Spike-train description: process or Gibbs? (Overview of issues in stochastic modeling)

Transcription

1 Spike-train description: process or Gibbs? (Overview of issues in stochastic modeling) Contributors: S. Berghout (Leiden) R. Cofré (Geneve) S. Gallo (São Carlos), G. Maillard (Aix-Marseille), E. Verbitskiy (Leiden) MathStatneuro workshop, Nice, September 2015

2 Issues Two modeling issues in neuroscience Spike train Sequence of spikes of one neuron ω n 1 ω n ω 1 ω 0 ω 1 ω n ω n+1 ω i = 0 or 1 (simplicity). Brain mapping Sequence of spike configurations of regions of N neurons (rasters) ω N (t 1 ). ω 1 (t 1 ) ω N (t n ). ω 1 (t n ) = ω(t 1 ) ω(t N )

3 Issues Two modeling issues in neuroscience Spike train Sequence of spikes of one neuron ω n 1 ω n ω 1 ω 0 ω 1 ω n ω n+1 ω i = 0 or 1 (simplicity). Brain mapping Sequence of spike configurations of regions of N neurons (rasters) ω N (t 1 ). ω 1 (t 1 ) ω N (t n ). ω 1 (t n ) = ω(t 1 ) ω(t N )

4 Tenets Basic tenets Stochastic description due to signal variability Full description = probability measure Spike trains: Measure on {0, 1} Z = stochastic process Brain mappings: Evolving approach: Stochastic process on Ω Z N with Ω N = {0, 1} N (e.g. Galves-Löcherbach). Equilibrium approach: Register has stabilized, law of ω(t) independent of t = measure on Ω N.

7 Modeling vs decoding Describing probability measures µ probability measure on a product space X L, L countable = Countable family of random variables (X i ) i L E.g: Spike trains: X = {0, 1}, L = Z (or part thereof) Brain mappings - evolving: X = Ω N, L = Z Brain mappings - equilibrium: X = {0, 1}, L = {1,..., N} Key issue: efficient characterization of µ. Complementary points of view: Modeling: Definition of µ on the basis of available info Decoding: Extracting info from realizations of µ

10 Averages First aspect: Averages Consider functions of finitely many random variables ( ) F j = F j Xi j,..., X 1 i j nj j = 1,, K [{i j 1,..., ij n j } = (finite) window for F j ] Consider the corresponding expected values α j = E(F j ) j = 1,, K (1) Issues: Modeling: α 1,..., α K known; choose µ satisfying (1) Decoding: Compute α 1,..., α K from realizations of µ

13 Averages Truncated averages Related aspect: correlation between far away regions: τ n = translation a distance n E(F τ n G) E(F ) E(G) Φ(n) Expected: Φ(n) 0 (asymptotic independence) N Form of Φ(n) mixing properties Fully exploited in ergodic theory (less so in stat. mech.)

14 Averages Truncated averages Related aspect: correlation between far away regions: τ n = translation a distance n E(F τ n G) E(F ) E(G) Φ(n) Expected: Φ(n) 0 (asymptotic independence) N Form of Φ(n) mixing properties Fully exploited in ergodic theory (less so in stat. mech.)

15 Issues for averaging Modeling issues for averages Finitely many averages do not uniquely determine a process Which of the possible measures should we choose? Standard answer: The one with maximal entropy Proposed by Boltzmann, re-proposed by Jaynes Succesful in statistical mechanics (ensembles!) Leads to variational approaches Problem: Well defined only for finite systems (finite train, finite brain) However: relevant information as size (thermodynamic limit)

18 Maximal entropy Review on entropy If ν is a probability measure on a finite space Γ S Γ (µ) = x Γ µ(x) ln µ(x) [0 ln 0 = 0]. In particular, if µ is the uniform measure: S(uniform) = ln Γ Want to find µ maximizing S among the family P(α 1,..., α K ) = { µ : E µ (F 1 ) = α 1,..., E µ (F K ) = α K }

21 Maximal entropy Canonical ensembles Lagrange multipliers λ 1,..., λ K lead to extremization of L [ {µ(x)} x Γ, (λ i ) i=1,...,k ] = x µ(x) ln µ(x) + K j=1 λ j [ α j x ] F j (x) µ(x) The solution is µ(x) = 1 Z exp [ K j=1 ] λ j F j (x) (2) with λ j tuned so that α j = x Γ F j (x) µ(x) j = 1,, K (3)

