Using persistent homology to reveal hidden information in neural data

Transcription

1 Using persistent homology to reveal hidden information in neural data Department of Mathematical Sciences, Norwegian University of Science and Technology ACAT Final Project Meeting, IST Austria July 9, 2015

2 Collaborators and inspiration Benjamin Adric Dunn; Kavli Institute for Systems Neuroscience Magnus Bakke Botnan; NTNU dept. math. sci. Yasser Roudi; Kavli Institute for Systems Neuroscience Nils Baas; NTNU dept. math. sci.

3 Collaborators and inspiration Benjamin Adric Dunn; Kavli Institute for Systems Neuroscience Magnus Bakke Botnan; NTNU dept. math. sci. Yasser Roudi; Kavli Institute for Systems Neuroscience Nils Baas; NTNU dept. math. sci. Builds on ideas by C. Giusti and collaborators Y. Dabaghian and collaborators

4 Place cells Fundamental question: How do mammals navigate space? In 1971, John O Keefe discovered the first component of this positioning system. He found a type of nerve cell in an area of the brain called the hippocampus that was always activated when a rat was at a certain place in a room. Other nerve cells were activated when the rat was at other places. O Keefe concluded that these place cells formed a map of the room. The Nobel Committee, October 2014

5 Place cells idealized 1 Animal enters room (loosely defined environment)

6 Place cells idealized 1 Animal enters room (loosely defined environment) 2 Place cells establish a cover by place fields

7 Place cells idealized 1 Animal enters room (loosely defined environment) 2 Place cells establish a cover by place fields 3 Animal moves around, place cells fire Neuron 1 Neuron 2 Neuron 3 Time

8 Place cells idealized 1 Animal enters room (loosely defined environment) 2 Place cells establish a cover by place fields 3 Animal moves around, place cells fire Indicative of triple intersection Indicative of double intersection

9 Can we see the environment from neuron recordings? Definition Let U = {U u } u be an open cover. Its nerve NU is a simplicial complex with simplices {i 0, i 1,, i u } NU U u 0 U u 1 U u u. Theorem (Nerve theorem) Let X be a paracompact space and U an open cover of X such that every covering set and every non-empty intersection of covering sets is contractible. Then NU X.

10 Can we see the environment from neuron recordings? Definition Let U = {U u } u be an open cover. Its nerve NU is a simplicial complex with simplices {i 0, i 1,, i u } NU U u 0 U u 1 U u u. Theorem (Nerve theorem) Let X be a paracompact space and U an open cover of X such that every covering set and every non-empty intersection of covering sets is contractible. Then NU X. Firing data intersection data homotopy type.

11 Firing data intersection data homotopy type

12 Firing data intersection data homotopy type

13 Firing data intersection data homotopy type

14 Firing data intersection data homotopy type

15 Firing data intersection data homotopy type (Embedding is of course not known)

16 Reminder: Biology is messy Real place fields (and these are nice): Generated using data from Mizuseki, Sirota, Pastalkova, Diba, and Buzsáki: Multiple single unit recordings from different rat hippocampal and entorhinal regions while the animals were performing multiple behavioral tasks (CRCNS.org, 2013).

17 Pretend biology is nice and clean Neurons Obstruction Neuroscientists observe Time Construct from cofiring

18 Pretend biology is nice and clean Neurons Obstruction Neuroscientists observe Time Construct from cofiring Made without spatial data!

19 Pretend biology is nice and clean Neurons Obstruction Neuroscientists observe Time Construct from cofiring But what qualifies as cofiring?

20 Firing data correlation Record firing data ( spike trains ) from N neurons for a time T (T 10 min.) Bin data, keeping fire (+1) or no fire (0 or 1) for each time interval [t, t + Δt) (Δt 1 ms) Data thus consists of S 1,, S u (Z/2Z) u /Δu For each pair (i, j), xcorr u (S u, S u ) = 1 S u 1 S u 1 S u, S u [n] xcorr(s u, S u ) = 1 L max ( u u =0 xcorr u (S u, S u ), u u =0 xcorr u (S u, S u ))

21 Order complex Have complete graph with weights C u,u = xcorr(s u, S u ), so could consider its flag complex.

22 Order complex Have complete graph with weights C u,u = xcorr(s u, S u ), so could consider its flag complex. But neuron firing is a highly non-linear process. We re really observing C u,u = f(real relationship between neuron i and j) for some (highly non-linear) monotone f.

23 Order complex Definition Let G be a complete graph with real edge weights W u,u. Sort the edges W u (0) < W u (1) < < W u (( u 2 ) 1) (breaking ties arbitrarily), and let G have edge weights W u (u ) = k ( u 2 ). The order complex of G is OC(G) = flag( G).

