Presentation on theme: "Lecture 20081 Bioinspired Computing Lecture 3 Biological Neural Networks and Artificial Neural Networks Based on slides from Netta Cohen."— Presentation transcript:
lecture 20081 Bioinspired Computing Lecture 3 Biological Neural Networks and Artificial Neural Networks Based on slides from Netta Cohen
lecture 20082 We introduced swarm intelligence. We saw how many simple agents can follow simple rules that allow them to collectively perform more complex tasks. Last week: Today... Biological systems whose manifest function is information processing: computation, thought, memory, communication and control. We begin a dissection of a brain: How different is a brain from an artificial computer? How can we build and use artificial neural networks?
lecture 20083 Investigating the brain The computer Input program Output Summon Scottie, your engineer to disassemble the machines into component parts, test each part (electronically, optically, chemically…), decode the machine language, and study how components are connected. to connect to the input & output ports of a machine, find a language to communicate with it & write computer programs to test the systems response by measuring its speed, efficiency & performance at different tasks. Summon Data - your software wiz Imagine landing on an abandoned alien planet and finding thousands of alien computers. You and your crews mission is to find out how they work. What do you do? part #373a Inputs Outputs
lecture 20084 The brain as a computer Higher level functions in animal behaviour Gathering data (sensation) Inferring useful structures in data (perception) Storing and recalling information (memory) Planning and guiding future actions (decision) Carrying out the decisions (behaviour) Learning consequences of these actions Hardware functions and architectures 10 billion neurons in human cortex 10,000 synapses (connections) per neuron Machine language: 100mV, 1-2msec spikes (action potential) Specialised regions & pathways (visual, auditory, language…)
lecture 20085 Special task: program often hard-coded into system. Hardware not hard: plastic, rewiring. No clear hierarchy. Bi-directional feedback up & down the system. Unreliable components. Parallelism, redundancy appear to compensate. Output doesnt always match input: Internal state is important. Development & evolutionary constraints are crucial. Universal, general-purpose. Software: general, user-supplied. Hardware is hard: Only upgraded in discrete units. Obvious hierarchy: each component has a specific function. Once burned in, circuits run without failure for extended lifetimes. Input-output relations are well- defined. Engineering design depends on engineer. Function is not an issue. The brain as a computer versus
lecture 20086 Neuroscience pre-history 200 AD: Greek physician Galen hypothesises that nerves carry signals back & forth between sensory organs & the brain. 17th century: Descartes suggests that nerve signals account for reflex movements. 19th century: Helmholtz discovers the electrical nature of these signals, as they travel down a nerve. 1838-9: Schleiden & Schwann systematically study plant & animal tissue. Schwann proposes the theory of the cell (the basic unit of life in all living things). Mid-1800s: anatomists map the structure of the brain. but… The microscopic composition of the brain remains elusive. A raging debate surrounds early neuroscience research, until...
lecture 20087 1) Neurons are cells: distinct entities (or agents). 2) Inputs & outputs are received at junctions called synapses. 3) Input & output ports are distinct. Signals are uni-directional from input to output. Today, neurons (or nerve cells) are regarded as the basic information processing unit of the nervous system. The neuron doctrine Ramon y Cajal (1899) neuron Inputs Outputs
9 Neuron details
lecture 200810 Organisation of neurons
lecture 200811 Ion channels and spiking Membrane potential negative (inside /outside) Na+ would like to rush in but cant Depolarisation opens Na+ channels, Na+ flows in Chain reaction! More Na+ flows in! This opens K+ channels, K+ flows out: hyperpolarisation
lecture 200812 Macaque brain (Felleman & van Essen 1991)
lecture 200814 Both have well-defined inputs and outputs. Both are basic information processing units that comprise computational networks. If transistors can perform logical operations, maybe neurons can too? The neuron as a transistor Neuronal function is typically modelled by a combination of a linear operation (sum over inputs) and a nonlinear one (thresholding). input neuron output This simple representation relies on Cajals concept of
lecture 200815 The basic bit of information is represented by neurons in spikes. The cell is said to be either at rest or active. A spike (action potential) is a strong, brief electrical pulse. Since these action potentials are mostly identical, we can safely refer to them as all-or-none signals. Machine language Why Spikes? Why dont neurons use analog signals? One answer lies in the network architecture: signals cover long distances (both within the brain and throughout the body). Reliable transmissions requires strong pulses.
