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6.9 Concluding Remarks

The inevitable presence of noise in a neural assembly permits the neurons to change their internal states in a random manner. The relevant state-transitional stochastical dynamics is governed by relaxational equation(s) of Langevin or Fokker-Planck types.

In general, the noise or intracell disturbances cited above could be gaussian, but need not be white. Such band-limited (colored) properties are intrinsic properties of the disturbances and are not influenced by the switching action of the state transition. Considering the colored noise situation, the Langevin and/or Fokker-Planck equation(s) can be solved by a scaling approximation technique.

The colored nature of the cellular noise also refers implicitly to the markovian nature of the temporal statistics of the action potentials which assume bistable values at random intervals. The weighting times in each state are exponentially distributed. Correspondingly, the onset of macroscopic order of neuronal state transition is simply delayed (in extensive terms) as the correlation time increases. The correlation time does not, however, alter the qualitative aspects of intracellular disturbances.

The effect of intracellular disturbances when addressed to artificial neural networks refers to stochastical instability in solving optimization problems. Such noise-induced effects would render the problem suboptimal with increased computational time.

Considering Hopfield networks, the presence of intracellular noise may not permit the network to settle at a global minimum of the energy function.

In terms of the Lyapunov condition, this nonrealization of a global minimum refers to the instability in the state transition process with specified lower and upper statistical bounds.

In the presence of noise, linear estimates of the input/output vectors of the neuronal network can be obtained via linear regression techniques. The corresponding estimate of the energy function indicates that the effect of intracellular disturbances can be implicitly dictated by modifying the constant (external) input bias to an extent proportional to the strength of the randomness.

The implications of this modified bias parameters are:

a.  For small values of noise, a linear approximation of input-output relation leads to the feasibility of a recursive search for stable states via appropriate feedback techniques.
b.  The modified bias parameter also alters the saturated neuronal state population randomly.


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