Learning the dynamics of biological networks

Transcription

1 Learning the dynamics of biological networks Thierry Mora ENS Paris A.Walczak Università Sapienza Rome A. Cavagna I. Giardina O. Pohl E. Silvestri M.Viale Aberdeen University F. Ginelli Laboratoire de physique statistique École normale supérieure, Paris & CNRS IST Austria G.Tkacik Vision Institute S. Deny O. Marre Princeton University W. Bialek M. Berry

2 statistical mechanics as a tool to describe correlated systems interacting spins (any) interacting agents spontaneous magnetization collective behaviour

3 statistical mechanics as a tool to describe correlated systems interacting spins (any) interacting agents A E C R Q D R E N D G R N A spontaneous magnetization collective behaviour

4 statistical mechanics as a tool to describe correlated systems interacting spins (any) interacting agents spontaneous magnetization collective behaviour

5 statistical mechanics as a tool to describe correlated systems interacting spins (any) interacting agents spontaneous magnetization collective behaviour

6 two modeling approaches bottom-up C(r), phenomena solve model H = X ij J ij s i s j

7 two modeling approaches bottom-up C(r), phenomena top-down C(r), observation solve solve model model H = X ij J ij s i s j H = X ij J ij s i s j

8 two modeling approaches bottom-up C(r), phenomena top-down C(r), observation solve solve inverse problem model model H = X ij J ij s i s j H = X ij J ij s i s j

9 how to fit models to data: the maximum entropy approach =( 1, 2,..., N )

10 how to fit models to data: the maximum entropy approach N agents / units described by a variable σ =( 1, 2,..., N )

11 how to fit models to data: the maximum entropy approach N agents / units described by a variable σ =( 1, 2,..., N ) Maximize the entropy

12 how to fit models to data: the maximum entropy approach N agents / units described by a variable σ =( 1, 2,..., N ) Maximize the entropy under the constraint that observables have the same average as the data O 1, O 2,... O a model = O a data O a is typically a moment, e.g. h i i, h i j i

13 how to fit models to data: the maximum entropy approach N agents / units described by a variable σ =( 1, 2,..., N ) Maximize the entropy under the constraint that observables have the same average as the data O 1, O 2,... O a model = O a data O a is typically a moment, e.g. h i i, h i j i # P ( )= 1 Z exp " X a J a O a ( ) e.g. P ( )= 1 Z ep i h i i+ P ij i j E/k B T

14 how to fit models to data: the maximum entropy approach N agents / units described by a variable σ =( 1, 2,..., N ) Maximize the entropy under the constraint that observables have the same average as the data O 1, O 2,... O a model = O a data O a is typically a moment, e.g. h i i, h i j i # P ( )= 1 Z exp " X a J a O a ( ) e.g. P ( )= 1 Z ep i h i i+ P ij i j E/k B T (disordered) Ising model!

15 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) #

16 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) #

17 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best?

18 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best? Bayes rule

19 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best? Bayes rule P ({J a } data) = P (data {J a})p ({J a }) P (data)

20 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best? Bayes rule P ({J a } data) = P (data {J a})p ({J a }) P (data) " M Y m=1 P ( m {J a }) # P ({J a })

21 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best? Bayes rule P ({J a } data) = P (data {J a})p ({J a }) P (data) over datapoints " M Y m=1 P ( m {J a }) # P ({J a })

22 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best? Bayes rule P ({J a } data) = P (data {J a})p ({J a }) P (data) over datapoints " M Y m=1 P ( m {J a }) # flat prior P ({J a })

23 maximum likelihood formulation Given the functional form P ( )= 1 Z exp " X a J a O a ( ) # what parameters Ja explain the data best? Bayes rule P ({J a } data) = P (data {J a})p ({J a }) P (data) over datapoints " M Y m=1 P ( m {J a }) # flat prior P ({J a }) log P J a =0 M log Z J a + X a MX m=1 O a ( m )=0 (maximum likelihood) O a ( ) model = O a ( ) data satisfies the constaints

24 maximum entropy in biology collective activity of neural populations Schneidman et al. Nature 2006 Shlens et al J. Neuroscience 2006 Tang et al J. Neuroscience 2008 Tkacik et al PLoS CP 2014

