Statistical Modeling of Temporal Evolution in Neuronal Activity. Rob Kass

Transcription

1 . p.1 Statistical Modeling of Temporal Evolution in Neuronal Activity Rob Kass Department of Statistics and The Center for the Neural Basis of Cognition Carnegie Mellon University

2 . p.2 Statistical Collaborators: Sam Behseta, Can Cai, Cari Kaufman, Alex Rojas, Liuxia Wang and Anthony Brockwell, Emery Brown, Satish Iyengar, Valérie Ventura Neurophysiological Collaborators: Tai-Sing Lee, Carl Olson, Andy Schwartz, Tony Reina Peter Strick, Donna Hoffman, Natalie Picard

3 . p.3 REFERENCES Brown, E.N., Barbieri, R., Ventura, V., Kass, R.E., and Frank, L.M. (2002) The time-rescaling theorem and its application to neuronal spike train data analysis, Neural Comput., 14: DiMatteo, I., Genovese, C.R., and Kass, R.E. (2001) Bayesian curve-fitting with free-knot splines. Biometrika, 88: Kass, R.E., and Ventura, V. (2001) A spike-train probability model. Neural Comput., 13: Olson, C.R., Gettner, S.N., Ventura, V., Carta, R. and Kass, R.E. (2000) Neuronal activity in macaque supplementary eye field during planning of saccades in response to pattern and spatial cues. J. Neurophys., 84: Ventura, V., Carta, R., Kass, R.E., Gettner, S.N., and Olson, C.R. (2001) Statistical analysis of temporal evolution in single-neuron firing rates. Biostatistics, 3: 1-20.

4 . p.4 Also An on-line, self-contained talk Statistical Smoothing of Neuronal Data is available at

5 . p.5 Stochastic Models for 1. Instantaneous firing rate of single neurons; 2. Variation in firing rate across many neurons; 3. Decoding for movement prediction; 4. Within-trial firing rate: Non-Poisson spiking; 5. Correlated spiking across pairs of neurons.

6 . p.6 Statistical Motivation for Modeling Probability models offer efficiency; flexibility; formal inferences.

7 . p.7 Statistical Motivation for Modeling Probability models offer efficiency; flexibility; formal inferences. Comment: Probability models require assumptions; so does every well-justified statistical method; of course, some make stronger assumptions than others.

8 . p.8 Statistical Motivation for Modeling Efficency: toy problem; smoothing the PSTH; Bayesian decoding; Flexibility: smoothing with BARS (and MBARS); non-poisson firing; Formal inferences: all of the methods discussed here.

9 . p.9 Some Data From Chris Baker, in Carl Olson s lab. Recorded from IT. Passive fixation task: Visual stimulus (simple geometric figure, white on dark background) presented (in receptive field) for 500 ms; if fixation maintained, reward given. Time = 0 is onset of stimulus.

10 Time (ms). p.10 Firing rate Trial number

11 p(xjμ) = e μ μx x! :. p.11 Assume number of spikes x in given time interval (a; b) follows Poisson distribution, with mean μ: Obvious estimator of μ: μx, mean number of spikes in (a; b) across observed trials Alternative: s 2, sample variance

12 . p.12 Compare μx to s 2 using Mean Squared Error (MSE): MSE(^μ) = E((^μ μ) 2 ) Result: For n = 100 trials and μ = 10, s 2 has about 21 times larger MSE than μx. E.g., need about 2100 trials using s 2 to get accuracy of 100 trials using μx.

13 . p.13 Maximum Likelihood Estimators Poisson μx case: is MLE. Fisher (1922, 1925; and subsequent theoreticians): Quite generally, for large samples, the MLE has the minimal possible Mean Squared Error. Applies to parametric models; formally, number of parameters remains finite while sample size grows indefinitely.

