Cellular Automaton, Genetic Algorithms, and Neural Networks

Transcription

1 Cellular Automaton, Genetic Algorithms, and Neural Networks Catherine Beauchemin, Department of Physics, University of Alberta January 30, 2004

2 Overview Cellular Automaton What is a cellular automaton? A simple cellular automaton: Wolfram s 1-D cellular automaton A more complex cellular automaton: a viral infection model MASyV: a cellular automaton visualization tool Genetic Algorithms What are genetic algorithms? A simple genetic algorithm Other applications for genetic algorithms Artificial Neural Networks What are artificial neural networks? Applications for artificial neural networks

3 What is a cellular automaton? Cellular automata (CA) were originally introduced by John von Neumann and Stanislas Ulam under the name of cellular spaces as possible idealization of biological systems. A typical CA consists of a regular lattice. The lattice contains a discrete variable or cell at each site, that can assume m possible discrete values. The state of a CA is completely specified by the values of the variables at each site. The CA evolves in discrete time steps, with the value of a variable at a given site being affected by the values of variables at sites in its neighborhood on the previous time step. At a new time step, the variables at all sites are updated simultaneously, based on the values of the variables and their defined neighborhood at the preceding time step, and according to a definite set of local rules.

4 A simple CA: Wolfram s 1-D CA Wolfram concentrated most of his work on 1-D CA with only two possible state at each site: dead or alive (0 or 1). The neighborhood was composed of the cell itself plus its two immediate neighbours. Rules can be expressed as 8 digit binary numbers in the following manner: The above rule is Rule 90. Recall that a binary number can be expressed in base 10 in the following manner: = = 90 Of the 256 possible rules, only 32 rules are considered legal. biological origins of the CA impose certain restrictions The The rule must be symetrical (110 and 011 must yield the same thing) Only life can generate life (000 must yield 0)

5 Evolution of various rules in time When initially seeded with only one non-zero seed, the behaviour of the 32 legal rules can be divided into 4 types (disapear with time, merely copies the non-zero cell forever, yields a completely uniform type of row or pairs of rows, develop nontrivial patterns). Each pattern below is the result of rules 128 ( ), 36 ( ), 250 ( ), and 90 ( ) respectively.

6 A more complex CA: a viral infection model A model I developed to explore whether spatial inhomogeneities such as localized populations of dead cells might adversely affect the spread of infection, similar to the manner in which a counter-fire can stop a forest fire from spreading The model considers 2 species: epithelial cells, which are the target of the viral infection, and immune cells which fight the infection The CA is run on a two-dimensional square lattice where each site represents one epithelial cell Immune cells are mobile, moving from one lattice site to another, and their population size is not constant: the CA lattice is therefore like a tissue of immobile cells which is patrolled by the mobile immune cells The CA is updated synchronously and the boundary conditions for both the epithelial and immune cells are toroidal The virus particles are not explicitly considered, rather the infection is modelled as spreading directly from one epithelial cell to another

7 The CA model: the epithelial cells An epithelial cell can be in any of five states: healthy, infected, expressing, infectious, or dead. Transitions between epithelial cell states occur as follows: Epithelial cells of all states become dead when they are older than CELL LIFESPAN. A healthy epithelial cell becomes infected with probability INFECT RATE/(8 nearest neighbours) for each infectious nearest neighbour. An infected cell becomes expressing, i.e. begins expressing the viral peptide, after having been infected for EXPRESS DELAY. An expressing cell becomes infectious after having been infected for INFECT DELAY > EXPRESS DELAY. Infected, expressing, and infectious cells become dead after having been infected for INFECT LIFESPAN.

8 The CA model: the epithelial cells Continued... Expressing and infectious cells become dead when recognized by an immune cell. See below for the meaning of recognition. A dead cell is revived at a rate DIVISION TIME 1 # healthy/# dead. When revived, a dead cell becomes a healthy cell or an infected cell with probability INFECT RATE/(8 nearest neighbours) for each infectious nearest neighbour. A simulation is initialized with each epithelial cell being assigned a random age between 0 and CELL LIFESPAN inclusively. All epithelial cells start in the healthy state with the exception of a fraction IN- FECT INIT of the total number of epithelial cells which, chosen at random, are set to the infected state.