25 maximum entropy in biology collective activity of neural populations Schneidman et al. Nature 2006 Shlens et al J. Neuroscience 2006 Tang et al J. Neuroscience 2008 Tkacik et al PLoS CP 2014 co-variations in protein families contact prediction Weigt et al. PNAS 2009; Morcos et al. PNAS 2011 Marks et al PLoS ONE 2012; Sulkowska et al PNAS 2012

26 maximum entropy in biology collective activity of neural populations Schneidman et al. Nature 2006 Shlens et al J. Neuroscience 2006 Tang et al J. Neuroscience 2008 Tkacik et al PLoS CP 2014 co-variations in protein families contact prediction Weigt et al. PNAS 2009; Morcos et al. PNAS 2011 Marks et al PLoS ONE 2012; Sulkowska et al PNAS 2012 diversity of antibody repertoires in the immune system Mora Walczak Callan Bialek PNAS 2010

27 maximum entropy in biology collective activity of neural populations Schneidman et al. Nature 2006 Shlens et al J. Neuroscience 2006 Tang et al J. Neuroscience 2008 Tkacik et al PLoS CP 2014 co-variations in protein families contact prediction Weigt et al. PNAS 2009; Morcos et al. PNAS 2011 Marks et al PLoS ONE 2012; Sulkowska et al PNAS 2012 diversity of antibody repertoires in the immune system Mora Walczak Callan Bialek PNAS 2010 DNA motifs of transcription factor binding sites Santolini Mora Hakim Plos ONE 2014

28 maximum entropy in biology collective activity of neural populations Schneidman et al. Nature 2006 Shlens et al J. Neuroscience 2006 Tang et al J. Neuroscience 2008 Tkacik et al PLoS CP 2014 co-variations in protein families contact prediction Weigt et al. PNAS 2009; Morcos et al. PNAS 2011 Marks et al PLoS ONE 2012; Sulkowska et al PNAS 2012 diversity of antibody repertoires in the immune system Mora Walczak Callan Bialek PNAS 2010 DNA motifs of transcription factor binding sites Santolini Mora Hakim Plos ONE 2014 collective behaviour of mice Shemesh et al. elife 2013

29 maximum entropy in biology collective activity of neural populations Schneidman et al. Nature 2006 Shlens et al J. Neuroscience 2006 Tang et al J. Neuroscience 2008 Tkacik et al PLoS CP 2014 co-variations in protein families contact prediction Weigt et al. PNAS 2009; Morcos et al. PNAS 2011 Marks et al PLoS ONE 2012; Sulkowska et al PNAS 2012 diversity of antibody repertoires in the immune system Mora Walczak Callan Bialek PNAS 2010 DNA motifs of transcription factor binding sites Santolini Mora Hakim Plos ONE 2014 collective behaviour of mice Shemesh et al. elife 2013 collective behaviour of bird flocks Bialek et al. PNAS 2012; Cavagna et al. PRE 2014 Bialek et al. PNAS 2014

30 dynamical maximum entropy maximum entropy gives a steady-state picture. what about the dynamics? ad hoc dynamics such as Glauber, Metropolis may be wrong

31 dynamical maximum entropy maximum entropy gives a steady-state picture. what about the dynamics? ad hoc dynamics such as Glauber, Metropolis may be wrong (a) solution: maximum entropy over trajectories P ( 1,..., T ) t t+1 t+2 constraints on cross-time correlations, e.g. h t i t 0 j i

32 dynamical maximum entropy maximum entropy gives a steady-state picture. what about the dynamics? ad hoc dynamics such as Glauber, Metropolis may be wrong (a) solution: maximum entropy over trajectories P ( 1,..., T ) t t+1 t+2 constraints on cross-time correlations, e.g. P ( 1,..., T )= 1 Z exp 0 h t X J t ij i,j,t,t 0 t 0 j i 0 t t i A action t 0 j 1 A

33 dynamical maximum entropy maximum entropy gives a steady-state picture. what about the dynamics? ad hoc dynamics such as Glauber, Metropolis may be wrong (a) solution: maximum entropy over trajectories P ( 1,..., T ) t t+1 t+2 constraints on cross-time correlations, e.g. not the same as: P ( i,t { j,t 0} t0 <t) = P ( 1,..., T )= 1 Z exp 0 1 Z({ j,t 0} t0 <t) exp h t X J t ij i,j,t,t 0 2 4h i t 0 j i 0 t t i A action t 0 j i,t + X j,t 0 <t 1 A J t t0 ij i,t j,t 0 3 5

