Statistical models of visual neurons

Transcription

1 Statistical models of visual neurons Final Presentation Anna Sotnikova Applied Mathematics and Statistics, and Scientific Computation program Advisor: Dr. Daniel A. Butts Department of Biology 1

2 The power of transistor-based computing barely competes with brainpower of a mouse Inspired by Ray Kurzweil s book The Singularity is Near 2

3 Visual system of a neuron light electrical spikes dt stimulus neural response computation stimulus number of spikes time (sec) General goal: find a functional link between stimulus and response time time (sec) time

4 Statistical modeling of a neuron s response? stimulus firing rate Poisson process: average # of spikes is given by the firing rate <n(t)> = r(t)dt Prob(n(t)) = r(t)n(t) n(t)! exp( r(t)) n(t) - # of spikes between t, t+dt n(t) Identify a model for the firing rate 4

5 Project goals Implement 5 specific models: 1. Linear models Spike Triggered Average (STA) 2. Quadratic models Generalized Linear Model (GLM) (Fall semester) Spike Triggered Covariance (STC) 3. Cascade model Generalized Quadratic Model (GQM) (Spring semester) Nonlinear Input Model (NIM) Test models on 3 data sets: 1. Model-specific synthetic data to validate all algorithms 2. Synthetic Retina ganglion cells (RGC) data to test NIM model 3. Experimental Lateral geniculate body (LGN) to test GLM model 5

6 Project goals Implement 5 specific models: 1. Linear models Spike Triggered Average (STA) 2. Quadratic models Generalized Linear Model (GLM) (Fall semester) Spike Triggered Covariance (STC) 3. Cascade model Generalized Quadratic Model (GQM) (Spring semester) Nonlinear Input Model (NIM) Test models on 3 data sets: 1. Model-specific synthetic data to validate all algorithms 2. Synthetic Retina ganglion cells (RGC) data to test NIM model 3. Experimental Lateral geniculate body (LGN) to test GLM model 6

7 Generalized Linear Model (GLM): a single linear filter (k) + history filter (h) s(t, ) =(S(t ),...,S(t)) n obs (t, ) =(n(t ),...,n(t)) k linear filter: models receptive field of a neuron h history filter: models neuron s memory of spikes Spiking non-linearity: models neuron s non-linear processing firing rate model r(t) =F (k s(t, )+h n obs (t, )+b) 7 Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression.

8 How to fit parameters to a probabilistic data? number of spikes number of spikes another run Neural response time(sec) 8

9 Maximum Likelihood estimation P (N ) = Y t (r(t)) n(t) n(t)! exp( r(t)) Poisson distribution N = {n(t)} = {r(t)} LL( ) =log(p (N )) = X t n(t)log(r(t)) X r(t) t GLM model specifically: r(t) =F (k s(t, )+h n obs (t, )+b) = {k, h,b} F () = exp() Maximize log-likelihood using gradient ascent method LL( ) = X t X n(t)(k s(t)+h n obs (t)+b) exp(k s(t)+h n obs (t)+b) t 9

10 Synthetic data for GLM algorithm validation Step 1: generate white noise stimulus s(t) stimulus time (sec) Step 2: calculate r(t) using test filters and test function F: F(x) = Exp(x) Step 3: generate Poisson spikes n(t) using calculated r(t) 1

11 Use synthetic GLM data without history to recover a decaying linear k-filter created filter optimization (no history) Created filter vs optimization result Stimulus more important to the neuron.7 filter value Stimulus less important to the neuron time step time (j in the formula) 11 spike event

12 Use synthetic GLM data without history to recover oscillating linear k-filter 1 Created filter vs optimization result created filter optimization (no history) filter value time step time (j in the formula) 12 spike event

13 Use synthetic GLM data with history to recover k filter vs optimization result for generated data k filter optimization both k (linear) & h (history) filters time step time linear filter k history filter (h<) h filter and optimization result for generated data h filter optimization Neuron cannot fire after it just did time step 13 time

14 History cannot be ignored! (synthetic GLM data with exponential history filter) 1 used filter optmization result with history optimization result without history used k and found k (k,b,h/k,b,h case) h filter and optimization result for generated data h filter optimization -.6 linear k-filter value history filter time step time 14 spike event

15 RGC data (no history): GLM optimal filter matches STA STA = spiked-triggered average stimulus 3 STA filter vs optimization result for filter k 2 STA found filter k filter value STA = X j s(t j, )n(t j ) time step spike event True for any Gaussian fluctuating variable (see report for details) 15

16 LGN data: regularization of the search algorithm.4 High resolution, k filter, no regularization Use a priori information that k-filter must be a smooth filter. Penalize LL for large gradients time step RLL( ) = X t n(t)(k s(t)+h n obs (t)+b) X exp(k s(t)+h n obs (t)+b) X (k i k i 1 ) 2 t i Find optimal lambda by cross validation 16

