![]() |
|
|||
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
|
![]() |
[an error occurred while processing this directive]
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 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 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.
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 5.12 Stochastical Response of Neurons under ActivationBy 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 Langevins 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]:
Copyright © CRC Press LLC
![]() |
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |