EarthWeb   
HomeAccount InfoLoginSearchMy ITKnowledgeFAQSitemapContact Us
     

   
  All ITKnowledge
  Source Code

  Search Tips
  Advanced Search
   
  

  

[an error occurred while processing this directive]
Previous Table of Contents Next


The corresponding energy of the polarized molecule in the presence of an applied field Ε is constituted by: (1) The potential energy WPE due to the permanent dipole given by,

and (2) the potential energy due to the induced dipole given by:

Hence, the total energy is equal to WT = WPE + WiE. Further, the statistical average of μE can be specified by:

where dΩ is the elemental solid angle around the direction of . That is, dΩ = 2π sin(θ)dθ. By performing the integration of Equation (5.21) using Equation (5.18), it follows that:

where the quantity <cos2θ> varies from 1/3 (for randomly oriented molecules) to 1 for the case where all the molecules are parallel (or antiparallel) to the field . On the basis of the limits specified by <cos2θ>, the following parameter can be defined:

The parameter So which is bounded between 0 and 1 under the above conditions, represents the “order parameter” of the system [82]. Appropriate to the nematic phase, So specifies the long-range orientational parameter pertaining to a liquid crystal of rod-like molecules as follows: Assuming the distribution function of the molecules to be cylindrically symmetric about the axis of preferred orientation, So defines the degree of alignment, namely, for perfectly parallel (or antiparallel) alignment So = 1, while for random orientations So = 0. In the nematic phase So has an intermediate value which is strongly temperature dependent.


Figure 5.2  Types of disorders in spatial free-point molecular arrangement subjected to external electric field () (a) & (b) Completely ordered (total anisotropy): Parallel and antiparallel arrangements; (c) Partial long-range order (partial anisotropy): Nematic phase arrangement; (d) Complete absence of long-range order (total isotropy): Random arrangement

For So = 0, it refers to an isotropic statistical arrangement of random orientations so that for each dipole pointing in one direction, there is statistically a corresponding molecule in the opposite direction (Figure 5.2). In the presence of an external electric field , the dipoles experience a torque and tend to polarize along , so that the system becomes slightly anisotropic; and eventually under a strong field () the system becomes totally anisotropic with So = 1.

5.12 Stochastical Response of Neurons under Activation

By considering the neurons as analogous to a random, statistically isotropic dipole system, the graded response of the neurons under activation could be modeled by applying the concepts of Langevin’s theory of dipole polarization; and the continuous graded response of neuron activity corresponding to the stochastical interaction between incoming excitations that produce true, collective, nonlinear effects can be elucidated in terms of a sigmoidal function specified by a gain parameter λ = Λ/kBT, with Λ being the scaling factor of σi which depicts the neuronal state-vector.

In the pertinent considerations, the neurons are depicted similar to the nematic phase of liquid crystals and are assumed to possess an inherent, long-range spatial order. In other words, it is suggested that 0 < So < 1 is an appropriate and valid order function for the neural complex that So = 0. Specifying in terms of So = (3/2) <cos2θ> - 1/2, the term <cos2θ> should correspond to a value between 1/3 to 1 (justifying the spatial anisotropy).

To determine an appropriate squashing function for this range of <cos2θ> between 1/3 to 1 (or for 0 < So < 1), the quantity <cos2θ> can be replaced by (1/3 + 1/3q) in defining the order parameter So. Hence:

where q → ∞ and q = 1/2 set the corresponding limits of So = 0 and So = 1 respectively.

Again, resorting to statistical mechanics, q = 1/2 refers to dichotomous states, if the number of states are specified by (2q + 1). For the dipoles or neuronal alignments, it corresponds to the two totally discrete anisotropic (parallel or antiparallel) orientations. In a statistically isotropic, randomly oriented system, the number of (possible) discrete alignments would, however, approach infinity, as dictated by q → ∞.

For the intermediate (2q + 1) number of discrete orientations, the extent of dipole alignment to an external field or, correspondingly, the (output) response of a neuron to excitation would be decided by the probability of a discrete orientation being realized. It can be specified by [83]:


Previous Table of Contents Next

Copyright © CRC Press LLC

HomeAccount InfoSubscribeLoginSearchMy ITKnowledgeFAQSitemapContact Us
Products |  Contact Us |  About Us |  Privacy  |  Ad Info  |  Home

Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction in whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement.