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6.4 Fokker-Planck Equation of Neural Dynamics

The state of neural dynamics as indicated earlier is essentially decided by the intrinsic disturbances (noise) associated with the weighting function W. Due to the finite correlation time involved, the disturbance has bandlimited (colored) gaussian statistics.

Ideally in repeated neuronal cells, there could be no coherence between state transitions induced by the disturbance/noise, or even between successive transitions. Such complete decorrelation is valid only if the noise or disturbance level is very small. However, inasmuch as the correlation does persist, the state-variable W specified before in an M-dimensional space (Wi, i = 1, 2, 3, …, M), can be modeled as a simple version of Equation (63). It is given by [89]:

where η(t) is the noise term such that <η(t)> = 0 and <η(t)η’ (t)> = (kBT)Γexp{-Γ|t - t’|} where Γ → ∞ sets the limit that the above Fokker-Planck relation corresponds to the white noise case.

In the state-transition process, the relevant instability dynamics can be dictated by a set of stochastical differential equations, namely:

where ηi(t) refers to the noise/disturbance involved at the ith cell and Di(W) is an arbitrary function of W. Here as W approaches a representative value, say W0, at t = 0 so that Di(WO) = 0, then the state-transition is regarded as unstable. An approximate solution to Equation. (6.9) can then be sought by assuming that P(W, 0) = δ(W - W0) as the initial condition. This can be done by the scaling procedure outlined by Valsakumar [89]. Corresponding to Equation (6.9), a new stochastic process defined by a variable ζ(t) can be conceived such that in the limit of vanishing noise Equation (6.9) would refer to the new variable ζ(t), replacing the original variable W(t). The correspondence between ζ(t) and W(t) can then be written as [89]:

At the enunciation of instability (at t = 0), the extent of disturbance/noise is important; however, as the time progresses, the nonlinearity associated with the neuronal state transition overwhelms. Therefore, the initial fluctuations can be specified by replacing ∂ζ(t)/∂Wj in Equation (6.10) by its value at the unstable point. This refers to a scaling approximation and is explicitly written as:

The above approximation leads to a correspondence relation between the probability distribution of the scaled variable ζs, and distribution function PW (W, t). The scaling solution to Equation (6.8) is hence obtained as [89]:

where

The various moments (under scaling approximation) are:

where [τ’/Td(Γ)] = 2 (β/kBT[exp(2t) -1] and Td(Γ) is the switching-delay given by:

The second moment <W2> as a function of time is presented in Figure (6.5) for various values of Γ, namely, ∞, 10, 1, 0.1, and 0.01, which span very small to large correlation times. Further, (β/kBT) refers to the evolution of the normalized pseudo-thermodynamic energy level, and is decided by Equation (6.12a).


Figure 6.5  Evolution of the mean-squared value of W(t) for various discrete extents of correlation time (Γ)
(1. Γ = 10-2; 2. Γ = 10-1; 3. Γ = 100; 4. Γ = 10+1; 5. Γ = 10+4)

From Figure (6.5), it can be observed that the correlation time does not alter the qualitative aspects of the fluctuation behavior of the noise/disturbance. That is, in extensive terms, the onset of macroscopic order of neuronal state-transition is simply delayed when the correlation time increases.

6.5 Stochastical Instability in Neural Networks

Typically (artificial) neural networks are useful in solving a class of discrete optimization problems [34] in which the convergence of a system to a stable state is tracked via an energy function E, where the stable state presumably exists at the global minimum of E as mentioned in Chapter 4.

This internal state (in biological terms specified as the soma potential) of each neuron i is given by a time-dependent scalar value Si; the equilibrium state is assumed as 0. The output of the cell (corresponding to spike or action potential frequency) σi is a continuous, bounded, monotonic function F. That is, σi = F(Si); and, in general, F is nonlinear. Thus, the output of the cell is a nonlinear function of the internal state.

Typically F(x) is a sigmoid, taken conventionally in the hyperbolic tangent form as (1/2) [1 + tanh(Λx)], or more justifiably as the Langevin function Lq(Λx) as described in Chapter 5. The coefficient Λ is the scaling factor which includes a pseudo-temperature corresponding to the Boltzmann (pseudo) energy of the system. Ideally, the rate of change of internal state is decided by the sum of the inputs from other neurons of the network in the form of weighted sum of firing rates by external sources (such as a constant bias) and by the inhibiting internal state:

where ηi(t) represents the intracell disturbance/noise.

Upon integration (corresponding to a first-order low-pass transition with a time constant τ0), Equation (6.16) reduces to:

With a symmetric weighting (Wij = Wji), Hopfield [31,36] defines an energy function (E) relevant to the above temporal model of a neuron-cell as:


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