Models of visual neuron function. Quantitative Biology Course Lecture Dan Butts

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1 Models of visual neuron function Quantitative Biology Course Lecture Dan Butts 1

2 What is the neural code"? neurons 1,000-10,000 inputs Electrical activity: nerve impulses 2

3 ? How does neural activity relate to brain function? Use visual system: Well characterized Intuitive function Lateral Geniculate Nucleus (LGN) Population input to the cortex 3

4 Starting point: recordings in the LGN Retina 1. Still relatively simple non-linear transforms on stimulus 2. Population input to the visual cortex The Visual Cortex Lateral Geniculate Nucleus (LGN) Extracellular recordings (Jose-Manuel Alonso Lab) 4

5 Visual stimulus Cat-cam movies from Peter Koenig s Lab (Kayser et al, 2004) 5

6 LGN neuron responses 500 ms 6

7 Understanding and Decoding Neural Signals 7

8 Outline 1. Introduction to receptive fields 2. Building a visual neuron model The LN (Linear-Non-linear) model 3. The problem of temporal precision and the need for new statistical methods Maximum-likelihood modeling 4. Research: Application of maximum-likelihood modeling to explain precise timing of neuronal responses 8

9 Coding like the muscle Little contraction Muscle picture with motoneuron Lots of contraction 9

10 The receptive field LGN + RF Stim Center Surr Retina Multiplication LGN responses related to how much the stimulus matches the receptive field Linear comparison: Spatial Spatial receptive stimulus field R = x K sp ( x) S( x) 10

11 The receptive field LGN + RF Stim Center Surr Retina Multiplication LGN responses related to how much the stimulus matches the receptive field Linear comparison: Spatial Spatial receptive stimulus field R = x K sp ( x) S( x) 11

12 The receptive field LGN + RF Stim Center Surr Retina Multiplication Spatial features of image matter in relation to RF Spatial Spatial receptive stimulus field Neuron is tuned for a given stimulus over a certain range. 12

13 Circuitry of the retina 13

14 ...but vision involves motion Motion in the visual scene/ self motion Eye movements saccades, microsac. ocular drift 14

15 LGN response during movie Raster plot PSTH: peri-stimulus time histogram Firing Rate (Hz) Time (sec) 15

16 What does neural activity look like during a time-varying stimulus? Raster Plot Time (sec) 16

17 Visual neuron function: the spatiotemporal receptive field What stimuli are represented by a neuron s response? Spike-Triggered Average (STA) stimulus: receptive field -200 ms

18 Visual neuron function: the spatiotemporal receptive field What stimuli are represented by a neuron s response? Spike-Triggered Average (STA) stimulus: receptive field -200 ms

19 Implicit Problems with Modeling > Too many parameters ! ! ! ! ! !0.2!0.3! ! ! ! !0.1!0.15!0.2! !0.01!0.02! !0.3 30!

20 The temporal receptive field Spatiotemporal Stimuli -100 Average temporal stim Time (ms) Full-field stimuli -100 Average temporal stim Time (ms) 20

21 Outline 1. Introduction to receptive fields 2. Building a visual neuron model The LN (Linear-Non-linear) model 3. The problem of temporal precision and the need for new statistical methods Maximum-likelihood modeling 4. Research: Application of maximum-likelihood modeling to explain precise timing of neuronal responses 21

22 The temporal receptive field Full-field stimuli -100 Average temporal stim Time (ms) 22

23 How measure the receptive field? The spike-triggered average Full-field stimuli -100 Average temporal stim Stimulus-response cross-correlation k(τ) dt r(t) s(t τ) Time (ms) 23

24 Linear model predictions ms Temporal Stimulus (full-field) x Temporal Receptive Field = Filtered stimulus r est (t) =r 0 + dτ k(τ) s(t τ) 24

25 Mathematical result (take functional derivative of MSE) Layman s summary: In the presence of Gaussian noise (uncorrelated) stimuli, the best linear model for the neuron is proportional to the spike triggered average. r est (t) =r 0 + dτ k(τ) s(t τ) Mean Squared Error (MSE) MSE = t [r(t) r est (t)] 2 Stimulus-response cross-correlation k(τ) dt r(t) s(t τ) 25

26 Linear model predictions ms Temporal Stimulus (full-field) x Temporal Receptive Field = Filtered stimulus Firing Rate 500 Hz Observed firing rate ms 26

27 How map linear function to firing rate? LN (Linear-Non-Linear) model of encoding L inear Kernel K(τ) g(t) N on-linearity ν(g) Firing Rate λ(t) Stimulus s(t) Firing Rate 500 Hz Observed firing rate ms 27

28 Measuring reliability of RF model : the non-linearity ms 3 2 Pr{g spike} 1 g Pr{g} -3 28

29 Measuring reliability of RF model : the non-linearity Pr{g spike} Pr{g} LN Model of Encoding Firing Rate (Hz) ν 0 (g) Pr{spike g} α θ Generating Function g L inear Kernel K(τ) g(t) N on-linearity ν(g) Firing Rate λ(t) Stimulus s(t) 29

