (depicting the dynamics of the cellular automata) can be described in terms of a stochastically justifiable sigmoidal function and statistical mechanics considerations as presented in the pursuant sections.

5.11 Free-Point Molecular Dipole Interactions

Suppose a set of polarizable molecules are anisotropic with a spatial long-range orientational order corresponding to the nematic liquid crystal in the mesomorphic phase. This differs from the isotropic molecular arrangement (as in a liquid) in that the molecules are spontaneously oriented with their long axes approximately parallel. The preferred direction or orientational order may vary from point-to-point in the medium, but in the long-range, a specific orientational parallelism is retained.

In the nematic phase, the statistical aspects of dipole orientation in the presence of an externally applied field can be studied via Langevin’s theory with the following hypotheses:

1.  The molecules are point-dipoles with a prescribed extent of anisotropy.
2.  The ensemble average taken at an instant is the same as the time average taken on any element (ergodicity property).
3.  The characteristic quantum numbers of the problem are so high that the system obeys the classical statistics of Maxwell-Boltzmann, which is the limit of quantum statistics for systems with high quantum numbers. The present characterization of paraelectricity differs from spin paramagnetism, wherein the quantum levels are restricted to two values only.
4.  The dipole molecules in general when subjected to an external electric field , experience a moment μE = αE , where αE by definition refers to the polarizability of the molecule. The dipole orientation contributing to the polarization of the material is quantified as P = N<μE> where N is the dipole concentration.
5.  In an anisotropic system such as the liquid crystal, there is a permanent dipole moment μPE, the direction of which is assumed along the long axis of a nonspherical dipole configuration. Consequently, two orthogonal polarizability components exist, namely, αE1 along the long axis and αE2 perpendicular to this long axis.

The dipole moments in an anisotropic molecule are depicted in Figure 5.1. Projecting along the applied electric field the net-induced electric polarization moment is:

where ΔαE is a measure of anisotropy.


Figure 5.1  Free-point dipole and its moments : Applied electric field; : Permanent dipole moment; : Induced dipole moment


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