14 P (AjB) = P (BjA)P (A) P (BjA)P (A) + P (BjA c )P (A c ). p.14 Bayes s Theorem:

15 P (AjB) = P (BjA)P (A) P (BjA)P (A) + P (BjA c )P (A c ) R p(dataj )p( ) p(dataj )p( )d. p.14 Bayes s Theorem: p( jdata) =

16 . p.15 Likelihood function: L( ) = p(dataj ) Bayes Estimators 1. Often: Bayes ß MLE; both are optimal for large n. 2. But, sometimes prior information is also important. 3. Bayes Theorem provides very powerful machinery, especially via Monte Carlo.

17 . p.16 Instantaneous Firing Rate May be inherently interesting. (E.g., SEF data)

18 (A) (B) Time (ms). p.17 Firing rate Firing rate Trial number

19 . p.18 Instantaneous Firing Rate May be inherently interesting. (E.g., SEF data) May be necessary (or helpful) for subsequent calculations.

20 1 ;:::;s n ) = e R T 0 (t)dt Π n k=1 (s k) p(s. p.19 Probability density for set of spikes s 1 ;:::;s n (in non-poisson case (t) is generalized)

21 . p.20 In Estimating Instantaneous Firing Rate Smoothing is a Good Idea Illustration: small simulation study.

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24 . p.21

25 MISE = Time. p.22 Firing rate

26 . p.22 Firing rate MISE = 4.68 MISE = times smaller Time

27 . p % simulation bands for unsmoothed PSTH with 14 times more trials

28 . p.23 For SEF data above, doesn t matter much how the smoothing is accomplished, e.g., Gaussian filtering is fine. Sometimes fixed-bandwidth methods are not adequate.

29 Time (ms). p.24 Firing rate

30 . p.25 BARS (Bayesian Adaptive Regression Splines) Fits firing rate curve with cubic splines, finding optimal knot locations via Monte Carlo simulation. Seems to be the most powerful existing method of generalized curve-fitting (small MSE in various examples). In many contexts (firing rate, fmri, EEG, EMG) BARS tracks the signal very well, effectively capturing the high-frequency changes in signal while filtering out high-frequency noise.

31 (a) Original EEG and EOG Orig EEG Orig EOG Time(sec) (b) Original EOG with Fixed Knot (dashed) and Free Knot (bold) Splines volts -1.5*10^-4-5*10^ p.26

32 . p.27 BARS (Bayesian Adaptive Regression Splines) Provides formal statistical inferences. See DiMatteo, Genovese, Kass (2001, Biometrika); Kass and Wallstrom (2002, Statistical Science). Applies to usual curve-fitting setting, and to generalized curve-fitting of intensity functions (in Poisson and non-poisson settings).

33 . p.28 Consider first the usual curve-fitting setting Example 2 f(x) x

34 Y i = f (x i ) + " i f is (approximated by) cubic spline with k knots at ο 1 ;:::;ο k. p.29 Usual curve-fitting framework With free-knot splines

35 . p.30 0 T 1 2 ο ο ο 3 1

36 Y i = f (x i ) + " i f (x) = b j (x)fi j. p.31 X k+2 j=1 b j (x) is j-th spline basis function. If knots (ο; k) were pre-determined, would have a linear regression problem. Hard (statistically interesting) problem is to determine (ο; k).

37 . p.32 BARS algorithm: first integrate (fi; ff), then apply Monte Carlo to find knots. Get: (i) estimated curve and (ii) uncertainty.

38 . p.33 Average MSE (with simulation SE) SARS DMS BARS Example (0.001) (0.002) (0.001) SARS: Zhou and Shen (J. Amer. Statist. Assoc., 2001) DMS: Denison, Mallick, and Smith (J. Royal Statist. Soc., B, 1998)

39 . p.34 SARS DMS modified-dms BARS

40 . p.35 Generalization to Firing-Rate Intensity Function Straightforward (because of approximation to integrals studied by Kass and Wasserman, 1995, J. Amer. Statist. Assoc.). Simulation study based on IT neuron shown above: Gaussian filtering reduces MSE compared to PSTH by factor of 3 and BARS reduces further by additional factor of 3.

41 Time (ms). p.36 Firing rate

42 . p.37 Assessing Population Variability Want: variability of intensity functions and also of particular characteristics (e.g., location of maximal firing rate; peak minus trough difference). Standard principal components analysis (Optican and Richmond, J. Neurophys., 1987) useful descriptively.

