1. Chapter 14 Spatial Threshold Systems
including Self-Organized Criticality

2. Cellular DEVS Models of Spatial Threshold Systems
DEVS abstractions of continuous systems with threshold behavior allow simulations that would otherwise be too complex/time consuming to be feasible
An interesting class of such systems are the Self Organized Criticality (SOC) natural systems which depend on threshold properties of continuous systems
These properties allow using DEVS to model them with appropriate abstractions
We break space into cells and model these with DEVS models; the cells are placed into 1,2,3 or higher dimensional grids and coupled with neighborhoods form a cellular space.

3. SOC Earthquake expressed as a One Dimensional Cellular Space
Due to compression on tectonic plates, each cell accumulates energy at a
constant rate until a critical threshold is exceeded. At this point a fraction of its accumulated energy is equally distributed among its neighbors. This reduces the time to achieve their critical levels. If the additional energy causes a cell to exceed its threshold, then it will release energy to its neighbors. The size of an avalanche is the number of such propagated releases that occur until all cells are below the threshold. Releases occur with ta = 0, so the size of an avalanche is the length of a transitory sequences of external transitions.
Power law distribution of
avalanches (~1/x)
Per Bak, “How Nature Works”
H. Jensen, “Self Organizing Complexity”

4. SOC Earthquake expressed as a One Dimensional Cellular Space
earthquakeCell inherits from oneDimCell. To insert digraph models as cells they must satisfy the cell interface
Neighborhood coupling
i-2
i-1 i
i+1
public class earthquakeCellSpace extends oneDimCellSpace{
earthquakeCell.numCells = numCells;
for (int i = 0;i<numCells;i++)
addCell(i,new earthquakeCell(i,energyScale));
hideAll(); //hides only components so far
doNeighborCoupling(+1,"outEnergy","in");
//nearest neighbor
doNeighborCoupling(-1,"outEnergy","in")
coupleAllTo("out",this,"out");
coupleOneToAll(this,"stop","stop");
coupleAllTo("outPair",this,"outPair");
EarthquakeCellSpace inherits from OneDimCellSpace which provides methods
for easily creating a one dimentional cellular space by adding
addCell(index, oneDimCell)
and coupling oneDImCells:
doNeighborCoupling(neighborIndex,outPort, inPort)
coupleAllTo(sport,devs,dport)
coupleOneToAll(sport,devs,dport)
Random.
earthquakeCellSpace

5. Self Organizing Critical Earthquake Cell
public void deltext(double e,message x){
energy += e*accumulationRate;
//updating energy
Continue(e);
if (somethingOnPort(x,"in")){
energy += getRealValueOnPort(x,"in");
//if you are neighbor get energy
if (energy >=energyThresh)
holdIn("active",0);
else holdIn("active“,
(energyThresh – energy)/accumRate);
//every cell increases at same rate }
else if (somethingOnPort(x,"stop"))
passivate();}
energyThresh
energy
released
Energy e
Updated
total ta
ta = time to
threshold crossing
Release
cell
neigh
bor
public void deltint(){
super.deltint();
energy = 0;//release all energy
holdIn("active",energyThresh
/accumRate);)
//schedule yourself to reach threshold}
public message out(){
message m = super.out();
m.add(makeContent("outEnergy",
new doubleEnt(
alpha*(energy + sigma*accumRate))));
// alpha small implies non conservation
// of accumulated energy
return m;}
Random.
earthquakeCellSpace

6. output= Time advance = s s’ Make a transition
Internal Transition /Output Generation -- a cell reaches its threshold energy and distributes its energy to its neighbors; the cell starts accumulating energy starting from zero
output=
alpha*(energy
+ sigma*accumRate)
Generate output
Time advance =
(energyThresh – energy)/accumRate) s
Make a transition s’

