Bioinspired Computing Lectures Week 3
Biological Neural Networks and Artificial Neural Networks
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We introduced swarm intelligence.
We saw how many simple agents can follow simple rules that allow them to collectively perform more complex tasks.
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?
Investigating the brain
Imagine landing on an abandoned alien planet and finding thousands of alien computers. You and your crew’s mission is to find out how they work. What do you do?
Engineering approach
– disassemble the machines into component parts, test each part (electronically, optically, chemically…), decode the machine language, and study how components are connected.
Inputs Outputs
Software approach
– connect to the input & output ports of a machine, find a language to communicate with it & write computer programs to test the system’s response by measuring its speed, efficiency & performance at different tasks.
Input Output program
part #373a
The computer
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
• 10-100 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…)
The brain as a computer
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 doesn’t 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.
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…
The neuron doctrine Ramon y Cajal (1899)
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.
Neuron details
Organisation of neurons
Ion channels and spiking
Membrane potential negative (inside /outside) Na+ would like to rush in but can’t
Depolarisation opens Na+ channels, Na+ flows in
Chain reaction! More Na+ flows in!
hyperpolarisation
This opens K+ channels, K+ flows out:
Macaque brain (Felleman & van Essen 1991)
The neuron as a transistor
• 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?
Neuronal function is typically modelled by a combination of • a linear operation (sum over inputs) and
• a nonlinear one (thresholding).
This simple representation relies on Cajal’s concept of
input neuron output
Machine language
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.
Why Spikes?
Why don’t 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.
Computation of a pyramidal neuron
Many inputs (dendrites)
Single all-or-none
From transistors to networks
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’.
Information codes
Temporal code
Neural code
Population code/ Distributed code
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 content
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 words could be represented.
In this sense, temporal coding is much more powerful.
51 different
250 different
Circuitry depends on neural code
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.
Neuronal computation
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?
If read 1, write 0, go right, repeat. If read 0, 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?
Association
an example of bio-computation
One recasting of biological brain function in these computational terms was proposed by in the 1980s as a model for associative memory.
Question: How does the brain associate some memory with a given input?
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.
Trajectories in a schematic
state space
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) Hopfield’s 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).
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’?
body Motor
outputs environment
Learning (cont.)
Learning can take many forms:
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.
Supervised learning Reinforcement learning Association Conditioning
Learning from experience
How do the neural networks form in the brain? Once formed, what determines how the circuit might change?
In 1948, , 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?
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
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.
Today… From biology to information processing
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.
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.
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.
For computer scientists, many natural systems appear to share many attractive properties:
dynamic activity
speed, tolerance, robustness, flexibility, self-driven
In addition, they are very good at some tasks that computers are typically poor at:
motor coordination, interaction with the
recognising patterns, balancing conflicts, sensory-
environment, anticipation, learning… even curiosity, creativity & consciousness.
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.
Just like biological neurons, this artificial neuron neuron will have: • Inputs (like biological dendrites) carry signal to cell body.
• A body (like the soma), sums over inputs to compute output, and
• outputs (like synapses on the axon) transmit the output downstream.
The artificial neuron is a cartoon model that will not have all the biological complexity of real neurons. How powerful is it?
Early history (1943)
McCulloch & Pitts (1943). “A logical calculus of the ideas immanent in nervous activity”, Bulletin of Mathematical Biophysics, 5, 115-137.
In this seminal paper, Culloch and 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.
The McCulloch-Pitts (MP) neuron
• Inputs x are binary: 0,1.
• Each input has an assigned weight w.
• Weighted inputs are summed in the cell body.
• Neuron fires if sum exceeds (or equals) activation threshold . • If the neuron fires, the output =1.
• Otherwise, the output=0.
The “computation” consists of “adders” and a threshold.
x1 * w x2 * 1
xw2 3 * w3
= 1 if 0 if<
Note: an equivalent formalism assigns =0
& instead of threshold introduces an extra bias input, such that
bias * wbias = -
over all i ()
inputs weights
Logic gates with MP neurons For binary logic gates, with only one input, possible outputs are described
by the following truth tables:
Always 0 IDENTITY NOT Always 1
For example:
NOT x = -0.5
Exercise: Findwandforthe3remaininggates.
Logic gates with MP neurons
With two binary inputs, there are 4 possible inputs and 24 = 16 corresponding truth tables (outputs)!
For example, the AND gate implemented in the MP neuron:
= +1.5 IN 1
Excercise:
Find w and for OR & NAND.
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 let’s 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.
Even one neuron can successfully handle simple classification problem.
Classification in Action
A set of patients may have a medical problem. Blood samples are analysed for the quantities of two trace elements.
trace 1 trace 2 problem?
2.4 1.0 yes
9.8 8.3 no
1.2 0.2 yes 0.4 2.1 yes 7.9 8.8 no 6.7 7.2 no
∑xi wi ∑xi wi
∑xi wi ∑xi wi ∑xi wi ∑xi wi
sum output
+ive output = problem
etc. etc. etc. etc.
w1=-1, w2=-1, w3=+10 & bias=+1
With correct weights, this MP neuron consistently classifies patients.
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.
(1962). Principles of Neurodynamics, Spartan,
The perceptron algorithm
Take wj random
START:TakeXεF+ UF-
CHECK: ifxεF+ and Σwjxj>0gotoSTART ifxεF+ and Σwjxj≤0gotoADD
ifxεF- andΣwjxj≤0 gotoSTART
ifxεF- andΣwjxj>0 gotoSUB ADD: wj → wj + xj
goto START SUB: wj→wj- xj
goto START:
The Perceptron Theorem
converge on a set of weights in a finite number of steps if w* exists
Says that the previous algorithm will
Learning Rule:
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:
• Compile a training set of N (say 100) sick and healthy patients.
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.
Repeatedly run through training set until all outputs agree with targets.
wiwi + xi if output too low wiwixi 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.
Related idea • Minimize E = Σi(ti – oi )2
– o is the observed output
Find weights that minimize E Steepest gradient descent
t is the desired output
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 can’t supervised learning be used? Are biological neurons supervised?
A simple example
Let’s try to train a neuron to learn the logical OR operation:
Decision ( le 0 or gt 0)
desired output
wiwi + xi 0
if output low if output high
wwx 1i i i
The power of learning rules
The rule is guaranteed to converge on a set of appropriate weights, if a solution exists. While it might not be the most efficient of algorithms, this proven convergence is crucial.
What can be done to improve the convergence rate? Some common variations on this learning rule:
Adding a learning rate 0
• XεF- →Σwjxj<0
• If there are wj* for which this is true, then the algorithm fnds wj
(possibly diferent ones) which also do the trick
Suppose there are two sets: F+ andF-;F+∩F- empty
The Fall of the Artificial Neuron
• Before long researchers had begun to discover the neuron’s 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.
Few Hours in the Gym per Week
Many Hours in the Gym per Week
Footballers Academics
Successful
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?
Unsuccessful
Marvin Minsky & (1969). Perceptrons, MIT Press, Cambridge.
Connectionism Reborn
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 train
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