43 . p.38 M1 Data from Peter Strick s Lab Button-touch task in either learned, sequential order or random order. Standard analysis is useful. We extend BARS to MBARS to get automated smoothing, then can also make formal statistical inferences.

44 . p.39 neuron 1 neuron 2 neuron 3 neuron 4 neuron 5 rates(hz) rates(hz) rates(hz) rates(hz) rates(hz) Time(ms) Time(ms) Time(ms) Time(ms) Time(ms) neuron 6 neuron 7 neuron 8 neuron 9 neuron 10 rates(hz) rates(hz) rates(hz) rates(hz) rates(hz) Time(ms) Time(ms) Time(ms) Time(ms) Time(ms) neuron 11 neuron 12 neuron 13 neuron 14 neuron 15 rates(hz) rates(hz) rates(hz) rates(hz) rates(hz) Time(ms) Time(ms) Time(ms) Time(ms) Time(ms) neuron 16 neuron 17 neuron 18 neuron 19 neuron 20 rates(hz) rates(hz) rates(hz) rates(hz) rates(hz) Time(ms) Time(ms) Time(ms) Time(ms) Time(ms) neuron 21 neuron 22 neuron 23 neuron 24 neuron 25 rates(hz) rates(hz) rates(hz) rates(hz) rates(hz) Time(ms) Time(ms) Time(ms) Time(ms) Time(ms) neuron 26 neuron 27 neuron 28 neuron 29 neuron 30 rates(hz) rates(hz) rates(hz) rates(hz) rates(hz) Time(ms) Time(ms) Time(ms) Time(ms) Time(ms)

45 . p.40 neuron 1 neuron 2 neuron 3 neuron 4 neuron rates (Hz) rates (Hz) rates (Hz) rates (Hz) rates (Hz) Time (msec) Time (msec) Time (msec) Time (msec) Time (msec) neuron 6 neuron 7 neuron 8 neuron 9 neuron rates (Hz) rates (Hz) rates (Hz) rates (Hz) rates (Hz) Time (msec) Time (msec) Time (msec) Time (msec) Time (msec) neuron 11 neuron 12 neuron 13 neuron 14 neuron rates (Hz) rates (Hz) rates (Hz) rates (Hz) rates (Hz) Time (msec) Time (msec) Time (msec) Time (msec) Time (msec) neuron 16 neuron 17 neuron 18 neuron 19 neuron rates (Hz) rates (Hz) rates (Hz) rates (Hz) rates (Hz) Time (msec) Time (msec) Time (msec) Time (msec) Time (msec) neuron 21 neuron 22 neuron 23 neuron 24 neuron rates (Hz) rates (Hz) rates (Hz) rates (Hz) rates (Hz) Time (msec) Time (msec) Time (msec) Time (msec) Time (msec) neuron 26 neuron 27 neuron 28 neuron 29 neuron rates (Hz) rates (Hz) rates (Hz) rates (Hz) rates (Hz) Time (msec) Time (msec) Time (msec) Time (msec) Time (msec)

46 . p.41 First Principal Component Prop.of.Var=0.63 Second Principal Component Prop.of.Var=0.88 Third Principal Component P.of.V=

47 . p Comp Comp Comp

48 . p.43 Repeating Mode Movement 2 to 3 - Muscle Activity Random Mode Movement 2 to 3 - Muscle Activity EMG Recording (Hz) EMG Recording (Hz) time (msec) time (msec) Repeating Mode Movement 2 to 3 - Neuron Activity Random Mode Movement 2 to 3 - Neuron Activity Rate (Hz) Rate (Hz) time (msec) time (msec)

49 . p.44 Assessing Population Variability Issues: Want formal inferences. Should keep in mind intensity functions are estimated, from limited data, with uncertainty. Methodology: View (t) as realization from random process, want to estimate Cov( (u); (v)). Extend BARS to MBARS (M for Multiple).

50 . p.45 Proportion of Variance Results From 30 neurons First principal component: :697 ± :092 Compare standard PCA:.78.

