Tools from Computational Topology with applications in the life sciences

Transcription

1 Tools from Computational Topology with applications in the life sciences Sarah Day Department of Mathematics College of William & Mary February 10, 2016

2 Data can be Messy (recurrent dynamics, chaos) Noisy (measurement error, stochastity) High dimensional Sparse

3 (Topology, Wikipedia) Topological Zoo Anatoly Fomenko 1967

4 Euler Characteristic = V E + F Wikipedia Wolfram (g) =2 2g

5 Homology H k (X) = Z k =0, 1 0 otherwise. H k (S n ) = Z k =0,n 0 otherwise. H k (X) = 8 < : or Z k =0, 2 Z 2 k =1 0 otherwise. H k (K) = 8 < : Z k =0, Z Z/2Z k =1, 0 otherwise. H k (X) = Z 2 k =0, 1 0 otherwise.

6 Betti numbers H k (X) = 0 =1 1 =1 Z k =0, 1 0 otherwise. or 0 =1 1 =2 2 =1 H k (X) = 8 < : Z k =0, 2 Z 2 k =1 0 otherwise. 8 H k (K) < = : Z k =0, Z Z/2Z k =1, 0 otherwise. 0 =2 1 =2 H k (X) = Z 2 k =0, 1 0 otherwise.

7 Betti numbers and Euler Characteristic H k (X) = Z k =0, 1 0 otherwise. 0 =1 1 =1 = X ( 1) n k n = X ( 1) n n or 0 =1 1 =2 2 =1 H k (X) = 8 < : Z k =0, 2 Z 2 k =1 0 otherwise. H k (K) = 8 < : Z k =0, Z Z/2Z k =1, 0 otherwise. 0 =2 1 =2 H k (X) = Z 2 k =0, 1 0 otherwise.

8 population density patterns

9 Project I: Coupled-Patch Model (with Ben Holman) Ricker map f(n) =rne N N ij fitness parameter growth phase N ij (t + 1) = f(n ij (t)) dispersal phase N ij (t + 1) = (1 d) N ij (t)+ d 4 X i i 0 + j j 0 =1 N i0 j 0 (t) dispersal parameter

10 N Ricker map orbit diagram T =lnr e 0 r r = 22

11 Example 1: dispersal and smoothing d =0 d =0.3 n = 100,r = 22

12 Example 1: dispersal and smoothing 800 d =0 800 d =

13 Example 2: global extinction event t =0 total abundance: knk 1 = 142,380 # decoupled subsystems: 0 =1 # enclosed extinct regions: 1 =0

14 Example 2: global extinction event t =1 total abundance: knk 1 = 133,186 # decoupled subsystems: 0 = 222 # enclosed extinct regions: 1 = 1,115

15 Example 2: global extinction event t =5 total abundance: knk 1 = 10,339 # decoupled subsystems: 0 = 388 # enclosed extinct regions: 1 =0

16 Example 2: global extinction event t = 10 total abundance: knk 1 = 10 # decoupled subsystems: 0 = 22 # enclosed extinct regions: 1 =0

17 Three scenarios The system decouples into a few small subsystems before rebounding and re-coupling. (r = 22,d=0.15,e o =0.2) The system decouples into persistent subsystems. (r = 22,d=0.15,e o =0.6) The system decouples into subsystems but only as a transient stage prior to extinction. (r = 22,d=0.15,e o =0.77)

18 global extinction local extinction threshold e o topological transition (where 0 and 1 cross) dispersal d

19 What are the effects of threshold choice, noise, and measurement error?

20 From Homology to Persistent Homology X := {x f(x) apple } We will focus on computing the homology of sublevel sets X over a continuous range of thresholds. For each generator (hole) we record its birth threshold b and death threshold d. The importance of a homology generator (topological feature) is correlated to the generator s lifespan ( ). d b 1 =2 1 =2 1 =1

21 Persistent Homology H 1 Persistence Diagram death birth

22 Persistent Homology H 1 Persistence Diagram death birth

23 Persistent Homology H 1 Persistence Diagram death birth

24 Persistent Homology H 1 Persistence Diagram death birth

25 Adding noise H 1 Persistence Diagram death birth

26 Project II: Red Blood Cells and Flickering with Jesse Berwald, Kelly Spendlove, Madalena Costa, Ary Goldberger Red blood cells flicker. As red blood cells age their membranes lose plasticity and this flickering changes (as noted in Costa, et al). Membrane changes have implications for oxygen transport. Blood banks want to know RBC age for this reason. We study the change in membrane structure using persistent homology.

27 Phase Contrast Microscopy of RBCs

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30 infinite generator robust generators noisy generators

31 The largest feature of interest represents a deeper (intensity) depression in young cells.

32 Initial experiments: Topological features change more rapidly for young cells.

33 The number of robust generators varies in a more complex way for young cells.

34 Is there a way to measure dynamics without passing to time series analysis?

35 Computational topology can extract information from data that is Messy (recurrent dynamics, chaos) Noisy (measurement error, stochastity) High dimensional Sparse

36 students currently working on projects in dynamics and computational topology: Martin Salgado-Flores, Liam Bench, Matthew Andriotty Thank you!