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U of M PSY 5038 - Lateral inhibition

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Introduction to Neural NetworksU. Minn. Psy 5038Lateral inhibitionIntroductionLast time‡Developed a "structure-less, continuous signal, and discrete time" generic neuron model and from there built a network. ‡Basic linear algebra review. Motivated linear algebra concepts from neural networks.TodayWe are going to look at an explanation of a human perceptual phenomenon called Mach bands, that involves a linear approximation based on a real neural network. This is an example of neural filtering found in early visual coding. We will study two types of network that may account for Mach bands: 1) feedforward; 2) feedback. The feedback system will provide our first example of a dynamical system. The system we will look at was developed as a model of the neural processing in the horseshoe crab (limulus) compound eye. Despite the (apparent) enormous difference between your visual system and that of the horseshoe crab, our visual system shares a fundamental image processing function with that of this lowly crustacean (and virtually all other animals that process image patterns).‡An application of a simple linear model for visual spatial filtering‡Add some dynamics for visual spatial filtering‡Winner-take-all network: Add a threshold non-linearityMach bands & perceptionErnst Mach was an Austrian physicist and philosopher. In addition to being well-known today for a unit of speed, and Mach's Principle in theoretical physics, he is also known for several visual illusions. One illusion is called "Mach bands". Let's make and experience some.In[13]:=width = 256;y@x_, hix_D := Module@8low, hi, lowx<,low = 0.2; hi = 0.8; lowx = .35 * width;Piecewise@88low , x < lowx<,8HHhi - lowLê Hhix - lowxLL x - HHhi - lowL lowxL êHhix - lowxL + low ,x >= lowx && x < hix<, 8hi, x >= hix<<DD;In[16]:=Manipulate@e3 = Table@y@i, hixD, 8i, 1, width<D; picture2 = Table@e3, 8i, 1, 60<D;GraphicsGrid@88Graphics@Raster@picture2, 881, 1<, 8120, 60<<, 80, 1<DD,Plot@y@x, hixD, 8x, 1, width<, PlotRange Ø 80, 1<D<<D,88hix, 87<, width ê 3., width * 3 ê4<DOut[16]=hix0501001502002500.20.40.60.81.0 PlotRange is used to scale the brightness.What Mach noticed was that the left knee of the ramp looked too dark, and the right knee looked too bright to be explained by the light intensity. Objective light intensity did not predict apparent brightness.2 Lect_5_LatInhibition.nb‡Mach's explanation in terms of lateral inhibitionIn general, lateral inhibition increases contrast at edges.Neural basis?Early visual neurons (e.g. ommatidia in horseshoe crab, ganglion cells in the mammalian retina and even later cells in the lateral geniculate neurons of the thalamus, and some cells in V1 or primary visual cortex of the monkey) have receptive fields with Mach's center surround organization. I.e. approximately circular excitatory centers and inhibitory surrounds. Or the opposite polarity, inhibitory centers and excitatory surrounds.Some history:Limulus (horseshoe crab)--Hartline won the 1967 Nobel prize for this work that began in the 20's with publications up to the 70's. (See http : êêwww.mbl.edu êanimalsêLimulus êvision êindex.html).Frog -- Barlow, H. B. (1953). Summation and inhibition in the frog's retina. J Physiol, 119, 69-88. Cat --S. W. Kuffler (1953). Discharge patterns and functional organization of mammalian retina . Journal of Neurophysiol-ogy, 16:37--68.Feedforward modelTwo types of models: feedforward and feedback (in our context, "recurrent lateral inhibition")f = w¢.ewhere e is a vector representing the input intensities (the e1 or y[] values above), w¢ is a suitably chosen set of weights (i.e. excitatory center and inhibitory surround as shown in the above figure), and f is the output. Lect_5_LatInhibition.nb 3‡Programming implementationBecause changes in the stimulus are one-dimensional, we'll simulate the response in one dimension. We specify the input vector e as above:In[17]:=e := Table@y@i, 170D, 8i, 1, width<D;ListPlot@eDOut[18]=501001502002500.30.40.50.60.70.8Let's assign weights consistent with Ernst Mach's 19th century hypothesis. We can model the receptive field for one output unit to be represented by 5 weights, with a center value of 6, and surround values of -1:4 Lect_5_LatInhibition.nbIn[19]:=wp = Table@0, 8i, 1, Length@eD<D; wpP1T = - 1; wpP2T = - 1; wpP3T = 6;wpP4T = -1; wpP5T = - 1;ListPlot@ wp, Joined Ø True, PlotRange Ø 880, 10<, 8- 2, 7<<,Filling Ø AxisDOut[19]=...or we could get a little more sophisticated, and use a common formula to model center-surround organization, the difference of two Gaussian functions, with different widths. Here is a specific choice that specifies a wider filter than above:In[20]:=wp = [email protected] * Exp@-HHi - 10L ê3L^2D - Exp@- HHi - 10Lê 6L^ 2D,8i, 1, Length@eD<D;ListPlot@ wp, Joined Ø True, PlotRange Ø 880, width ê8<, 8- 2, 2<<,Filling Ø AxisDOut[21]=The plot shows the "center-surround" organization of the filter. It is sometimes refered to as a "Mexican hat" filter.Now assume that all units have the same weights, and calculate the response at each point by shifting the weight filter wp right one by one, and taking the dot product with the input pattern e, each time:In[22]:=response = Table[RotateRight[wp,i].e,{i,1,width-30}];This way we can mimic the response we want:Lect_5_LatInhibition.nb 5In[23]:=ListPlot@response, Joined Ø True, Axes Ø FalseDOut[23]=Note that we cut the rotation short to eliminate boundary effects.Show that you can do this operation as matrix multiplication, where each subsequent row of a matrix W is the vector wp shifted over by one. ‡ConvolutionThis kind of operation where the same filter gets applied repeatedly at different positions is common in signal processing. Mathematica has a function ListConvolve[ ] that does this for you. It has additional arguments that allow for handling of the boundaries. What should you do when the filter gets close to the end of the stimulus? A common default is to let the filter wrap around. Another common solution is to "pad" the ends of e with fixed values, such as zero. What does the retina do?.Here's ListPlot[] with the simpler center-surround receptive field, {-1, -1, 6, -1, -1}. In mathematics, this filter vector is sometimes called the "kernel" of the filter.In[24]:=ListPlot@ListConvolve@8-1, - 1, 6, - 1, - 1<, eD, Joined Ø TrueDOut[24]=501001502002500.60.81.01.21.41.66 Lect_5_LatInhibition.nbWhat is the response to the ramp if the sum of the weights is zero? Build a


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