51 . p.46 Population Coding for Movement Prediction M1 neurons are broadly directionally tuned. Each tuning curve may be characterized (reasonably well) by that cell s preferred direction! D. By combining firing from hundreds of neurons, movement can be predicted reasonably well.

52 . p.47

53 Direction of movement (2 pi radians) Firing Counts in 200 ms Cosine Regression: Deviance = BARS: Deviance = p.48

54 . p.49 The Population Vector Algorithm (PVA) The PVA! P = P w i! D i is simple and reasonably effective.! is predicted movement direction P! is i-th neuron s preferred direction Di w i scaled version of firing rate of i-th neuron

55 . p.50 Population Vector Algorithm vs. Bayesian Decoding The PVA! P = P w i! D i is simple and reasonably effective. Problem is to improve the prediction and be able to use a small set of M1 neurons more reliably. Our Bayesian decoding scheme uses (i) a flexible probability model for the firing rate as a function of movement (velocity) (ii) a prior that assumes movement at time t is similar to movement at time t 1. (Similar to Brown et al. for hippocampus; our particle filter implements a generalized Kalman filter.) We performed a small simulation study, assuming 100 neurons with directional tuning curves similar to those from real data, to compare PVA with Bayesian decoding.

56 . p.51

57 . p.52 4 X and Y Components Time

58 5 Population Vector Reconstruction X and Y components of Velocity. p Time Bin

59 5 PV Reconstruction with Smoothing X and Y components of Velocity. p Time Bin

60 5 Particle Filter Reconstruction X and Y components of Velocity. p Time Bin

61 . p.56 Population Vector Algorithm vs. Bayesian Decoding In our simulation study, Bayesian decoding was 7 times more efficient than the plain PVA, and 2:6 times more efficient than the PVA with smoothing. Also 2:3 times more efficient than OLE (not shown). Other advantages: Bayesian decoding can accommodate non-uniformly distributed preferred directions, variable time lags for behavior, non-cosine tuning functions, fine resolution in time, etc.

62 Peri-Stimulus Time Histogram PSTH. p.57

63 Time (ms). p.58 Firing rate Trial number

64 . p.59 Non-Poisson neuron firing Data pooled across trials ) Poisson Within trials, not Poisson Probability of spike in interval (t; t + dt) Poisson: (t)dt General: (tjs 1 ;s 2 ;:::;s n )dt Here, s 1 ;s 2 ;:::;s n are spikes up to time t and (tjs 1 ;s 2 ;:::;s n ) is conditional intensity of process. In Poisson case, intensity does not depend on the past.

65 . p.60 Non-Poisson neuron firing Data pooled across trials ) Poisson Within trials, not Poisson Probability of spike in interval (t; t + dt) Poisson: (t)dt General: (tjs 1 ;s 2 ;:::;s n )dt Here, s 1 ;s 2 ;:::;s n are spikes up to time t and (tjs 1 ;s 2 ;:::;s n ) is conditional intensity of process. In Poisson case, intensity does not depend on the past. Difficulty: (tjs 1 ;s 2 ;:::;s n ) is function of n + 1 variables. Need some simplifying assumption.

66 . p.61 Inhomogeneous Markov interval (IMI) processes Assume j s 1 ;s 2 ;:::;s n ) = (t; t s Λ (t)) (t s where (t) Λ is last spike time preceding t.

67 . p.62 Inhomogeneous Markov interval (IMI) processes Assume j s 1 ;s 2 ;:::;s n ) = (t; t s Λ (t)) (t s where (t) Λ is last spike time preceding t. Special cases: Barbieri et al. (2001), Miller and Mark (1992). Likelihood approximated as in Poisson case (but now within trials) and splines may be used again to estimate this function of two variables.

68 . p.63 (A) (B) Spikes/sec Poisson IMI Time (ms) Inter-spike time (ms) (C) 50 Spike Previous spike at t = 50 ms Time (ms) Spikes/sec (D) Poisson IMI Time (ms)

69 IP IMI Model Quantiles. p.64 Empirical Quantiles

70 We have also used a version of BARS in IMI models.. p.65

71 . p.66 PSTH Counts number of times a spike occurs in bin at time t (then divides by time interval and number of trials to get rate). Displays time course of firing rate.

