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The above function, Lq(x), is a modified Langevin function and is also known as the Bernoulli function. The traditional Langevin function L(x) is the limit of Lq(x) for q → ∞. The other limiting case, namely, q = 1/2, which exists for dichotomous states, corresponds to L1/2(x) = tanh(x).

Thus, the sigmoidal function FS(x) which decides the neuronal output response to an excitation has two bounds: With FS(x) = tanh(x), it corresponds to the assumption that there exists a total orientational long-range order in the neuronal arrangement. Conventionally [16], FS(x) = tanh(x) has been regarded as the squashing function (for neuronal nets) purely on empirical considerations of the input-output nonlinear relation being S-shaped (which remains bounded between two logistic limits, and follows a continuous monotonic functional form between these limits). In terms of the input variate xi and the gain/scaling parameter Λ of an ith neuron, the sigmoidal function specified as the hyberbolic tangent function is tanh(Λxi). The logistic operation that compresses the range of the input so that the output remains bounded between the logical limits can also be specified alternatively by an exponential form, FS(y) = 1/[(1 + exp(-y)] with y = Λxi.

Except for being sigmoidal, the adoption of the hyperbolic tangent or the exponential form in the neural network analyses has been purely empirical with no justifiable reasoning attributed to their choice. Pursuant to the earlier discussion, L(y) = Lq→∞ (y) specifies the system in which the randomness is totally isotropic. That is, the anisotropicity being zero is implicit. This, however, refers to rather an extensive situation assuming that the neuronal configuration poses no spatial anisotropicity or long-range order whatsoever. Likewise, considering the intuitive modeling of FS(y) = tanh(y), as adopted commonly, it depicts a totally anisotropic system wherein the long-range order attains a value one. That is, tanh(y) = Lq→1/2(y) corresponds to the dichotomous discrete orientations (parallel or antiparallel) specified by (2q + 1) → 2.

In the nematic phase, neither of the above functions, namely, tanh(y) nor L(y), is commensurable since a partial long-range order (depicting a partial anisotropicity) is rattier imminent in such systems. Thus, with 1/2 < q < ∞, the true sigmoid of a neuronal arrangement (with an inherent nematic, spatial long-range order) should be Lq(y).

Therefore, it can be regarded that the conventional sigmoid, namely, the hyperbolic tangent (or its variations) and the Langevin function, constitute the upper and lower bounds of the state-vector squashing characteristics of a neuronal unit, respectively.

Relevant to the above discussions, the pertinent results are summarized in Table 5.3.


Figure 5.3  Sigmoidal function

Table 5.3 Types of Spatial Disorder in the Neural Configuration

5.13 Hamiltonian of Neural Spatial Long-Range Order

In general, the anisotropicity of a disorder leads to a Hamiltonian which can be specified in two ways: (1) Suppose the exchange Hamiltonian is given by:

where Wxx, Wyy and Wzz are diagonal elements of the exchange matrix W (with the off- diagonal elements being zero). If Wxx = Wyy = 0 and Wzz ≠ 0, it is a symmetric anisotropy (with dichotomous states as in the Ising model). Note that the anisotropy arises if the strength of at least one of the exchange constants is different from the other two. If Wxx = Wyy ≠ 0 and Wzz = 0, it corresponds to an isotropic xy model; and, if Wxx = Wyy = Wzz, it is known as the isotropic Heisenberg model. (2) Given that the system has an anisotropy due to partial long-range order as in the nematic phase representation of the neuronal arrangement, the corresponding Hamiltonian is:

where Ha refers to the anisotropic contribution which can be specified by an inherent constant hio related to the order parameter, So, so that

While the interactions Wij are local, HN refers to an extensive quantity corresponding to the long-range orientational (spatial) interconnections in the neuronal arrangement.

5.14 Spatial Persistence in the Nematic Phase

The nematic-phase modeling of the neuronal arrangement specifies (as discussed earlier) a long-range spatial anisotropy which may pose a persistency (or preferred, directional routing) of the synaptic transmission. Pertinent analysis would be similar to the time-domain persistency demonstrated by Little [33] as existing in neuronal firing patterns.

Considering (2q + 1) possible spatial orientations (or states) pertaining to M interacting neurons as represented by Ψ(α), then the probability of obtaining the state Ψ(α’), having started with a preceding Ψ(α)m spatial intervals {x}, can be written in terms of a transfer matrix as:

where Ψ(α) can be expressed in terms of (2q + 1)M orthonormal eigenvectors (with eigenvalues λr) of the operator TM. Each has (2q + 1)M components, one for each configuration #945;; that is:

Hence

Analogous to the time-domain persistent order analysis due to Little, it is of interest to find a particular state α1 after in spatial steps, having started at an arbitrary commencement (spatial location) in the neuronal topology; and hence the probability of obtaining the state α2 after spatial steps, given α1 after in spatial steps from the commencement location, can be written as:


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