9 The CA model: the immune cells An immune cell can be in any of two states: virgin or mature. A virgin cell is an immune cell that has no specificity. A mature cell is an immune cell that has either already encoutered an infected cell or has been recruited by another mature immune cell. Immune cells move randomly on the CA lattice at a speed of one lattice site per time step. An immune cell is removed if it is older than IMM LIFESPAN. An encounter between an immune cell and an expressing or infectious epithelial cell requires the immune cell to be in the same site as the expressing or infectious cell. A virgin immune cell becomes a mature immune cell if the lattice site it is occupying is in the expressing or infectious states.

10 The CA model: the immune cells Continued... A mature immune cell occupying an expressing or infectious lattice site recognizes the epithelial cell and causes it to become dead. Each recognition event causes RECRUITMENT mature immune cells to be added at random sites on the CA lattice after a delay of RE- CRUIT DELAY. Virgin immune cells are added at random lattice sites as needed to maintain a minimum density of BASE IMM CELL virgin immune cells. A simulation is initialzed with a density of BASE IMM CELL virgin immune cells at random locations on the CA lattice, each with a random age.

11 Viral infection model: quantitative results % total cells time (h)

12 MASyV: a visualization tool MASyV stands for Multi-Agent System Visualization. It is a powerful general purpose user interface accompanied by a message passing library. It enables one to write cellular automaton simulations programs and visualize them with very little additional time and effort. The simulation (client) and the user interface (server) interact via UNIX domain sockets using the message passing library. Simulation runs can be saved as movies through the use of transcode. MASyV requires: GTK+ 2.0 or higher an OpenGL compatible library as well as GtkGLExt, an OpenGL widget for GTK+. MASyV comes with a few client modules which stand as good examples of how to write a client module and how to use the message passing library. MASyV is distributed under the GNU General Public License and is hosted on SourceForge (

13 Visualizing a few simulations The simple Wolfram 1-D CA A simple model of gas diffusion The viral infection CA model Ants moving around depositing pheromone trails

14 Applications of CA When to use a CA? ODEs are better when simple enough to be analytically solved such that the parameters space can be explored theoretically. With a CA, adding complexity such as delays, nonlinearities, and stage-classes does not introduce greater difficulty in solving it as is the case with ODE but the parameter space has to be explored by brute force. A CA offers a natural description of the physical system. There is good correspondance between the CA s parameters and measurable quantities. Why would a Medical Physics student want to use a CA? To model the dynamics of a particular disease where spatial inhomogeneities are important (e.g. Cancer) Where non-linearities and age classes would make a differential equation model too messy

15 What is a genetic algorithm? Genetic algorithms (GA) were invented by John Holland in the 1960s in order to: formally study the phenomenon of adaptation as it occurs in nature develop ways in which the mechanisms of natural adaptation might be imported into computer systems. Now, GAs have evoluved into a class of algorithms typically used to solve optimization problems. Biological terms are used in GA research in the spirit of analogy with the real biological system. A population consists of chromosomes which are strings of genes each of which can take on different values called alleles. Each gene is located at a particular locus (position or site) on the chromosome and the particular sequence of genes of a chromosome is called a genotype. Chromosome from a population are selected for the fitness of their genotype to a particular problem and the most fit pair of individuals get to reproduce. Each pair of individuals will produce a pair of offspring and the process can involve crossovers or mutations.

16 A simple genetic algorithm 1. Start with a randomly generated population of n l-bit chromosomes. 2. Calculate the fitness of each chromosome in the population. 3. Repeat the following until n chromosomes have been created: (a) Select a pair of parent chromosome from the population (with replacement) with the probability of selection being an increasing function of fitness. (b) With a probability p C, cross over the pair at a randomly chosen locus. If no crossover occurs, the offspring will be exact copies of the parents. (c) Mutate the 2 offsprings at each locus with probability p M and place the resulting offspring in the new population 4. If n is odd, one new population member is discarded at random. 5. Replace the current population with the new population. 6. Go back to step 2. Each iteration of this process is called a generation and the entire set of generations is called a run.

17 A simple GA example Let us consider a population of n = 4 chromosomes made up of l = 8 genes, each of which have 2 alleles: 0 and 1. With a goal to evolve the chromosome , let us define the fitness function as being the number of 1s in the chromosome. Chromosome label Chromosome string Fitness A B C D A common selection method is fitness-proportionate selection where the probability that a chromosome will reproduce is equal to its fitness divided by the average fitness of the population. Let s say the pair (B,D) and (B,C) were selected. Say parents B and D cross over after the 1st bit to yield offsprings E= and F= and parents B and C do not.