lecture 200816 Computation of a pyramidal neuron Single all-or-none output Many inputs (dendrites) soma axon
lecture 200817 We can now summarise our working principles: The basic computational unit of the brain is the neuron. The machine language is binary: spikes. Communication between neurons is via synapses. However, we have not yet asked how information is encoded in the brain, how it is processed in the brain, and whether what goes on in the brain is really computation. From transistors to networks
lecture 200818 Examples of both neural codes and distributed representations have been found in the brain. Example in the visual system: colour representation, face recognition, orientation, motion detection, & more… Information codes Temporal codeNeural code Rate codePopulation code/ Distributed code noise http://www.cs.stir.ac.uk/courses/31YF/Notes/Notes_NC.html
lecture 200819 Example. A spike train produced by a neuron over an interval of 100ms is recorded. Neurons can produce a spike every 2ms. Therefore, rates (individual code words) can be produced by this neuron. In contrast, if the neuron were using temporal coding, up to 2 50 different words could be represented. In this sense, temporal coding is much more powerful. Information content 51 different 2 50 different
lecture 200820 Temporal codes rely on a noise-free signal transmission. Thus, we would expect to find very few redundant neurons with co-varying outputs in that network. Accordingly, an optimal temporal coding circuit might tend to eliminate redundancy in the pattern of inputs to different neurons. On the other hand, if neural information is carried by a noisy rate-based code, then noise can be averaged out over a population of neurons. Population coding schemes, in which many neurons represent the same information, would therefore be the norm in those networks. Experiments on various brain systems find either coding systems, and in some cases, combinations of temporal and rate coding are found. Circuitry depends on neural code
lecture 200821 Having introduced neurons, neuronal circuits and even information codes with well defined inputs and outputs, we still have not mentioned the term computation. Is neuronal computation anything like computer computation? Neuronal computation 101111 If read 1, write 0, go right, repeat. If read 0, write 1, HALT! If read , write 1, HALT! In a computer program, variable have initial states, there are possible transitions, and a program specifies the rules. The same is true for machine language. To obtain an answer at the end of a computation, the program must HALT. Does the brain initialise variables? Does the brain ever halt?
lecture 200822 Answer: The input causes the network to enter an initial state. The state of the neural network then evolves until it reaches some new stable state. The new state is associated with the input state. One recasting of biological brain function in these computational terms was proposed by John Hopfield in the 1980s as a model for associative memory. Association an example of bio-computation Question: How does the brain associate some memory with a given input?
lecture 200823 Association (cont.) Whatever initial condition is chosen, the system will follow a well-defined route through state-space that is guaranteed to always reach some stable point (i.e., pattern of activity) Hopfields ideas were strongly motivated by existing theories of self-organisation in neural networks. Today, Hopfield nets are a successful example of bio-inspired computing (but no longer believed to model computation in the brain). Trajectories in a schematic state space
lecture 200824 Learning No discussion of the brain, or nervous systems more generally is complete without mention of learning. What is learning? How does a neural network know what computation to perform? How does it know when it gets an answer right (or wrong)? What actually changes as a neural network undergoes learning? brain Sensory inputs Motor outputs body environment
lecture 200825 Learning (cont.) Learning can take many forms: Supervised learning Reinforcement learning Association Conditioning Evolution At the level of neural networks, the best understood forms of learning occur in the synapses, i.e., the strengthening and weakening of connections between neurons. The brain uses its own learning algorithms to define how connections should change in a network.
lecture 200826 How do the neural networks form in the brain? Once formed, what determines how the circuit might change? Learning from experience In 1948, Donald Hebb, in his book, "The Organization of Behavior", showed how basic psychological phenomena of attention, perception & memory might emerge in the brain. Hebb regarded neural networks as a collection of cells that can collectively store memories. Our memories reflect our experience. How does experience affect neurons and neural networks? How do neural networks learn?
lecture 200827 Synaptic Plasticity Definition of Learning: experience alters behaviour The basic experience in neurons is spikes. Spikes are transmitted between neurons through synapses. Hebb suggested that connections in the brain change in response to experience. Pre-synaptic cell Post-synaptic cell delay time Hebbian learning: If the pre-synaptic cell causes the post-synaptic cell to fire a spike, then the connection between them will be enhanced. Eventually, this will lead to a path of least resistance in the network.
lecture 200828 Today... From biology to information processing Next time... Artificial neural networks (part 1) Focus on the simplest cartoon models of biological neural nets. We will build on lessons from today to design simple artificial neurons and networks that perform useful computational tasks. At the turn of the 21st century, how does it work remains an open question. But even the kernel of understanding and simplified models we already have for various brain function are priceless, in providing useful intuition and powerful tools for bioinspired computation.