24 Order complex Definition Let G be a complete graph with real edge weights W u,u. Sort the edges W u (0) < W u (1) < < W u (( u 2 ) 1) (breaking ties arbitrarily), and let G have edge weights W u (u ) = k ( u 2 ). The order complex of G is OC(G) = flag( G). As filtered simplicial complexes OC(C) = OC(f(real neuron relations)) = OC(real neuron relations).

25 Toy example

26 Toy example 0.20 u Death Birth

27 Real data? Similar techniques have been applied to real data: Curto and Itskov (2008): Cell groups reveal structure of stimulus space

28 Real data? Similar techniques have been applied to real data: Curto and Itskov (2008): Cell groups reveal structure of stimulus space Curto, Giusti, Itskov and Pastalkova (2015): Clique topology reveals intrinsic geometric structure in neural correlations

29 Place cells care about more than space Neuroscientists suspect (know) that firing is influenced also by Theta phase preference/precession

30 Place cells care about more than space Neuroscientists suspect (know) that firing is influenced also by Theta phase preference/precession Head orientation

31 Place cells care about more than space Neuroscientists suspect (know) that firing is influenced also by Theta phase preference/precession Head orientation Neuron couplings

32 Place cells care about more than space Neuroscientists suspect (know) that firing is influenced also by Theta phase preference/precession Head orientation Neuron couplings Tactile sensory input

33 Place cells care about more than space Neuroscientists suspect (know) that firing is influenced also by Theta phase preference/precession Head orientation Neuron couplings Tactile sensory input Olfactory sensory input

34 Place cells care about more than space Neuroscientists suspect (know) that firing is influenced also by Theta phase preference/precession Head orientation Neuron couplings Tactile sensory input Olfactory sensory input???

35 Uncovering such hidden information Suppose we have a list of candidate stimuli influencing place cell firing, and a recording of the corresponding animal state x(δt), x(2δt),, x(t ) M, as well as spike trains S 1,, S u recorded from N neurons.

36 Uncovering such hidden information Suppose we have a list of candidate stimuli influencing place cell firing, and a recording of the corresponding animal state x(δt), x(2δt),, x(t ) M, as well as spike trains S 1,, S u recorded from N neurons. Two questions: 1 Is our list of candidate stimuli complete? 2 If not, what can we say about the topology of remaining (unknown) stimuli?

37 Kinetic Ising model for neuron activity The Ising model has seen success as a model of neuron firing. Definition G complete graph on vertices 1,, N. Edge weights J u,u. Let E 1,, E u R R. A set of ±1-valued random variables S u (t), with 1 i N and t = kδt are said to obey the kinetic Ising model with couplings J and external fields E 1,, E u if they have the conditional probabilities P (S u (t + Δt) = s S 1 (t) = s 1,, S u (t) = s u ) = exp (s (E u (t) + u J u =1 u,u s u )). 2cosh (E u (t) + u J u =1 u,u s u )

38 Kinetic Ising model for neuron activity The Ising model has seen success as a model of neuron firing. Definition G complete graph on vertices 1,, N. Edge weights J u,u. Let E 1,, E u R R. A set of ±1-valued random variables S u (t), with 1 i N and t = kδt are said to obey the kinetic Ising model with couplings J and external fields E 1,, E u if they have the conditional probabilities P (S u (t + Δt) = s S 1 (t) = s 1,, S u (t) = s u ) = exp (s (E u (t) + u J u =1 u,u s u )). 2cosh (E u (t) + u J u =1 u,u s u ) F u (t) = E u (t) + u u =1 J u,u s u is called the system s Hamiltonian.

39 F u (t) = E u (t) + u u =1 J u,u s u 1.0 Firing probability u u (u ) Internal couplings J u,u capture activiation and inhibition by neighbor neurons. External fields E u capture stimuli.

40 F u (t) = E u (t) + u u =1 J u,u s u 1.0 Firing probability u u (u ) Internal couplings J u,u capture activiation and inhibition by neighbor neurons. External fields E u capture stimuli. Assume E u (t) = E spatial,u (x(t)) + E head,u (x(t)) + E u,u (x(t)) + +??? where each term is a sum of Gaussians on factors of some nice

41 Kinetic Ising model for neuron activity; example Space and head only. Physical space [0, 1] 2, head configuration S 1. x R M = [0, 1] 2 S 1.

