Background Stochastic Cellular Automaton (CA) Mean Field (MFA) Model Pair Approximation (OPA) Model Numerical Results Conclusion Glial Cell Defense Mechanisms in Response to Ischemic Hypoxia in the Brain Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani Graduate Advisors: Kamal Barley & Adrian Smith Faculty Advisors: Benjamin Morin & Anuj Mubayi Simon A. Levin Mathematical, Computational and Modeling Sciences Center Arizona State University July 24, 2014 Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 1 / 35
1. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Glial Cell Defense Mechanisms in Response to Ischemic
Hypoxia in the Brain Matthew Buhr, Oscar Garcia, Tiffany Reyes,
Hasan Sumdani Graduate Advisors: Kamal Barley & Adrian Smith
Faculty Advisors: Benjamin Morin & Anuj Mubayi Simon A. Levin
Mathematical, Computational and Modeling Sciences Center Arizona
State University July 24, 2014 Glial Cell Defense Mechanisms
Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 1 /
35
2. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion 1 Background Ischemic Stroke and Research Goal 2
Stochastic Cellular Automaton (CA) CA Assumptions CA Stochastic
Realizations 3 Mean Field (MFA) Model MFA System of ODEs MFA
Stability Analysis 4 Pair Approximation (OPA) Model Development of
the Pair Approximation ODE System 5 Numerical Results Qualitative
Results Parameter Sweeps 6 Conclusion Discussion Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 2 / 35
3. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Background and Research Goal: Ischemic Stroke Glial Cell
Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan
Sumdani MCMSC 3 / 35
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4. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Background and Research Goal: Ischemic Stroke Glial Cell
Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan
Sumdani MCMSC 4 / 35 (Courtesy of medmovie.com)
5. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Denition of State Variables and Parameters State Meaning
H Healthy cells D Damaged cells S Scar cells G Glial cells
Parameter Denition The rate of scar cell formation The rate that
damaged cells are repaired The rate of damage to healthy cells
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 5 / 35
6. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Assumptions MFA OPA CA Spatial Complexity 100 100
lattice of cells is created Glial Cell Defense Mechanisms Matthew
Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 6 / 35
7. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Assumptions MFA OPA CA Spatial Complexity von Neumann
neighborhood Glial Cell Defense Mechanisms Matthew Buhr, Oscar
Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 7 / 35
8. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Assumptions MFA OPA CA Spatial Complexity Poisson
process Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia,
Tiffany Reyes, Hasan Sumdani MCMSC 8 / 35
9. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Assumptions MFA OPA CA Spatial Complexity Periodic
boundary conditions Glial Cell Defense Mechanisms Matthew Buhr,
Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 9 / 35
10. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Assumptions MFA OPA CA Spatial Complexity Damage must
be small enough to avoid self-intersection through the boundary
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 10 / 35
11. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Stochastic Realizations CA realizations at initial
state (top) and steady state (bottom) with clustered (left),
semi-clustered (middle), and random (right) initial damage Glial
Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes,
Hasan Sumdani MCMSC 11 / 35
12. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Realization: Random Damage Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 12 / 35
13. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Two Systems of ODE Approximations of Stochastic
Simulation Mean Field (MFA) model Pair Approximation (OPA) model
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 13 / 35
14. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Mean Field Model to Approximate Stochastic Simulation
MFA OPA CA Spatial Complexity d dt P[H] = P[G]P[D] P[H]P[D] d dt
P[D] = P[G]P[D] P[G]P[D] + P[H]P[D] d dt P[G] = P[G]P[D] d dt P[S]
= 2P[G]P[D] Glial Cell Defense Mechanisms Matthew Buhr, Oscar
Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 14 / 35
15. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Reduction of MFA MFA OPA CA Spatial Complexity
Invariants of motion: P[H] + P[D] + P[G] + P[S] = 1 2P[G] + P[S] =
k | k R 2-Dimensional system: d dt P[H] =P[D]((P[H] + P[D] + k 1)
P[H]) d dt P[D] =P[D](P[H] ( + )(P[H] + P[D] + k 1)) Equilibria:
(0, 1 k), (H, 0) Glial Cell Defense Mechanisms Matthew Buhr, Oscar
Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 15 / 35
16. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Local Stability of Healthy Cell Free Equilibrium MFA OPA
CA Spatial Complexity J(0, 1 k) = ( )(1 k) (1 k) ( )(1 k) ( )(1 k)
. Eigenvalues: 1 = (1 k), 2 = (1 k) Glial Cell Defense Mechanisms
Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 16 /
35
17. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Local Stability of Healthy Cell Free Equilibrium MFA OPA
CA Spatial Complexity J(0, 1 k) = ( )(1 k) (1 k) ( )(1 k) ( )(1 k)
. Eigenvalues: 1 = (1 k), 2 = (1 k) (H, D) = (0, 1 k) is
asymptotically stable when (1 k) > 0 Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 16 / 35
18. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Local Stability of Healthy Cell Free Equilibrium MFA OPA
CA Spatial Complexity J(0, 1 k) = ( )(1 k) (1 k) ( )(1 k) ( )(1 k)
. Eigenvalues: 1 = (1 k), 2 = (1 k) (H, D) = (0, 1 k) is
asymptotically stable when (1 k) > 0 This implies: 2P[G] + P[S]
< 1 P[G]0 < 1 2 Glial Cell Defense Mechanisms Matthew Buhr,
Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 16 / 35
19. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Global Stability of Healthy Cell Free Equilibrium MFA
OPA CA Spatial Complexity Phase Plane, Case 1: 1 k > 0 and >
+ Positive invariance Glial Cell Defense Mechanisms Matthew Buhr,
Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 17 / 35
20. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Global Stability of Healthy Cell Free Equilibrium MFA
OPA CA Spatial Complexity Phase Plane, Case 1: 1 k > 0 and >
+ Dulac criterion Glial Cell Defense Mechanisms Matthew Buhr, Oscar
Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 17 / 35
21. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Global Stability of Healthy Cell Free Equilibrium MFA
OPA CA Spatial Complexity Phase Plane, Case 1: 1 k > 0 and >
+ Global stability when > + Glial Cell Defense Mechanisms
Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 17 /
35
22. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Parameter Space MFA OPA CA Spatial Complexity + 1/2
P[G0] Healthy Cell-Free Equilibrium Global Stability Unstable
Unstable 0 Local Stability Glial Cell Defense Mechanisms Matthew
Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 18 / 35
23. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Two Systems of ODE Approximations of Stochastic
Simulation Mean Field (MFA) model Pair Approximation (OPA) model
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 19 / 35
24. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Development of the Pair Approximation ODE System MFA OPA
CA Spatial Complexity Assumptions Development dP[HH] dt = Inow
Outow Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia,
Tiffany Reyes, Hasan Sumdani MCMSC 20 / 35
25. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Development of the Pair Approximation ODE System MFA OPA
CA Spatial Complexity DH ? G ? dP[HH] dt = P[HD] 3P[DG] 4P[D] + ...
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 21 / 35
26. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Development of the Pair Approximation ODE System MFA OPA
CA Spatial Complexity ? ? G D H dP[HH] dt = 2P[HD] 3P[DG] 4P[D] +
... Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia,
Tiffany Reyes, Hasan Sumdani MCMSC 22 / 35
27. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Development of the Pair Approximation ODE System MFA OPA
CA Spatial Complexity D H? ? H dP[HH] dt = ... P[HH] 3P[HD] 4P[H]
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 23 / 35
28. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Development of the Pair Approximation ODE System MFA OPA
CA Spatial Complexity D H ? ? H dP[HH] dt = ... 2P[HH] 3P[HD] 4P[H]
Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany
Reyes, Hasan Sumdani MCMSC 24 / 35
29. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Development of the Pair Approximation ODE System MFA OPA
CA Spatial Complexity dP[HH] dt = 2P[HD] 3P[DG] 4P[D] 2P[HH] 3P[HD]
4P[H] Glial Cell Defense Mechanisms Matthew Buhr, Oscar Garcia,
Tiffany Reyes, Hasan Sumdani MCMSC 25 / 35
31. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Qualitative Results > + Glial Cell Defense Mechanisms
Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 27 /
35
32. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Qualitative Results > + Glial Cell Defense Mechanisms
Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 28 /
35
33. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Parameter Sweep: Clustered Damage Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 29 / 35
34. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Parameter Sweep: Semi-Clustered Damage Glial Cell
Defense Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan
Sumdani MCMSC 30 / 35
35. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Parameter Sweep: Random Damage Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 31 / 35
36. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion CA Initial Conditions Steady State Min/Max in Varying
and Clustered Semi-Clustered Random H 42.4 43.7% 38.8 40.4% 0.0
26.0% D 11.0 13.4% 11.7 17.2% 9.0 50.0% G 42.0 44.0% 44.0 41.5%
14.0 44.0% S 0.0 2.8% 0.0 9.0% 0.0 61.0% Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 32 / 35
37. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Steady States vs. Initial Glial Cells Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 33 / 35
38. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Steady States vs. Initial Glial Cells Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 34 / 35
39. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Discussion If P[G]0 < 1 2 and > + , then P[H] = 0.
The spatial stochastic model predicts that P[G0] < 1 2 for
non-zero P[H]. Increased values of result in decreased amounts scar
tissue. Variance of steady states increases with randomness of
initial damage. Damage is contained with less scarring when initial
damage is more clustered. Glial Cell Defense Mechanisms Matthew
Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani MCMSC 35 / 35
40. Background Stochastic Cellular Automaton (CA) Mean Field
(MFA) Model Pair Approximation (OPA) Model Numerical Results
Conclusion Acknowledgments We would like to thank Dr. Carlos
Castillo-Chavez, Executive Director of the Mathematical and
Theoretical Biology Institute (MTBI), for giving us this
opportunity to participate in this research program. We would also
like to thank Co-Executive Summer Directors Dr. Omayra Ortega and
Dr. Baojun Song for their efforts in planning and executing the day
to day activities of MTBI. This research was conducted in MTBI at
the Simon A. Levin Mathematical, Computational and Modeling
Sciences Center (SAL MCMSC) at Arizona State University (ASU). This
project has been partially supported by grants from the National
Science Foundation (DMS-1263374 and DUE-1101782), the National
Security Agency (H98230-14-1-0157), the Ofce of the President of
ASU, and the Ofce of the Provost of ASU. Glial Cell Defense
Mechanisms Matthew Buhr, Oscar Garcia, Tiffany Reyes, Hasan Sumdani
MCMSC 36 / 35