7. input Make a transition Time advance, ta
Response to External Input – a neighbor cell receives energy and schedules itself to reach the threshold earlier than before (or it may exceed threshold and become critical)
input
Make a transition
energy +=e*accumRate;
if (energy >=energyThresh) ta = 0;
else
ta =(energyThresh –energy)/accumRate);
elapsed
time, e
Time advance, ta

8. Sand Pile Avalanche model vs Salt Pile model
Cells communicate to discover the state (color coded) they are in. The 8 cell Mealy neighborhood is needed. Since signals travel at different speeds, classical CA are not (easily) able to do this.
black - falling grain
gray -track of falling grain
pink - cells with open side
blue - cell about to destabilize –
open on at least one side and with weight = threshold (here 4)
red - cell walled in on all sides
Avalanche size –
Number of lateral moves in one event – very close to power law = 53/x
Salt grain model – grains fill up empty slots, come to rest on top of a red (walled in) cell.
Avalanche – sand grains fall down center and pile up to threshold, as determined by weight on bottom cell. This destabilized cell moves and destabilizes those above, which fall with equal probability to the left or right.
This process continues with the same rules applying to distributed grains.
Large avalanches occur when a number of adjacent columns have cells near the threshold.
Sand and Salt Piles show distinct avalanche size distributions

9. Sand and Salt Piles show distinct avalanche distributions
Avalanche size –
Number of lateral moves in one event – very close to power law = 53/x
Sand pile

10. Threshold-based watershed
rain
Discrete flow is injected at the top, middle cell. It randomly diffuses down and to the sides under threshold constraint. That is, it cannot transmit the flow until its accumulated water exceeds a threshold. When it overflows it acts like a burst dam. The distinct increasing-with-depth lateral extents may be related to the threshold value.
Avalanches are the propagated series of dam bursts – difficult to measure.
depth

11. Chapter 15 Partial Differential Equation Models
Method of Lines
One Dimensional Diffusion
Example: Triangular Initial Data Distribution
Number of Transitions = Activity

12. Example: One Dimensional Diffusion (PDE)
cell tries to move
toward average of
its neighbors
lower
than
neighbors in
between
neighbors
higher
than
neighbors

13. One Dimensional Diffusion (Con’d)
higher
than
neighbors
lower
than
neighbors
tend to flatten out u in
between
neighbors x x
over time x x

14. DEVS Implementation of 1D Diffusion
oneDimCellSpace
cell 1D
DEVS
Integrator
left
right
time derivative
= k*spatial
derivative
cell 1D
DEVS
Integrator
left
right
time derivative
= k*spatial
derivative
cell 1D
DEVS
Integrator
left
right
time derivative
= k*spatial
derivative

15. DEVS Implementation of 1D Diffusion : sloping initial state
oneDimCellSpace
length and height = 11 and speed = 10, 11 cells

16. Definition and Measurement of Activity
Continuous segment
Piecewise Continuous Segment t1 tn ti m1 mi mn T
Activity = sum of ranges.
Avg Activity ~ A*f
for wave with amplitude A
and frequency f
computational efficiency occurs when number of transitions reflects number of threshold crossings
activity in monotonic region reflects avg rate of change

17. activity distribution11
5.5
activity distribution
activity = 35.5
=(area of space with arrows)
length and height = 11 and speed = 10, 11 cells
Since there is some oscillartion arround 5.5, I used the first time to hit 5.5 in the table.The oscillation is about 3 times the quantum.
quantum
activity/quantum for cell 0
Actual number
of cell 0 transitions to
reach 5.5
.05
110
122
.01
550
561
.005
1100
1101
.001
5500
5501
DEVS transitions
= Activity/quantum

18. q = .01
q = .001
q = .0001
As quantum gets smaller, the #DEVS transitions gets closer to the predicted activity. The reason can be found in the distribution over cells which moves from uniform to the sharp curve predicted. The uniform distribution, at lower quanta is due to increased oscillation as cells adjust to their neighbors in big steps.