18 A simple GA example (continued) Next mutation is applied, E= is mutated at the 6th locus to yield E = , B= is mutated at the 1st locus to yield B = , and F= and C= are not mutated. The new generation is Chromosome label Chromosome string Fitness E F C B Although the best string was lost (B= ), the average fitness rose from 12/4 to 14/4. Repeating the procedure will eventually result in a string with all 1s.

19 Common GA terminology (Hamming) Distance The distance between two chromosomes is defined as the number of loci at which the chromosome s genotypes differ. When the chromosomes are composed of bits, this distance is refered to as the Hamming distance. E.g. Chromosomes and have a Hamming distance of 2. Fitness Landscape Originally defined by the biologist Sewell Wright (1931) in the context of population genetics, a fitness landscape is a representation of the space of all possible genotypes along with their fitness. In other word, a fitness landscape for l-gene chromosomes would be an l +1-dimensional plot in which each genotype is a point in l dimensions and its fitness is plotted along the (l + 1)st axis. The GA provides the way in which to patrol this fitness landscape to find local minima.

20 Applications of GAs When to use a GA? When you want to take advantage of evolutionary rules to evolve a solution to an optimization problem When you want to model the process of evolution of a system Preferably when your problem can be expressed in terms of a character string (although aparently, even this problem can be overcome) Why would a Medical Physics student want to use a GA? To find a sequence of amino acid that would fold to a desired threedimensional protein structure To model the phenomenon of escape mutants with a given pathogen in order to determine the mutation parameters p C and p M specific to that pathogen

21 What are artificial neural networks? Artificial neural networks (ANNs) are inspired by the way biological nervous systems, such as the brain, process information. Like the brain an ANN is composed of a large number of highly interconnected processing elements (neurones) working in unison to solve specific problems. ANNs like people, learn by example. There are two types of training used in ANNs, with different types of networks using different types of training. These are supervised and unsupervised training, of which supervised is the most common. An ANN is configured for a specific application, such as pattern recognition or data classification, through a learning process. Just like biological systems, learning in ANNs involves adjustments to the synaptic connections that exist between the neurones.

22 Real neurons VS artificial neurons Dendrites Soma Axon Synaptic Terminal x 1 x 2 W 1 summation function u(w 1x 1, w 2 x 2,..., w nx n) transfer function f(u) W 2 u f x n Wn

23 A simple ANN example: the OR gate Let s say we want our ANN to learn to emulate an OR gate: x 1 x 2 d (desired output) Let s pick our summation and transfer (or activation) functions to be u = f(u) = 2 w i x i, i=1 { 0 u < θ 1 u θ respectively. Let us pick a correction function that adds (or subtracts) 0.1 to w i when f(u) is smaller (or greater) than d, whenever x i is non-zero.

24 Linear separation in single node networks Since the critical condition for classification occurs when the result of the summation equals the threshold of the transfer function, let us put the summation equal to the threshold, this means 2 w i=1 ix i = θ. In the 2-D case we are considering w 1 x 1 + w 2 x 2 = θ so that ( ) ( ) w1 θ x 2 = x 1 +. w 2 w 2 Yielding x 2 θ w 2 (0,1) (1,1) (0,0) (1,0) θ w 1 x 1

25 More complex separations For more complex separations, one needs a more complex ANN Y 1 hidden layer L1 L2 L1 L2 L3 L4 X1 X2 L4 L3 Y 2 hidden layers P1 P2 P1 P2 P3 P3 X1 X2

26 Applications of ANNs When to use ANNs? When you know (or strongly suspect) that there is a relationship between the proposed known inputs and unknown outputs. This relationship may be noisy but it must exist. When you don t know the exact nature of the relationship between inputs and outputs. Otherwise, you should model it directly. When you don t care how it works and you just want it to work. Why would a Medical Physics student want to use an ANN? Data mining, i.e. classifying large databases of protein sequences based on various criterion Recognition of patterns in X-Rays or MRI scans to enhance key information from apparently fuzzy images or to formulate diagnosis Recognition of complex combinations of changes in health indices (e.g. heart rate, levels of various substances in the blood) could allow one to predict the onset of a particular medical condition

27 Where to find information and links: On cellular automaton Cellular automaton and statistical mechanics, online at cbeau/docs/statmech.pdf MASyV on Sourceforge and movie files from ma immune On genetic algorithms M. Mitchell. An introduction to genetic algorithms, The MIT Press, Cambridge, Massachusetts, A good introduction to immunology for physicists A.S. Perelson, G. Weisbuch. Immunology for physicists, Reviews of Modern Physics, 69(4): , 1997.