72 . p.67 Joint PSTH Counts number of times a spike occurs in bin at time t for neuron 1 AND a spike occurs in bin at time u for neuron 2 (then divides by time interval and number of trials to get rate). Displays time course of joint firing rate, BUT hard to see time course of EXCESS firing rate above what would be expected from INDEPENDENT neurons.

73 Time (ms). p.68 Time (ms)

74 . p.69 prob. of spike at t time for neuron 1: prob. of spike at t time for neuron 2: 1 (t)ffit 2 (t)ffit prob. of simultaneous spike at time t: 12 (t; t)(ffit) 2 independence: 12 (t; t) = 1 (t) 2 (t)

75 12 (t; t) = (t) 1 (t) 2 (t). p.69 prob. of spike at t time for neuron 1: prob. of spike at t time for neuron 2: 1 (t)ffit 2 (t)ffit prob. of simultaneous spike at time t: (t; t)(ffit) 2 independence: (t; t) = 1 (t) 2 (t) replace independence with ratio of joint firing rate to firing rate under independence. (t) H Independence : (t) = 1 0 for t all

76 . p.70 Test H 0 : (t) = 1 for all t 12 (t; t) = (t) 1 (t) 2 (t) (i) (t) estimate (ii) form confidence bands under H 0 using the Bootstrap (iii) assess whether the excursion of (t) outside the bands may be due to chance, i.e., obtain p-value

77 . p.71 ζ estimate of ζ Time (ms)

78 . p.72 One Bootstrap sample Suppose we n have trials of paired recordings. 1. Shuffle the trials of neuron 1 data. 2. Shuffle the trials of neuron 2 data. 3. Pair the shuffled trials to n obtain new trials. 4. Compute an estimate of from the new data. Repeat 1000 times

79 . p.73 The Bootstrap Re-samples the data (in some modified form) repeatedly; Uses the many data replications (here, trials) to capture most of the information about relevant unknown data distributions (here, spike train distributions); Assumes sampled units (here, trials) are statistically identical;

80 . p.74 The Bootstrap Re-samples the data (in some modified form) repeatedly; Uses the many data replications (here, trials) to capture most of the information about relevant unknown data distributions (here, spike train distributions); Assumes sampled units (here, trials) are statistically identical; Is not hard to mis-use!

81 . p.75 Performance of Procedure vs. Bin-wise method We make a particular assumption about (t); We generate data artificially; We compute the probability of correctly rejecting H 0, i.e., we compute the power of the test.

82 . p.76 Performance of Procedure vs. Bin-wise method We make a particular assumption about (t); We generate data artificially; We compute the probability of correctly rejecting H 0, i.e., we compute the power of the test. Bin-wise method: obtain a threshold and reject H 0 whenever the normalized correlation exceeds the threshold for two contiguous bins. (A fair comparison of power requires probability of incorrectly rejecting H 0, i.e. probability of Type I error, to be same for two methods.)

83 . p.77 Power Bootstrap Bin wise need 4 times more trials to achieve the same power Maximum percentage of excess firing (%)

84 . p.78 Additional Work on Correlated Spiking Aim: Distinguish within-trial association from excitability. Model for r-th trial: r (t) = (t)g r (t) Two neurons can share excitability or have different excitabilities. We can incorporate IMI effects.

85 Time (ms) firing rate rate Time lag Time (ms) Average counts z Time lag Time (ms). p.79 Average counts z

86 . p.80

87 . p.81 Summary Smoothing methods increase efficiency. BARS smoothing can further increase efficiency. Methods may be adapted for within-trial analyses, using IMI models. Inferential framework can be useful for population analysis. Bayesian decoding increases efficiency of movement prediction. These ideas may be applied to understanding association of pairs of neurons.

88 . p.82 Current/Future Interests Further testing, refinements, and software. Algorithms for neural prostheses. Evolving correlation among multiple neurons.