22 Maximal entropy Canonical ensembles Lagrange multipliers λ 1,..., λ K lead to extremization of L [ {µ(x)} x Γ, (λ i ) i=1,...,k ] = x µ(x) ln µ(x) + K j=1 λ j [ α j x ] F j (x) µ(x) The solution is µ(x) = 1 Z exp [ K j=1 ] λ j F j (x) (2) with λ j tuned so that α j = x Γ F j (x) µ(x) j = 1,, K (3)

23 Maximal entropy Canonical ensembles (cont.) The system (2) (3) has (usually) a unique solution Comparison with statistical mechanics Measure µ = (macro)canonical ensemble Temperature =1 λ j = field conjugated to the observable F j. Examples: F = magnetization, λ = magnetic field F = density, λ = chemical potencial F = n-body term, λ = n-body coupling Approach valid for finite configurations of neurons (or spikes) E.g.: Equilibrium brain-mapping measure on Ω N

26 Thermodynamic limit Entropy in the thermodynamic limit Two problems appear if Γ : Entropy blows up The limit of the measures (2) (3) is not uniquely defined Solutions: Pass to entropy density s(µ) = lim S Γ Γ where S Γ computed with the projection of µ on Γ Consider the measures minimizing the free energy f(µ) = j µ(f j ) s(µ) with f j = lim F j / Λ.

27 Thermodynamic limit Entropy in the thermodynamic limit Two problems appear if Γ : Entropy blows up The limit of the measures (2) (3) is not uniquely defined Solutions: Pass to entropy density s(µ) = lim S Γ Γ where S Γ computed with the projection of µ on Γ Consider the measures minimizing the free energy f(µ) = j µ(f j ) s(µ) with f j = lim F j / Λ.

28 Thermodynamic limit Delicate points Entropy density = large deviation rate function = fluctuation control Normal fluctuations (CLT): s strictly convex s affine = abnormal fluctuations = critical points = 2nd-order phase transtions There may be several minimizing measures (1st-order phase transitions) These measures have pairwise zero relative entropy (low cost of setting a bubble of one into the other one)

29 Thermodynamic limit Delicate points Entropy density = large deviation rate function = fluctuation control Normal fluctuations (CLT): s strictly convex s affine = abnormal fluctuations = critical points = 2nd-order phase transtions There may be several minimizing measures (1st-order phase transitions) These measures have pairwise zero relative entropy (low cost of setting a bubble of one into the other one)

30 Ergodicity Decoding issues for averages Ergodic hypothesis: Empirical averages µ expectations Spike trains: 1 n n l=1 Brain mappings - evolving: 1 n n l=1 Brain mappings - equilibrium: 1 n n l=1 F (τ l ω) N µ(f ) F (ω(t l )) N µ(f ) F (τ l ω(t)) N µ(f )

34 Ergodicity Fine points If multiple measures, ergodicity only for extremal measures Ergodicity true for almost all realizations Good realizations can be studied with techniques of ergodic theory (symbolic dynamics) Example (Takens and Verbitskiy, 2003): If K(α 1,..., α K ) = { (x i ) : 1 n n i=1 F j (x i ) n } α j, j = 1,..., K then, topological entropy of K(α) = max. entropy of measures in P(α) (l.h.s. = multifractal analysis of dynamical systems)

35 Ergodicity Fine points If multiple measures, ergodicity only for extremal measures Ergodicity true for almost all realizations Good realizations can be studied with techniques of ergodic theory (symbolic dynamics) Example (Takens and Verbitskiy, 2003): If K(α 1,..., α K ) = { (x i ) : 1 n n i=1 F j (x i ) n } α j, j = 1,..., K then, topological entropy of K(α) = max. entropy of measures in P(α) (l.h.s. = multifractal analysis of dynamical systems)

