1. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1 Pattern Formation Goal: See how globally ordered spatial structures can arise from local interactions Outline: chemical reaction systems: stirred and unstirred reactors diffusion equation: motivation and simple cases reaction diffusion systems: 1D activator/inhibitor system, Turing instability 2D cellular automata for modeling development of animal coat patterns L-Systems based on grammars

2. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 2 Chemical Reactors Setup 1: well stirred reactor N different chemicals (reactants) with time-varying concentrations c i (t), i=1,..N vector notation: c(t) changes in concentrations due to reactions modeled by system of DEs: chemical reactor, well stirred

3. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3 Example System: three chemicals A, B, C; concentrations n A, n B, n C Reaction: A+B ↔ C assume rate of “→”reaction proportional to probability of encounter of an A and B molecule: k 1 n A n B assume rate of “←“ reaction is: k 2 n C A A B B C C a) b)

4. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4 Setup 2: unstirred reactor N different chemicals (reactants) with time-varying and space-varying concentrations c i (x,t), i=1,..N vector notation: c(x,t) changes in concentrations of one chemical due to reactions depends on concentrations of other chemicals at the same location changes in concentration due to diffusion depend on concentration at neighboring sites chemical reactor, unstirred

5. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 5 Derivation, part 1: random walk time between collisions: T characteristic velocity v distance traveled l=T v current J : particles/surface area/time Diffusion Equation (1D)

6. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6 Derivation, part 2: conservation of particles, consider small volume Adx: From above: Together: “diffusion equation” “diffusion constant”

7. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7 Simple Analysis of Diffusion Equation Consider point source of strength S at x 0, decay with rate μ: decay point sourcediffusion

8. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 8 Case 1: source inactive (S=0), look for homogenous solution In this case no diffusion: with the solution: The homogeneous solution exponentially decays to zero.

9. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9 Case 2: source active (S>0), look for stationary solution In this case: bi-exponential Ansatz: leads to solution: In particular: shape gets wider if μ smaller (slower decay) and/or D gets bigger (faster diffusion)

10. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 10

11. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 11 Reaction Diffusion System Consider 1D system with just two substances: Gierer&Meinhardt (1972), Haken (1990) inhibitor grows due to activator spontaneous creation diffusion “activator” “inhibitor” positive feedback limited by inhibitor spontaneous decay spontaneous decay

12. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 12 1) Simplify System: new set of variables to remove some constants 2) Find homogenous stationary solution 3) Linearize the system: Consider small deviations around homogeneous stationary solution in the linearized system 4) Check under what conditions instability occurs 5) Translate results from simplified system back into original system 6) Main Finding: inhibitor must diffuse faster than activator Analysis of System

13. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 13 H. Meinhardt (1998), The algorithmic beauty of sea shells. Simulation of (similar) system: Lyria planicostata taiwanica

14. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14 More Sea Shell Patterns H. Meinhardt (1998), The algorithmic beauty of sea shells. Sea shells have 1D growing edge, pigmentation of shell can be seen as historical record of 1D reaction diffusion system. Amoria ellioti

15. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15 H. Meinhardt (1998), The algorithmic beauty of sea shells. Oliva porphyria

16. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 16 H. Meinhardt (1998), The algorithmic beauty of sea shells. Conus marmoreus

17. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 17 example of self-organization: positive and negative feedback global pattern from local interactions spontaneous symmetry breaking basis of many models of biological form sea shells animal coats (cheetah, zebra, giraffe, salamander, tropical fish, …) basis for thinking about some more advanced problems embryogenesis: not just diffusion but active transport, inter and intra cell communication growing of arms, legs, wings different organs, nervous system Discussion of Reaction Diffusion Systems

18. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 18 2D Cellular Automaton Model

19. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 19 Setup: Regular grid of “cells” each in binary state s i in{0,1} asynchronous updating: pick single unit at random, update, pick next, … update rule: Parameters: strength of activation/inactivation: J 1, J 2 range of activation/inactivation: R 1, R 2 “bias”: h

20. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 20 % Parameters length = 200; % size of layer fraction = 0.2; % fraction of units updated each turn j1 = 1.0; % strength of activation j2 = -0.3; % strength of inactivation r1 = 3; % range of activation r2 = 6; % range of inactivation h = 0; % strength of bias % Initializations X = unidrnd(2,length,length)*2-3; f_size = 2*ceil(r2)+1; f = j2*ones(f_size); for(x=1:f_size) for(y=1:f_size) rr = (x-(f_size+1)/2.0)^2+(y-(f_size+1)/2.0)^2; if rr <= r1^2 f(x,y)=j1; elseif rr > r2^2 f(x,y)=0; end figure(1) imagesc(f); % display filter colormap(gray); axis equal; % Main loop while(1) figure(2) imagesc(X); % display state colormap(gray); axis equal; axis off; pause % synchronous update leading to oscillations, so update % only a fraction of the units New = ((h+conv2(X,f,'same'))>0)*2-1; Update = unifrnd(0,1,length,length)<fraction; X(Update)=New(Update); end

21. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 21 j1=1, j2=-0.3, r1=3, r2=6, h=6j1=1, j2=-0.3, r1=3, r2=6, h=-6j1=1, j2=-0.3, r1=3, r2=6, h=0 Simulations Varying bias field h:

22. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 22 j1=1, j2=-0.3, r1=3, r2=18, h=-6 Simulations Varying inactivation range r2: j1=1, j2=-0.3, r1=3, r2=12, h=-6j1=1, j2=-0.3, r1=3, r2=6, h=-6

23. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 23 j1=1, j2=-0.01, r1=3.5, r2=6, h=0 j1=1, j2=-0.1, r1=3.5, r2=6, h=0 Using smaller J2:

24. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 24 nice example of formation of patterns from local interactions fairly abstract compared to reaction diffusion DE same scheme of local activation/global inhibition range of qualitatively different patterns from same dynamics simple models for formation of animal coat patterns role of boundary conditions: periodic, non-square lattice number of extensions Discussion of 2D Cellular Automaton

25. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 25 Idea (Aristid Lindenmayer, 1968): model development of tree-like structures w/ grammar Example Grammar: A→A B→AC C→ [B][B] (brackets mean branching) Example updates: 1.[B] 2.[AC] 3.[A[B][B]] 4.[A[AC][AC]] 5.[A[A[B][B]] [A[B][B]]] L-Systems

26. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 26 Example: (http://shakti.trincoll.edu/~cschneid/lsystems.html) This is a simple two-dimensional L-systems graphical representation of an oppositely branched red alga: Axiom = A A → F[-A][+A]FA F → FF http://spanky.triumf.ca/www/fractint/lsys/tutor.html Parameters: Initial angle = 90 degrees Angle increment = 30 degrees Iteration depth = 8

27. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 27 Other Examples: Developmental sequence of Mycelis muralis. Copyright © 1987 P. Prusinkiewicz and J. Hanan. The Garden of L. Copyright © 1988. P. Prusinkiewicz, F.D. Fracchia, J. Hanan, and D. Fowler

28. Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 28 Modeling Neurons: Ascoli and Krichmar (2000): top: real hippocampal pyramidal neurons, bottom: artificial “L-neurons”