42 Kinetic Ising model for neuron activity; example Space and head only. Physical space [0, 1] 2, head configuration S 1. x R M = [0, 1] 2 S 1. Then in assume E u (t) = E spatial,u (x(t)) + E head,u (x(t)), E spatial,u (x, y, α) = e (u u u ) 2 (u u u ) 2 E head,u (x, y, α) = e (u u u ) 2. I.e. spatial place field in (p u, q u ), head direction place field in a u.

43 Inference Will just write E u (t) = u u =1 A u,u exp ( d(x(t), c u ) 2 ). Easy to see that the likelihood function of observed data under the model is convex in A. Then we can 1 Maximize (log-)likelihood using say BFGS. 2 Get contribution of E spatial,u, E head,u and any other suspected covariates.

44 Inference Will just write E u (t) = u u =1 A u,u exp ( d(x(t), c u ) 2 ). Easy to see that the likelihood function of observed data under the model is convex in A. Then we can 1 Maximize (log-)likelihood using say BFGS. 2 Get contribution of E spatial,u, E head,u and any other suspected covariates. 3 Remove said (expected) contribution from data.

45 Inference Will just write E u (t) = u u =1 A u,u exp ( d(x(t), c u ) 2 ). Easy to see that the likelihood function of observed data under the model is convex in A. Then we can 1 Maximize (log-)likelihood using say BFGS. 2 Get contribution of E spatial,u, E head,u and any other suspected covariates. 3 Remove said (expected) contribution from data. 4 Get new spike trains (residuals); can still compute correlations and persistence.

46 Stimuli list exhausted? No Yes Information about topology of hidden stimuli No Random? No more to learn Yes Persistence diagrams and Betti curves 1 Pick phenomenon/stimuli; e.g. spatial pos., head dir., u -phase pref.,???, 2 Infer its contribution to spike trains 3 Remove contribution Persistent homology Spike trains Experiment or simulation

47 Synthetic data Real recordings are now plentiful 1, but: 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

48 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

49 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. There is a lot of black magic that goes into postprocessing; data should say consult your neuroscientist before trusting. 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

50 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. There is a lot of black magic that goes into postprocessing; data should say consult your neuroscientist before trusting. So, synthesize data: Create firing data using the same (kinetic Ising) model as for the inference. 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

51 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. There is a lot of black magic that goes into postprocessing; data should say consult your neuroscientist before trusting. So, synthesize data: Create firing data using the same (kinetic Ising) model as for the inference. Covariates are limited now only by your imagination: Spatial environment can be anything. 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

52 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. There is a lot of black magic that goes into postprocessing; data should say consult your neuroscientist before trusting. So, synthesize data: Create firing data using the same (kinetic Ising) model as for the inference. Covariates are limited now only by your imagination: Spatial environment can be anything. Why stop at one head? ( Given an u -headed rat exploring a Klein bottle ) 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

53 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. There is a lot of black magic that goes into postprocessing; data should say consult your neuroscientist before trusting. So, synthesize data: Create firing data using the same (kinetic Ising) model as for the inference. Covariates are limited now only by your imagination: Spatial environment can be anything. Why stop at one head? ( Given an u -headed rat exploring a Klein bottle ) Theta phase preference/precession. 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

54 Synthetic data Real recordings are now plentiful 1, but: Experiments tend not to be topologically minded, and they take a long time to do. There is a lot of black magic that goes into postprocessing; data should say consult your neuroscientist before trusting. So, synthesize data: Create firing data using the same (kinetic Ising) model as for the inference. Covariates are limited now only by your imagination: Spatial environment can be anything. Why stop at one head? ( Given an u -headed rat exploring a Klein bottle ) Theta phase preference/precession. Neurons can be coupled at will. 1 E.g. Collaborative Research in Computational Neuroscience data sharing at

55 Proof of concept 1 Space is a an annulus (specifically square box with impassable disc in the middle). Spatial tuning quite strong. Head direction tuning quite strong.

56 Proof of concept 1 Space is a an annulus (specifically square box with impassable disc in the middle). Spatial tuning quite strong. Head direction tuning quite strong. We pretend that we are neuroscientists who have not heard of head direction tuning; we believe spatial location is the only thing influencing neural activity.

57 Proof of concept 1; original data

58 Proof of concept 1; original data Death H Birth

59 Proof of concept 1; space removed

60 Proof of concept 1; space removed Death H Birth

61 Proof of concept 1; head removed

62 Proof of concept 1; head removed Death H Birth

63 Proof of concept 1; space and head removed

64 Proof of concept 1; space and head removed Death H Birth

65 Proof of concept 1; space and head removed 0.5 H Death Betti number Birth

66 Proof of concept 1; it s random Betti number Original Spatial removed Head dir. removed Spatial + head dir. removed Shuffled corr Filtration

67 Thank you!