36 Process point of view Second aspect: Transition probabilities Machine-learning approach for spike or brain-mapping trains: Use first part of the train to develop rules to predict rest By recurrence: enough to predict next bit given history That is, estimate the conditional probabilities w.r.t. past P (X n X n 1, X n 2,...) through its law, defined by a function g such that P ( X 0 = ω 0 X 1 = ) ( ω 1 = g ω0 ω 1 ) Look for µ determined by (consistent with) this g: µ ( X 0 = ω 0 X 1 = ) ( ω 1 = g ω0 ω 1 )

39 Process point of view Regular g-measures Relevant transitions must be insensible to the farther past: g is a regular g-function if ɛ > 0 n 0 such that g ( ω 0 σ 1 n ) ( ω n 1 g ω0 σ ) 1 < ɛ (4) for every ω, σ (4) is equivalent to g(ω 0 ) continuous in product topology Additional, not relevant, non-nullness condition A probability measure µ is a regular g-measure if it is consistent with some regular g-function train µ thought as a process: past determines future (causality)

44 Fields point of view Conditional probabilities If the full train is available, why use only the past? Learn to predict a bit using past and future! X n determined by finite-window probabilities P(X n X n 1, X n 2,... ; X n+1, X n+2,...) through conditional laws determined by a function γ s.t. P ( ( ) X 0 = ω 0 X {0} c = ω {0} c) = γ ω0 ω {0} c Specification: γ satisfying certain compatibility condition Look for µ determined by (consistent with) this γ: µ ( ( ) X 0 = ω 0 X {0} c = ω {0} c) = γ ω0 ω {0} c

45 Fields point of view Conditional probabilities If the full train is available, why use only the past? Learn to predict a bit using past and future! X n determined by finite-window probabilities P(X n X n 1, X n 2,... ; X n+1, X n+2,...) through conditional laws determined by a function γ s.t. P ( ( ) X 0 = ω 0 X {0} c = ω {0} c) = γ ω0 ω {0} c Specification: γ satisfying certain compatibility condition Look for µ determined by (consistent with) this γ: µ ( ( ) X 0 = ω 0 X{0} c = ω {0} c) = γ ω0 ω{0} c

46 Fields point of view Conditional probabilities If the full train is available, why use only the past? Learn to predict a bit using past and future! X n determined by finite-window probabilities P(X n X n 1, X n 2,... ; X n+1, X n+2,...) through conditional laws determined by a function γ s.t. P ( ( ) X 0 = ω 0 X {0} c = ω {0} c) = γ ω0 ω {0} c Specification: γ satisfying certain compatibility condition Look for µ determined by (consistent with) this γ: µ ( ( ) X 0 = ω 0 X{0} c = ω {0} c) = γ ω0 ω{0} c

47 Fields point of view Gibbs measures A specification γ is regular if ɛ > 0 n, m 0 γ ( ω 0 ω m nσ [n,m] c) γ ( ω0 ω {0} c) < ɛ (5) for every σ, ω (5) is equivalent to γ(ω 0 ) continuous in product topology Additional, not relevant, non-nullness condition A probability measure µ is a Gibbs-measure if it is consistent with some regular specification Train µ thought as non-causal or with anticipation

51 Fields point of view Link with stat mech Theorem (Kozlov) A specification is Gibbsian iff it has the Boltzmann form γ ( ) { ω 0 ω{0} c = exp } φ A (ω A ) /Norm., A 0 where {φ A } (interaction) satisfy φ A <. A 0

52 Comparison Questions, questions Trains described/decoded as processes or as Gibbs? Both setups give complementary information: Processes: ergodicity, coupling, renewal, perfect simulation Fields: Gibbs theory Are these setups mathematically equivalent? Is every regular g-measure Gibbs and viceversa? Answers: It depends on the dependency on history (exterior): Yes, if it decays exponentially fast No if it decays too slowly There are regular g-measures that are not Gibbs Not known whether there are Gibbs that are not regular g (belief: yes!)

56 Practical questions For trains of spikes or evolving brain mapping What is more efficient: One-side or two-side conditioning? Neural interpretation of two-side conditioning? Possible use of multifractal analysis (spectrum)? For equilibrium brain mapping Only Gibbs setup available (interaction). Is it appropriate? More generally, are asymptotic phenomena relevant?: Gibbsianness 1st-order phase transitions (multiple measures) 2nd-order phase transitions (critical behavior)