Shape Formation Through Cell Growth and Gradient Exudation Growing arbitrary shapes with more fun than you can shake a caml at! UVa October 4 th, 2002 Christopher Frost Massachusetts Institute of Technology / University of Virginia

Overview ● Amorphous computing ● Goals ● Description of environment ● Creating shapes without direction ● Bootstrapping, determining the ‘goodness’ of cells ● Shapes to descriptions ● Future work

Amorphous Computing The defining characteristics of an amorphous system are: ● Large numbers of identically programmed elements ● Limited computing power of individual elements ● Limited communication radius ● No a priori or global knowledge of the system ● No positional information

Amorphous Computing: Why Spatial Organization? ● This is a general example of how to organize an amorphous system ● Spatial differentiation is a clean way of showing differentiation of function ● Prevalence in biology

Goals ● Develop a method of constructing arbitrary shapes using only cells which have no concept of direction and can: – Replicate – Commit suicide – Exude and react to gradients ● Create circles from circles creating and using ‘reference points’ ● Work in 2D

Environment Description ● Cell Primitives – Replication/Having children ● Defines one’s ‘family’ – Committing suicide ● Important because of positioning later on.. – Exuding and reacting to gradients ● Control amount of material exuded, determining how far a gradient will travel – Not as realistic as we’re shooting for, but easier for now ● When a cells reads a gradient it knows: corresponding reference id, the strength of the gradient here, and the exuder's phi value Child Area

Environment Description, cont ● Cells are immobile ● Cells can’t overlap ● No concept of orientation of a cell or a global notion of direction ● Gradients spread instantly ● Gradients allow measurement of exact distance

Environment Simulator ● Was: thread for every cell – Now: execs cell steps in random order (random subset of cells in each cycle) ● So methods can't depend on cells receiving gradients instantly – Later: reintroduce threads to exec fixed number of, but randomly chosen, cells at a time ● Cells sleep until a change in the environment ● Big speedup, most cells not executing most of the time ● Gradients can pass through ‘voids’, but we only use gradients that pass through cell areas

How Do We Create ‘Shapes’ Without The Notion of Direction? ● We can create local coordinate systems in circles composed of cells ● Creating five reference points allows you to triangulate the closest cell within a circle to some position – Including locations to grow new circles – Five? Indeed, we will use all five in locating the first three reference points of additional circles

Without Direction, How Do We Get The First Five Points? ● First cell becomes the center ● Second reference point is an arbitrary point on the circumference ● Third reference point: cell a certain distance from second reference point ● Fourth and fifth points: triangulate using three existing points and be at least some minimum distance from each other ● Can now locate any point within the circle and triangulate points in the creation of new circles

General Location ‘Goodness’ Algorithm ● Phi is the measure of how good a cell’s location is compared to the location we would like A is the set of gradient values a particular cell reads and B is the set of gradient values one is trying to find: ● Scale independent

Example of Defining a Reference Point Ref 0 Ref 3, defined above Ref 2 Ref 1 Ref 4 Create a reference point with the id 3 whose gradient travels the diameter of the circle uses reference points 4, 1, 0, and 2, the ratios..., and triangulation to position itself

Getting Cells to Figure Out Who Has the Best Phi Value: A Fierce Competition ● Two methods explored: ‘cell hopping’ and ‘listen and exude’ ● Cell hopping – A cell asks its family for their phis, tells family the best phi value heard – If you ask your family and you are the best, send the gradient and terminate – Cell familiage often has separations, so this method can get stuck – Many gradients sent

Getting Cells to Figure Out Who Has the Best Phi Value, cont ● Listen and exude – Every cell listens to reference points it can hear – Finds phi value for every gradient it could create using these hearable reference points – If this value is within a certain range, the cell tries to become the gradient emitter ● If if no one else is exuding or this gradient is not being exuded with a better phi than ours: – this cell becomes the winner, exudes ● Stop exuding if: – another, better phied, gradient is heard – we fall out of the ‘phi range’

Qualities of the Listen and Exude Method ● If a gradient disappears, someone else will pick it up – Currently the first two points of the first circle are not recreateable, but once the first circle is created this can be changed ● If a reference point moves, reference points which depend on this reference point can move

Shapes to Descriptions ● Describe an arbitrary shape by packing it with overlapping circles ● Three-phase compilation 1. Finding an efficient circle covering (initial network) 2. Constructing a tightly-linked network of circles by introducing intermediate circles 3. Specifying position information in terms of distance ratios

Future Work ● The near future – Finish implementing multiple circle creation (Implement ideas for cell death and growth restart) – Read Catie’s program’s datastructure ● Ideas to look into – Introducing more perturbations and testing robustness – Possibly optimizing when should cells die – Dynamic gradient dependent determination / robust shape recreation