lecture 200829 The Appeal of Neural Computing The only intelligent systems that we know of are biological. In particular most brains share the following feature in their neural architecture – they are massively parallel networks organised into interconnected hierarchies of complex structures. In addition, they are very good at some tasks that computers are typically poor at: recognising patterns, balancing conflicts, sensory- motor coordination, interaction with the environment, anticipation, learning… even curiosity, creativity & consciousness. speed, tolerance, robustness, flexibility, self-driven dynamic activity For computer scientists, many natural systems appear to share many attractive properties:
lecture 200830 The first artificial neuron model In analogy to a biological neuron, we can think of a virtual neuron that crudely mimics the biological neuron and performs analogous computation. The artificial neuron is a cartoon model that will not have all the biological complexity of real neurons. How powerful is it? Just like biological neurons, this artificial neuron neuron will have: Inputs (like biological dendrites) carry signal to cell body. inputs A body (like the soma), sums over inputs to compute output, and Σ cell body outputs (like synapses on the axon) transmit the output downstream. output
lecture 200831 Early history (1943) In this seminal paper, Warren McCulloch and Walter Pitts invented the first artificial (MP) neuron, based on the insight that a nerve cell will fire an impulse only if its threshold value is exceeded. MP neurons are hard-wired devices, reading pre-defined input-output associations to determine their final output. Despite their simplicity, M&P proved that a single MP neuron can perform universal logic operations. A network of such neurons can therefore do anything a Turing machine can do, but with a much more flexible (and potentially very parallel) architecture. McCulloch & Pitts (1943). A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 5, 115-137.
lecture 200832 * * * * * ( ) over all i Weighted inputs are summed in the cell body. The McCulloch-Pitts (MP) neuron The computation consists of "adders" and a threshold. Each input has an assigned weight w. w1w1 w2w2 w3w3 wnwn weights inputs x1x1 x2x2 x3x3 xnxn inputs Inputs x are binary: 0,1. 1 if 0 if < output = Otherwise, the output=0. If the neuron fires, the output =1. Neuron fires if sum exceeds (or equals) activation threshold. Note: an equivalent formalism assigns =0 & instead of threshold introduces an extra bias input, such that bias * w bias = - bias * wbwb
lecture 200833 IN 1 OUT 1 0 0 1 0 Always 0 IN 1 OUT 2 0 0 1 1 IDENTITY IN 1 OUT 3 0 1 1 0 NOT IN 1 OUT 4 0 1 1 1 Always 1 For binary logic gates, with only one input, possible outputs are described by the following truth tables: For example: Logic gates with MP neurons NOT x x w = -0.5 w = -1 Excercise: Find w and for the 3 remaining gates.
lecture 200834 IN 1 IN 2 Here is a compact, graphical representation of the same truth table: Logic gates with MP neurons (cont.) With two binary inputs, there are 4 possible inputs and 2 4 = 16 corresponding truth tables (outputs)! For example, the AND gate implemented in the MP neuron: = +1.5 1 1 x 1 AND x 2 x1x1 x2x2 Excercise: Find w and for OR & NAND.
lecture 200835 Computational power of MP neurons Universality: NOT & AND can be combined to perform any logical function; MP neurons, circuited together in a network can solve any problem that a conventional computer could. But lets examine the single neuron a little longer. Q: Just how powerful is a single MP neuron? A: It can solve any problem that can be expressed as a classification of points on a plane by a single straight line. Generalisation to many inputs: points in many dimensions are now classified, not by a line, but by a flat surface. IN 1 IN 2 AND Even one neuron can successfully handle simple classification problem.
lecture 200836 trace 1 2.4 9.8 1.2 0.4 7.9 6.7 etc. trace 2 1.0 8.3 0.2 2.1 8.8 7.2 etc. problem? yes no yes no etc. output sum x i w i * inputs x1x1 bias x2x2 weights w1w1 w3w3 w2w2 x i w i etc. output +6.6 -8.1 etc. +8.6 +7.5 -6.7 -3.9 w 1 =-1, w 2 =-1, w 3 =+10 & bias=+1 output Yes No etc. Yes No Classification in Action A set of patients may have a medical problem. Blood samples are analysed for the quantities of two trace elements. With correct weights, this MP neuron consistently classifies patients. +ive output = problem
lecture 200837 The missing step The ability of the neuron to classify inputs correctly hinges on the appropriate assignment of the weights and threshold. So far, we have done this by hand. Imagine we had an automatic algorithm for the neuron to learn the right weights and threshold on its own. In 1962, Rosenblatt, inspired by biological learning rules, did just that. Frank Rosenblatt (1962). Principles of Neurodynamics, Spartan, New York
lecture 200838 The perceptron algorithm Take w j random START: Take X ε F + U F - CHECK: if x ε F + and Σ w j x j > 0 goto START if x ε F + and Σ w j x j 0 goto ADD if x ε F - and Σ w j x j 0 goto START if x ε F - and Σ w j x j > 0 goto SUB ADD: w j w j + x j goto START SUB:w j w j - x j goto START:
lecture 200839 The Perceptron Theorem Says that the previous algorithm will converge on a set of weights in a finite number of steps if w* exists
lecture 200840 Imagine a naive, randomly weighted neuron. One way to train a neuron to discriminate the sick from the healthy, is by reinforcing good behaviour and penalising bad. This carrot & stick model is the basis for the learning rule: Initialise the neuronal weights (random initialisation is the standard). Run each input set in turn through the neuron & note its output. Whenever a wrong output is encountered, alter responsible weights. Learning Rule: Repeatedly run through training set until all outputs agree with targets. w i w i + x i if output too low w i w i x i if output too high When training is complete, test the neuron on a new testing set of patients. If neuron succeeds, patients whose health is unknown may be determined. Compile a training set of N (say 100) sick and healthy patients.
