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6.7 Stable States Search via Modified Bias Parameter


Figure 6.7  Linear recursive search of stable states η: Noise; εr: Error; LPA: Low-pass action(integrator) F: Nonlinear estimator; Ii: Modified bias parameter [Ii → θi, as εr → 0, S1 → S and θ’i → θi as η → 0]

In the previous section, it is indicated that the presence of intracell disturbance ηi implicitly dictates the external bias parameter θi being modified to a new value specified as Ii. If the strength of randomness of the disturbance involved is small, an approximate (linear) recursive search for stable states is feasible. In general, the noise-perturbed vector a when subjected to F-1 transformation yields the corresponding noise-perturbed value of S1 as illustrated in Figure 6.7.

Hence, the summed input S and S1 can be compared, and the corresponding error εr can be used to cancel the effect of the intracellular noise which tends to alter the value of the input bias θi to Ii as shown in Figure 6.7. The corresponding correction leads to Ii → θi (≈θi). If necessary, a weighting WI (such as linear-logarithmic weighting) can be incorporated on θi’ for piecewise compatibility against the low to high strength of the randomness of the noise.

6.8 Noise-Induced Effects on Saturated Neural Population

The intracell disturbances could also affect implicitly the number of neurons attaining the saturation or the dichotomous values. Relevant considerations are addressed in this section pertaining to a simple input/output relation in a neuronal cell as depicted in Figure 6.8.


Figure 6.8  Network representation of neurocellular activity

The dynamics of neuron cellular activity can, in general, be written explicitly as [92]:

where τo is the time-constant (RC) of the integrator stated before and WI is the weighting factor on the external bias θi (modified to value Ii due to the noise, η). The neuronal state change is governed between its dichotomous limits by a nonlinear amplification process (with a gain Λ) as follows:

Over the transient regime of state change, the number of neurons attaining the saturation (or the dichotomous limits) would continuously change due to the nonlinear gain (Λ) of the system. Denoting the instants of such changes as a set of {tk}, k ∈ 0, 1, 2, …, at any instant tk, the number of neurons still at subdichotomous limiting values is assumed as μk. Therefore, during the period tk ≤ t ≤ tk+1, the following state dynamics can be specified:

assuming that Wij = WI and τo = 1. Further, χi = Ii + Λ Sign (Si) where:

The coupled relations of Equation (6.32) are not amenable for a single solution. However, as indicated by Yuan et al. [92], an intermediate function , with can be introduced in Equation (6.32) so as to modify it as follows:

If Λ > (M + 1) and |Si| ≤ 1, a relevant solution of Equation (6.34) indicates Si growing exponentially. However, if |Si| > 1, the dynamics of Si become stable provided Ii → θi with η = 0. At this stable state, considering the intermediate function Sign(Si) and |Si| > 1, for all 1 ≤ i ≤ M, the dynamic solution of Equation (6.34) can be written as:

As the network responds to an input vector Si to yield a dichotomous vector σi, the initial condition set as Si(0) and the external bias parameter Ii (→ θi) determine the division of neuronal states being “high” or “low”. Yuan et al. [92] point out that the binary output vector σ has M/2 neuronal high states corresponding to the M/2 high state components of the bias input; and there are M/2 neuronal low states corresponding to the rest of the components of the bias input. In the event of θi being corrupted by an additive noise, the resulting input bias, namely, Ii, will upset this division of high and low level states in the output vector σ in a random manner which manifests as the neuronal instability.


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