34 example 1: flocks of birds

35 example 1: flocks of birds

36 aligned collective motion ξ fluctuations around main orientation strong polarization r 1 vi φ = r ~ 0.95 N i vi domains

37 a maximum entropy model for birds velocity of bird v i, s i = v i /kv i k

38 a maximum entropy model for birds velocity of bird v i, s i = v i /kv i k constrain correlation functions C ij = hs i s j i 0 P (s 1,...,s N )= 1 Z X ij 1 J ij s i s j A = 1 Z exp( H) (Heisenberg model on lattice)

39 a maximum entropy model for birds velocity of bird v i, s i = v i /kv i k constrain correlation functions C ij = hs i s j i 0 P (s 1,...,s N )= 1 Z X ij 1 J ij s i s j A = 1 Z exp( H) (Heisenberg model on lattice)

40 a maximum entropy model for birds velocity of bird v i, s i = v i /kv i k constrain correlation functions 0 C ij = hs i s j i 1 P (s 1,...,s N )= 1 Z X ij J ij s i s j A = 1 Z exp( H) (Heisenberg model on lattice) derives from Langevin eqn, equilavent to social model, similar to Vicsek s d s i dt = H s i + i (t) = NX j=1 alignment J ij s j + i (t) noise (does not mean that s the only possible dynamics, or the true one)

41 parametrization J ij = J 0 if j is one i s nc first neighbors otherwise then symmetrized

42 parametrization J ij = J 0 if j is one i s nc first neighbors otherwise then symmetrized Equivalent to maximum entropy with constraint on C int = 1 N NX i=1 1 n c X j2v (i) hs i s j i single snapshot spatial averaging instead of ensemble averaging

43 predicting correlation functions interaction range correlation range perpendicular correlation long-range order from local interactions

44 predicting correlation functions interaction range correlation range perpendicular correlation Correlation C 4 (r 1,r 2 ) B 4-bird correlation function Data Model r 1 = Distance r 2 (m) i r 1 r 2 r1 j l k long-range order from local interactions

45 interaction range metric or topological?

46 interaction range rc r1 metric or topological?

47 interaction range rc r1 rc metric or topological? r1

48 interaction range rc nc = 6 r1 rc metric or topological? r1

49 interaction range rc nc = 6 r1 rc metric or topological? nc = 6 r1

50 interaction range rc nc = 6 r1 rc metric or topological? nc = 6 r1 nc -1/3 r1

51 interaction range rc nc = 6 r1 rc metric or topological? nc = 6 r1 nc ~ (rc / r1) 3 nc nc -1/3 r1 r1

52 answer: interaction is topological not metric Interaction range n c -1/ E sparseness r 1 (m) Bialek et al PNAS 2012

53 flock density answer: interaction is topological not metric nc ~ 21 does not depend on flock size Interaction range n c -1/ E sparseness r 1 (m) Interaction range n c D Flock size (m) Bialek et al PNAS 2012

54 dynamics (may) matter we ve assumed that neighborhoods are fixed but birds may exchange neighbors fast

55 dynamics (may) matter we ve assumed that neighborhoods are fixed but birds may exchange neighbors fast

56 dynamics (may) matter we ve assumed that neighborhoods are fixed but birds may exchange neighbors fast the effective number of interaction partners could be larger than the instantaneous one.

57 dynamics (on bird orientations) constrain hs t is t ji and hs t is t+1 i j P (s 1,...,s T )= 1 Ẑ exp ( A) action A = 1 2 X t X i6=j J (1) ij;t st is t j + J (2) ij;t st+1 i s t j

58 dynamics (on bird orientations) constrain hs t is t ji and hs t is t+1 i j P (s 1,...,s T )= 1 Ẑ exp ( A) action A = 1 2 X t X i6=j J (1) ij;t st is t j + J (2) ij;t st+1 i s t j in spin-wave approximation, equivalent to collective random walk n s A and M functions of J (1) and J (2) ~

59 dynamics (on bird orientations) constrain hs t is t ji and hs t is t+1 i j P (s 1,...,s T )= 1 Ẑ exp ( A) action A = 1 2 X t X i6=j J (1) ij;t st is t j + J (2) ij;t st+1 i s t j in spin-wave approximation, equivalent to collective random walk alignment strength n s temperature Langevin equation ~

