Chapter11.pdf - Chapter 11 Solutions to Review Questions 11.1 All cellular systems face the noise saturation dilemma If the network is sensitive to

Chapter11.pdf - Chapter 11 Solutions to Review Questions...

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Chapter 11 Solutions to Review Questions 11.1 All cellular systems face the noise saturation dilemma. If the network is sensitive to large inputs, the network tends to ignore small input as noise. If the network is sensitive to small inputs, large inputs tend to saturate the system. A limited operating range seems incompatible with an unlimited dynamical range of inputs. To solve this noise saturation dilemma first a simple additive model was introduced. Additive dynamics resemble linear dynamics. The change in neuronal activation equals a sum of activation and signals, not a product or other non linear function. In the additive model, each node receives an external input I i and that the individual nodes do not interact with each other. The basic requirement of the network is to remain sensitive to the input ratios θ i . the i th neuron receives input I i . the convex coefficient θ i defines the “reflectance” I i : I i = θ i I I I i i = θ The simplest additive activation model combines the feed forward input I i , the passive decay rate A and the activation bound [0, B ]. Then the additive model is as follows: ( ) i i i i I x B Ax x + = (1) where if B is the total number of cells then x i is the number of cells which are turned ON and ( B – x i ) cells are turned OFF. I i is the intensity which makes the OFF cells ON. Equation (1) can be written as, ( ) i i i i BI x I A x + + = x 1 x 2 x i x n + + + + I 1 I 2 I i I n
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Under equilibrium conditions i x = 0 ( ) 0 = + + i i i BI x I A or i i i I A BI x + = Now using I I i i = θ , we have I A I B x i i i θ θ + = B x I As i , Thus we see that x i approaches B even if the relative input intensity θ i is small. This is called saturation. Thus we see that additive models saturate. Now to preserve the sensitivity to the input signals, in the network, we consider a multiplicative model suggested by Grossberg. In it, the i th input excites the i th cell and inhibit all others. This is called an ON centre OFF surrounded kind of network or a cooperative competitive net. We know that, + = = i j j i i i i I I I I I θ Thus the equation of a cooperative competitive net is given by the following equation. ( ) n i I x I x B Ax x i j j i i i i i , 2 , 1 , L L = + = The term i j i i I x shows how the inputs I j j i inhibit the other excited nodes. - - - x 1 x 2 x i x n I 1 I i I n +
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( ) i i i i i j j i i BI x I A BI x I I A x + + = + + + = Under equilibrium condition 0 = i x 1 + = + = + = I A i i i i B I A BI I A BI x θ θ i i B x I θ as There we see that at equilibrium the neuronal activation measures the scaled pattern information θ i . Thus pattern sensitivity replaces activation saturation. Thus we see that according to the Grossberg Saturation Theorem: Additive Models saturate, multiplicative models do not.
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  • Summer '18
  • Shiv
  • Pattern matching

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