Neural Systems (1) Companion slides for the book Bio-Inspired - PowerPoint PPT Presentation

Neural Systems (1) Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 1 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Why Nervous Systems? Not all animals have nervous systems; some use only chemical reactions Paramecium and sponge move, eat, escape, display habituation QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Nervous systems give advantages : 1) Selective transmission of signals across distant areas (=more complex bodies) 2) Complex adaptation (=survival in changing environments) Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 2 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Biological Neurons QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 3 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Type of Neurons QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. Interneurons can be 1- Excitatory 2- Inhibitory Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 4 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

How Do Neurons Communicate? Firing rate Firing time McCulloch-Pitts Spiking neurons Computational Connectionism Biology 100 ms Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 5 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Synaptic Plasticity pre-synaptic neuron post-synaptic neuron A B synapse Hebb rule (1949): Synaptic strength is increased if cell A consistently contributes to firing of cell B This implies a temporal relation: neuron A fires first, neuron B fires second % synaptic modification Spike Time Dependent Plasticity (STDP): - Small time window - Strengthening (LTP) for positive time difference -Weakening (LTD) for negative time difference postsynaptic - presynaptic (ms) Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 6 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

An Artificial Neural Network A neural network communicates with the environments through input units and output units. All other elements are called internal or hidden units. Units are linked by uni-directional connections. A connection is characterized by a weight and a sign that transforms the signal. Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 7 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Biological and Artificial Neurons Biological Artificial (pyramidal) (McCulloch-Pitts)   N ( ) = Φ ∑   y i = Φ A i w ij x j − ϑ i   j Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 8 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Output functions Identity Step Sigmoid ( ) ( ) ( ) Φ x Φ x Φ x x x x Sigmoid function: 1 • continuous ( ) = Φ x − kx 1 + e • non-linear • monotonic ( ) = ( ) • bounded Φ x tanh kx • asymptotic Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 9 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Signalling Input Familiarity The output of a neuron is a measure of how similar is its current input pattern to its pattern of connection weights. 1. Output of a neuron in linear algebra notation:  N   ∑ y = w ⋅ x y = a a = 1 w i x i ,   i 2. Distance between two vectors is: cos ϑ = w ⋅ x 0 ≤ ϑ ≤ π w x , where the vector length is: 2 + x 2 ϑ = 0 o → cos ϑ = 1, x = x ⋅ x = 2 + ... + x n 2 x 1 o → cos ϑ = 0, ϑ = 90 3. Output signals input familiarity ϑ = 180 o → cos ϑ = − 1, w ⋅ x = w x cos ϑ Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 10 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Separating Input Patterns A neuron divides the input space in two regions, one where A>=0 and one where A<0. The separation line is defined by the synaptic weights: x 2 = ϑ − w 1 w 1 x 1 + w 2 x 2 − ϑ = 0 x 1 w 2 w 2 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. ϑ > 0 ϑ = 0 Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 11 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

From Threshold to Bias unit The threshold can be expressed as an additional weighted input from a special unit, known as bias unit, whose output is always -1.   N ∑ ( ) = Φ   y i = Φ A i w ij x j − ϑ i     j = 1   N Quick Time™ and a ∑ ( ) = Φ TIFF (LZW ) decompres s or   y i = Φ A i w ij x j are needed to s ee this picture.     j = 0 • Easier to express/program • Threshold is adaptable like other weights Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 12 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Architectures e a) feed-forward b) feedforward multilayer c, d) recurrent e) fully connected Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 13 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Input Encoding LOCAL One neuron stands for one item Grandmother cells Scalability problem Robustness problem DISTRIBUTED Neurons encode features One neuron may stand for >1 item One item may activate >1 neuron Robust to damage Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 14 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Learning Learning is experience-dependent modification of connection weights pre-synaptic neuron post-synaptic neuron x j y i w ij synapse ∆ w ij = x j y i Hebb’s rule (1949) t = w ij t − 1 + η ∆ w ij w ij Standard weight update learning rate (in the range [0,1]) Hebb’s rule suffers from self-amplification (unbounded growth of weights) Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 15 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Unsupervised Learning Biological synapses cannot grow indefinitely Oja (1982) proposed to limit weight growth by introducing a self-limiting factor ( ) ∆ w j = η y x j − w j y As a result, the weight vector develops along the direction of maximal variance of the input distribution. Neuron learns how familar a new pattern is: input patterns that are closer to this vector elict stronger response than patterns that are far away. Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 16 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Principal Component Analysis  N   ∑ ∆ w ij = η y i x j − w kj y k   Oja rule for N output units k = 1 develops weights that span the sub-space of the N principal components of the input distribution.  i   ∑ ∆ w ij = η y i x j − w kj y k   Sanger rule for N output k = 1 units develops weights that correspond to the N principal components of the input distribution. Useful for reduction of dimensionality and feature extraction Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 17 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Do brains compute PCA? Receptive field is the pattern of stimulation that activates a neuron. Equivalent to pattern of synaptic weights Example of visual RF An Oja network with multiple output units exposed to a large set of natural images develops receptive fields similar to those found in the visual cortex of all mammals [Hancock et al., 1992] However: a) PCA cannot detect spatial frequencies, brains do b) Cannot separate signal sources generated by independent signals Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 18 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press

Supervised Learning • Teacher provides desired responses for a set of training patterns • Synaptic weights are modified in order to reduce the error between the output y and its desired output t (a.k.a. teaching input) δ i = t i − y i Widrow-Hoff defined the error with the symbol delta: which is why this learning rule is also known as delta rule . ( ) w ij = rnd ± 0 .1 y, t initialize weights to random values linear ∑ present input pattern and units y i = w ij x j compute neuron output j = 0 x0 x1 x2 compute weight change using ( ) x j ∆ w ij = η t i − y i difference between desired repeat output and neuron output for every t − 1 + ∆ w ij input/output get new weights by adding w ij = w ij pair until computed change to previous error is 0 weight values Companion slides for the book Bio-Inspired Artificial Intelligence: Theories, 19 Methods, and Technologies by Dario Floreano and Claudio Mattiussi, MIT Press