60 inferring out-of-equilibrium behavior infering J, nc, and a third parameter, the temperature T Jn c = 1 t C int C s + G s G int 2C int C 0 int C s and similar eq. for T

61 inferring out-of-equilibrium behavior infering J, nc, and a third parameter, the temperature T Jn c = 1 t C int C s + G s G int 2C int C 0 int C s and similar eq. for T if equilibrium slowly evolving and symmetric nij then one recovers the same result as the Heisenberg model, with J J / T 0 P (s 1,...,s N )= 1 Z J T 1 X n ij s i s j A ij

62 test on simulated data simulation of 2D topological model with Voronoi neighbors µ is a parameter quantifying how fast birds change neighbors n c dynamical inference 3.8 static inference 3.6 n c static inference overestimates number of partners 5 true value at large µ, dynamic static µ Cavagna et al PRE 2014 dynamical maximum entropy works, static maximum entropy doesn t

63 the retina multielectrode array recordings

64 the stimulus

65 the stimulus

66 binary neurons raster binary variables i =0, 1 N ~ 150 neurons 10 ms Marre et al. J. Neurosci. 2012

67 neuron activities are correlated Schneidman et al, Nature 2005 independenc total number of spikes P 1 ( )= 1 Z ep i h i Ising model P 2 ( )= 1 Z ep i h i i+ P ij J ij i j i

68 neuron activities are correlated Schneidman et al, Nature 2005 independenc total number of spikes P 1 ( )= 1 Z ep i h i Ising model P 2 ( )= 1 Z ep i h i i+ P ij J ij i j i goal: build the thermodynamics of this correlated system from data

69 building the density of states evaluate P ( 1,..., N ) by modelling or by frequency counting

70 building the density of states evaluate P ( 1,..., N ) by modelling or by frequency counting define energy through Boltzmann law P = 1 Z e E/k BT E = log P

71 building the density of states evaluate P ( 1,..., N ) by modelling or by frequency counting define energy through Boltzmann law P = 1 Z e E/k BT E = log P now consider the distribution of energies E C(E) = number of states with E( ) <E

72 building the density of states evaluate P ( 1,..., N ) by modelling or by frequency counting define energy through Boltzmann law P = 1 Z e E/k BT E = log P now consider the distribution of energies E C(E) = number of states with E( ) <E define a microcanonical entropy : S(E) = log C(E) Mora Bialek J. Stat. Phys. 2011

73 density of states just counting states Maximum entropy model (under natural movie stimulus) Tkacik Mora Marre Amodei Berry Bialek, arxiv 2014

74 Zipf s law (interlude) Zipf 1949

75 Zipf s law (interlude) Probability P (E in log scale) ~ cumulative distribution Zipf 1949

76 what does S(E) = E mean?

77 what does S(E) = E mean? what s the probability of a given energy? how many states at E probability of a state at E P (E) e S E constant!

78 what does S(E) = E mean? what s the probability of a given energy? how many states at E P (E) e S E probability of a state at E constant! (note: only works if exponent is = 1)

79 what does S(E) = E mean? what s the probability of a given energy? how many states at E E scales with N its fluctuations scale with N heat capacity P (E) e S in usual thermodynamics E C = Var(E) N probability of a state at E constant! (note: only works if exponent is = 1) C / N diverges at 2nd order transition critical point (e.g. 2D Ising model)

80 what does S(E) = E mean? what s the probability of a given energy? how many states at E probability of a state at E P (E) e S in usual thermodynamics E constant! (note: only works if exponent is = 1) E scales with N its fluctuations scale with N heat capacity link to information theory E = log P surprise (Shannon 1948) equipartition theorem (valid for independent units): almost all codewords we see have the same surprise ~ entropy basis for compression C = Var(E) N C / N diverges at 2nd order transition critical point (e.g. 2D Ising model)

81 specific heat let s add a spurious temperature one direction in parameter space P T ( )= 1 Z(T ) e E/T C = Var T (E/T ) = Var T ( log P ) (T = 1 corresponds to the real ensemble)

82 specific heat let s add a spurious temperature one direction in parameter space P T ( )= 1 Z(T ) e E/T C = Var T (E/T ) = Var T ( log P ) (T = 1 corresponds to the real ensemble)

83 specific heat let s add a spurious temperature one direction in parameter space P T ( )= 1 Z(T ) e E/T C = Var T (E/T ) = Var T ( log P ) (T = 1 corresponds to the real ensemble) Neural network Tkacik Mora Marre Amodei Berry Bialek, arxiv 2014

