What is a receptive field? Why a sensory neuron has such particular RF How a RF was developed?

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2 What is a receptive field? Why a sensory neuron has such particular RF How a RF was developed?

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4 x 1 x 2 x 3 y f w 1 w 2 w 3 T x y = f (wx i i T ) i y

5 x 1 x 2 x 3 = = E (y y) (y f( wx T)) 2 2 o o i i i w 1 w 2 w 3 w = w η i + 1 i E w i y y = f (wx i i i T )

6 Optimization Cost E i Ei+1 E min W i W i+1 W opt

7 Optimization Cost E i Ei+1 E min W i W i+1 W opt

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9 Kohonen Network Model Neurons Inputs

10 Kohonen Self-organizing Feature Maps Order out of randomness

11 Human Brain 1. Organized spatially into regions according to sensory functions -performing special tasks, speech, analysis of sensory signals 2. Structure in brain -minimize the wiring between functions in close contact -separated (in location) responses to minimize crosstalk between function

12 Feature Maps What is the result of brain s selforganization? formation of feature maps in the brain that have a linear or planar topology, that is, they extend in one or two dimensions.

13 Feature Maps Retinotopic Map Somatosensory Map Tonotopic Map

14 Mapping of the visual field on the cortex

15 retinotopic map visual field is mapped in the visual cortex (occipital lobe) with higher resolution for the centre of the visual field. Feature Maps

16 Somatotopic Map The somatosensory cortex processes the information of the sensory neurons that lie below the skin. Note that both the skin and the somatosensory cortex can be seen as two-dimensional spaces Feature Map

17 Somatosensory Man Picture of the male body with the body parts scaled according to the area devoted to these parts in the somatosensory cortex

18 Feature Maps

19 tonotopic map sound frequencies are spatially mapped into regions of the cortex in an orderly progression from low to high frequencies. Feature Maps

20 Feature Maps Why are feature maps important? Sensory experience is multidimensional, e.g. sound is characterised by pitch, intensity, timbre, noise etc., The brain maps the external multidimensional representation of the world (including its spatial relations) into a similar 1 or 2 - dimensional internal representation.that is, the brain processes the external signals in a topology-preserving way.

21 Self-organization Simultaneous development of both structure and parameter in learning The evolution of a system into an organized form, under the control of the system itself

22 SOM with 2-D Topology j 2-D LATTICE OF NEURONS i w i1 w j2 x 1 x 2 INPUT VECTOR

23 Self Organizing Map Impose a topological order onto the competitive neurons (e.g., rectangular map) Let neighbors of the winner share the prize (The postcode lottery principle) After learning, neurons with similar weights tend to cluster on the map

24 SOM with 1-D Topology i w i1 j w i2 w j1 j w j2 1-D LATTICE OF NEURONS x 1 x 2 INPUT VECTOR

25 Winner Selection Let x = (x 1, x 2,..., x n ) T be the input vector in R n To each neuron i = (i 1, i 2 ) an associated weight vector w i = (w i1, w i2,..., w in ) T R n Neuron c = (c 1, c 2 ) whose weight vector w c is closest to the input (e.g. in Euclidean distance sense) is termed the winner of the competition: c = arg min { x w i } i

26 Weight Adaptation The weights of the winner and of neurons in its topological neighborhood N c (t) are adapted according to the following equation: w i (t+1) = w i (t) + (t) (i, N c (t)) [x(t) - w i (t)] i N c (t) with w i (t+1) = w i (t) i N c (t)

27 Graphical Interpretation of Adaptation Assume that i N c (t) and (i, N c (t)) = 1. In this case the weight adaptation equation becomes: w i (t+1) = w i (t) + (t) [x(t) - w i (t)] x 2 w i (t) x w i (t+1) x x(t) w j (t) x Input and Weight Space x 1

28 Example for 1-D Map x 2 x x x x INPUT x x x x x x x x x 1 Input and Weight Space

29 Self-Organization of a 1-D Map w i (0) (1) (2) ( ) x(1) x max x(2) Receptive field of neuron 5 x(0) x min neuron

30 MOST COMMON 2-D NEURON LATTICES (TOPOLOGICAL STRUCTURES) RECTANGULAR LATTICE HEXAGONAL LATTICE

31 Topological Neighborhoods N c (t) c c ROMBIC NEIGHBORHOODS SQUARE NEIGHBORHOODS

32 Well trained net should have same topology as that in the physical space, and will reflect the properties of the training set SOM-Graphics

33 2D examples

34 Self Organizing Map

35 Self Organizing Map

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37 World Poverty Map The SOM can be used to portray complex correlations in statistical data. Here the data consisted of World Bank statistics of countries in Altogether 39 indicators describing various quality-of-life factors, such as state of health, nutrition, educational services, etc, were used. The complex joint effect of these factors can can be visualized by organizing the countries using the self-organizing map. Countries that had similar values of the indicators found a place near each other on the map. The different clusters on the map were automatically encoded with different bright colors, nevertheless so that colors change smoothly on the map display. As a result of this process, each country was in fact automatically assigned a color describing its poverty type in relation to other countries. The poverty structures of the world can then be visualized in a straightforward manner: each country on the geographic map has been colored according to its poverty type.

38 Self Organizing Map

39 Self Organizing Map

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41 Phonetic Typewriter The phonetic typewriter is constructed by Tuevo Kohonen, see e.g. his book Self- Organizing Maps, Springer, 1995.

42 Phonotopic Map Input vectors are 15 dimensional speech samples from the Finnish language Each vector component represents the average output power over 10ms interval in a certain range of the spectrum (200 Hz 6400 Hz) Neurons are organized in a 8x12 hexagonal grid After formation of the map, the individual neurons were calibrated to represent phonemes The resulting map is called the phonetic typewriter

43 Phoneme Recognion