17 LGN data: cross-validation for the choice of regularization parameter 545 Lambda validation for dt/ Negative LogLikelihood good choice for lambda lambda 17

18 LGN data: GLM implementation with regularization, k filter.4 High resolution, k filter, no regularization time step high resolution no regularization high resolution optimal regularization High resolution, k filter, optimal regularization time step

19 Main result on GLM: recovered history term from LGN data 1 Filter h for dt/ spike event -5 refractory period of a neuron detected with a real LGN data time (sec) Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression. 19

20 Non-linear Input Model (NIM) motivation primary neurons secondary neuron NIM is based on the hypothesis that the dominant nonlinearities imposed by the physiological mechanisms arise from rectification of neuron s inputs Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression. 2

21 Non-linear Input Model (NIM) r(t) =F ( X i! i f(k i s(t, ))) w = 1 f F w = -1 f 21 Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression.

22 Synthetic data for NIM algorithm validation Step 1: generate white noise stimulus s(t) stimulus time (sec) Step 2: calculate r(t) using two test filters k1 & k2 and various combinations for test functions F and f: F(x) = Exp(x) f(x) = Log(1+Exp(x)) f(x) =, x<; x, x>= Step 3: generate Poisson spikes n(t) using calculated r(t) 22

23 Validation of NIM algorithm using synthetic NIM data F(x) =Exp(x) f(x) =, x<; x, x>= filter 2 filter 1 23

24 Details of rectifying non-linearity are not too important Generate: F(x) = Exp(x) f(x) = Log (1+Exp(x)) Recover: F(x) = Exp(x) f(x) =, x<; x, x>= filter 2 filter 1 24

25 Recovering model parameters from RGC synthetic data.6.4 filter 1 optimized filter 2 optimized true filter 2 true filter time step 25

26 RGC spiking output vs NIM-recovered spiking output 3 measured (given) estimated (optimization) firing rate time step 26 Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression.

27 Generalized Quadratic Model (GQM) r(t) =F (k L s(t, )+ X i! i (k i s(t, )) 2 ) Spiking non-linearity Linear filter A set of (a few) quadratic filters w = +1/-1 A competing model to NIM with minimal (quadratic) modification to simple linear filtering 27

28 Comparison of GLM, NIM, GQM for a single RGC data set Negative LL values GLM GQM NIM 28

29 Summary Realized GLM model on both real and synthetic data - full algorithm validation on synthetic data - recovered linear filter that matched STA - detected a short refractory period with a history term - results matched with the paper: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression. Realized GQM on synthetic data (1 linear/2 quadratic) Realized NIM model on synthetic data (2 rectified terms) - GQM does not recover correct NIM filters - NIM finds correct filters irrespective non-linearity details 29

30 Updated project schedule October - mid November November Implement STA and STC models Test models on synthetic data set and validate models on real data set November - December December - mid February Implement Generalized Linear Model (GLM) Test model on synthetic data set and validate model on LGN data set January - March mid February - mid April Implement Generalized Quadratic Model (GQM) and Nonlinear Input Model (NIM) Test models on synthetic data set and validate models on LGN data set April - May Mid April - May Collect results and prepare final report 3

31 Implementation Hardware MacBook Air, 1.4 GHz Intel Core i5, 4 GB 16 MHz DDR3 Software Matlab_R215b 31

32 Deliverables Code for STA and STC Code for GLM Code for GQM Code for NIM Validation codes for all models Reports and presentations 32

33 References 1. McFarland JM, Cui Y, Butts DA (213) Inferring nonlinear neuronal computation based on physiologically plausible inputs. PLoS Computational Biology 9(7): e Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (211) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression. J. Neurosci. 31: Simoncelli EP, Pillow J, Paninski L, Schwartz O (24) Characterization of neural responses with stochastic stimuli. In: The cognitive neurosciences (Gazzaniga M, ed), pp Cambridge, MA: MIT. 4. Paninski, L., Pillow, J., and Lewi, J. (26). Statistical models for neural encoding, decoding, and optimal stimulus design. 5. Shlens, J. (28). Notes on Generalized Linear Models of Neurons. 33

34 GQM algorithm validation is the same as GLM synthesize data with 2 (+) quadratic filters search for 2 (+) quadratic filters nd quadratic filter validation used quadratic filter optimization result st quadratic filter validation used quadratic filter optimization result time step used linear filter optimization result Linear filter validation time step 34 linear filter time step

35 GQM algorithm validation is the same as GLM synthesize data with 2 (+) quadratic filters search for 3 (+) quadratic filters nd quadratic filter validation original optimized 1 optimized 2.5 1st quadratic filter validation original optimized Found 2 identical filters consistent with 2nd filter time step Linear filter validation original optimized First q-filter matches time step 35 linear filter matches time step