84 dynamical criticality

85 dynamical approach 10 ms t

86 dynamical approach 10 ms t proposal: consider statistics over trajectories P ( 1,..., L ) t t+1 t+2

87 dynamical approach 10 ms t proposal: consider statistics over trajectories P ( 1,..., L ) define t t+1 t+2 E = log P ({ i,t }) calculate specific heat c = 1 NL Var(E)

88 link to dynamical criticality: branching process p ij P ({ i,t })= Y NY p i (t) i,t [1 p i (t)] 1 i,t t p i (t) =1 t i=1 Y (1 p ij ) i,t 1 j Beggs & Plenz 2003 branching parameter:! = 1 N X p ij ij

89 link to dynamical criticality: branching process p ij P ({ i,t })= Y NY p i (t) i,t [1 p i (t)] 1 i,t t p i (t) =1 t i=1 Y (1 p ij ) i,t 1 j Beggs & Plenz 2003 branching parameter:! = 1 N X p ij ij Shew Yang Petermann Roy Plenz 2009

90 model for multi-neuron spike trains sampling 2 N states is hard enough; here 2 NL states we need models let s do something simple total number of spikes K t = X i i,t is informative of collective behaviour Tkacik Marre Mora Amodei Berry Bialek, JSTAT 2013 independent neutrons

91 model for multi-neuron spike trains sampling 2 N states is hard enough; here 2 NL states we need models let s do something simple total number of spikes K t = X i i,t is informative of collective behaviour Tkacik Marre Mora Amodei Berry Bialek, JSTAT 2013 maximum entropy model with constrains on temporal correlations of K P (K t,k t 0) t t 0 <v all neurons behave the same independent neutrons

92 model for multi-neuron spike trains sampling 2 N states is hard enough; here 2 NL states we need models let s do something simple total number of spikes K t = X i i,t is informative of collective behaviour Tkacik Marre Mora Amodei Berry Bialek, JSTAT 2013 maximum entropy model with constrains on temporal correlations of K P (K t,k t 0) t t 0 <v all neurons behave the same independent neutrons Energy E = X t h(k t ) X vx J u (K t,k t+u ) t u=1

93 solving the problem E = X t h(k t ) X vx J u (K t,k t+u ) t u=1 define a super-variable X t =(K t,k t+1,...,k t+v 1 ) now becomes a 1D model P ({X t })= 1 Z exp " X t H(X t )+ X t W (X t,x t+1 ) # can be solved by transfer matrices (aka forward backward algorithm, or belief propagation in 1D)

94 model predicts avalanche dynamics b Probability observed =0 =1 =2 =3 =4 v = Avalanche size (number of spikes)

95 model predicts avalanche dynamics b Probability observed =0 =1 =2 =3 =4 v = Avalanche size (number of spikes) NB: no power laws in avalanche statistics

96 thermodynamics of spike trains i L t N E = log P ({ i,t }) P T ( )= 1 Z(T ) e E/T C = Var T (E/T ) C(T)/NL v = temporal range =0 =1 =2 =3 v =4 Temperature T 1/β b. C(T)/NL c(β) N =5 N = 10 N = 20 N = 30 N = 40 N = 50 N = 61 N = 97 N = /β T Mora Deny Marre PRL 2015

97 thermodynamics of spike trains i static (salamander) L! t N E = log P ({ i,t }) P T ( )= 1 Z(T ) e E/T C = Var T (E/T ) dynamic (rat) C(T)/NL =0 =1 =2 =3 v = v = temporal range Temperature T 1/β b. C(T)/NL c(β) N =5 N = 10 N = 20 N = 30 N = 40 N = 50 N = 61 N = 97 N = /β T Mora Deny Marre PRL 2015

98 scaling with network size b. Var(surprise)/N L =0 =1 =2 =3 =4 v = Network size N NL T Peak temperature 1/βc Network size N =0 =1 =2 =3 =4 v =5

99 conclusions stationary maximum entropy models may capture emergent behaviour in biological data but dynamic framework may be necessary to get parameters right in neural systems, heat capacity = useful indicator of critical properties critical signature enhanced by dynamical approach application to other biological contexts?

100 random flickering checkerboard

101 random flickering checkerboard static (salamander) dynamic (rat)