lecture 200841 Related idea Minimize E = Σ i (t i – o i ) 2 Here: – t is the desired output –o is the observed output Find weights that minimize E Steepest gradient descent will also yield
lecture 200842 Supervised learning The learning rule is an example of supervised learning. Training MP neurons requires a training set, for which the correct output is known. These correct or desired outputs are used to calculate the error, which in turn is used to adjust the input-output relation of the neuron. Without knowledge of the desired output, the neuron cannot be trained. Therefore, supervised learning is a powerful tool when training sets with desired outputs are available. When cant supervised learning be used? Are biological neurons supervised?
lecture 200843 A simple example Lets try to train a neuron to learn the logical OR operation: 0 1 0 x i w i 0 x 1 OR x 2 0 1 1 1 desired output x2x2 0 1 0 1 x1x1 0 0 1 1 x3x3 1 1 1 1 bias x1x1 x3 x3 x2x2 w1w1 w3w3 w2w2 x i w i output w i w i + x i if output low w i w i x i if output high Decision ( le 0 or gt 0) Example on white board
lecture 200844 Some common variations on this learning rule: Adding a learning rate 0
"name": "lecture 200844 Some common variations on this learning rule: Adding a learning rate 0
lecture 200845 Learning Rule Called rule because weight updates have the following form w w + x is a measure for the error: = 0, no weight change
lecture 200846 The Perceptron convergence Theorem Suppose there are two sets: F + and F - ; F + F - empty Goal: X ε F + Σ w j x j > 0 X ε F - Σ w j x j < 0 If there are w j * for which this is true, then the following algorithm finds w j (possibly different ones) which also do the trick
lecture 200847 The Fall of the Artificial Neuron Marvin Minsky & Seymour Papert (1969). Perceptrons, MIT Press, Cambridge. Before long researchers had begun to discover the neurons limitations. Unless input categories were linearly separable, a perceptron could not learn to discriminate between them. Unfortunately, it appeared that many important categories were not linearly separable. This proved a fatal blow to the artificial neural networks community. Successful Unsuccessful Many Hours in the Gym per Week Few Hours in the Gym per Week Footballers Academics In this example, an MP neuron would not be able to discriminate between the footballers and the academics… This failure caused the majority of researchers to walk away. Exercise: Which logic operation is described in this example?
lecture 200848 Connectionism Reborn David E. Rumelhart & James L. McClelland (1986). Parallel Distributed Processing, Vols. 1 & 2, MIT Press, Cambridge, MA. Most influential of these was a two-volume book by Rumelhart & McClelland, who suggested a feed-forward architecture of neurons: layers of neurons, with each layer feeding its calculations on to the next. The crisis in artificial neural networks can be understood, not as an inability to connect many neurons in a network, but an inability to generalise the training algorithms to arbitrary architectures. By arranging the neurons in an appropriate architecture, a suitable training algorithm could be invented. The solution, once found, quickly emerged as the most popular learning algorithm for nnets. Back-propagation first discovered in 1974 (Werbos, PhD thesis, Harvard) but discovery went unnoticed. In the mid-80s, it was rediscovered independently by three groups within about one year.
lecture 200849 This time… The appeal of neural computing From biological to artificial neurons Nervous systems as logic circuits Classification with the McCulloch & Pitts neuron Developments in the 60s: –The Delta learning rule & variations –Simple applications –The fatal flaw of linearity Next time… The disappointment with the single neuron dissipated as promptly as it dawned upon the AI community. Next time, we will see why the single neurons simplicity does not rule out immense richness at the network level. We will examine the simplest architecture of feed-forward neural networks and generalise the delta-learning rule to these multi-layer networks. We will also re-discover some impressive applications.
lecture 200850 Optional reading Excellent treatments of the perceptron, the delta rule & Hebbian learning, the multi-layer perceptron and the back-propagation learning algorithm can be found in: Beale & Jackson (1990). Neural Computing, chaps. 3 & 4. Hinton (1992). How neural networks learn from experience, Scientific American, 267 (Sep):104-109.