30 Bussgang s Theorem Layman s summary: In the context of simple non-linearities, the optimal receptive field is STIL given by the spike-triggered average in the context of Gaussian white noise *** 30

31 Receptive field predictions ms Temporal Stimulus (full-field) x Temporal Receptive Field = Filtered stimulus Firing Rate 500 Hz LN Model Observed firing rate ms 31

32 How quantify quality of model fit? Matlab Interlude 32

33 Problems with visual neuron modeling 1. Looking at higher time resolution reveals that the LN model is insufficient 2. Non-linearities force the use of Bussgang s theorem -> only use STA, and need noise stimuli 3. r2 is not the best measure for evaluating a nonlinear system (also, it requires multiple repeats to estimate a good firing rate) 33

34 Outline 1. Introduction to receptive fields 2. Building a visual neuron model The LN (Linear-Non-linear) model 3. The problem of temporal precision and the need for new statistical methods Maximum-likelihood modeling 4. Research: Application of maximum-likelihood modeling to explain precise timing of neuronal responses 34

35 Maximum Likelihood Approach Stimulus Likelihood: probability that model explains data Linear Kernel k Find the maximum likelihood : The probability that the spikes were generated by a model with a certain choice of parameters N on-linearity LL = t spk log Pr{spk t spk } t Pr{spk t} f P oisson process 500 Hz Firing rate when there is an observed spike Firing rate when there is no observed spike Response (Spike Train) Probability of spike 0 35

36 Problem: complicated function to Stimulus maximize! Linear Kernel The maximum likelihood: k LL = t spk log Pr{spk t spk } t Pr{spk t} N on-linearity P f oisson process Response (Spike Train) Paninski (2004): No local minima in likelihood surface given certain forms of non-linearity (f) Matlab can solve for optimal parameters in very little time! [e.g., 2 minutes of data, 0.5 ms resolution ~ 20 seconds] 36

37 LN model is fit optimally using maximum likelihood Quick Matlab Interlude 37

38 Outline 1. Introduction to receptive fields 2. Building a visual neuron model The LN (Linear-Non-linear) model 3. The problem of temporal precision and the need for new statistical methods Maximum-likelihood modeling 4. Research: Application of maximum-likelihood modeling to explain precise timing of neuronal responses 38

39 Receptive field predictions ms Temporal Stimulus (full-field) x Temporal Receptive Field = Filtered stimulus Neuronʼs Firing Rate 500 Hz LN Model Actual 0 39

40 Receptive field predictions Spike Rasters Neuron's Response "Function-based" Prediction Neuronʼs Firing Rate 500 Hz LN Model Actual 0 40

$during periods of stimulus that matches RF Refractoriness and Neural$

41 How to generate precision? 500 Hz 0 Neuron is tuned to the stimulus Need to suppress neuron s response during periods of stimulus that matches RF Refractoriness and Neural Precision e.g., Berry and Meister, 1998 Brillinger, 1992 Keat et al., 2001 Paninski, 2004 Pillow et al., Refractory effects 30 ms 41

42 Generalized Linear Model (GLM) Stimulus Paninski (2004): Linear Kernel k Optimal solution for model parameters (no matter how many parameters!) + N on-linearity P f oisson process Spike History Term w RP But, spike history term does not explain the temporal resolution of the data. Response (Spike Train) 42

43 GLM model does not explain LGN temporal precision 1200 Data (60 reps x-validated) LN GLM Despite matching ISI distributions Data LN Model +spike history 0.5 ms resolution Firing Rate (Hz) Interspike Interval (ms) Time (sec) 43

44 What about network suppression? 500 Hz 0 Refractoriness and precision model Network suppression model Stimulus Tuned Stimulus RF Spike generation Effects Spike-Refractory LGN neuron spikes Others neurons in network active Response (Spike Train) Suppression after spikes 44

45 What about network suppression? 500 Hz 0 Refractoriness and precision model Stimulus Network suppression model Stimulus RF RF 1 RF 2 Spike generation Effects Spike-Refractory Spike generation (Rectified) Suppression Response (Spike Train) Response (Spike Train) 45

46 Directly fit *multiple* receptive fields Stimulus RF 1 RF 2 -- Alternative to spiketriggered covariance Spike generation (Rectified) Suppression -- Simplest way to incorporate two RFs Response (Spike Train) -- Application to neurons that process stimuli non-linearly (e.g., on-off cells in mouse retina) 46

47 Precision explained by suppression Data (60 reps x-validated) LN GLM Suppression ms resolution

48 Cross-validation Fit model to 2 min Model test... 2 minutes of FF stimulation 30 sec unique sequence repeated 60 times 1.2 X-cells Y-cells (N = 13) (N = 10) GLM 0.4 GLM Sup Sup1 Response Time Scale (ms) Information (bits/spk) Significant improvement across all recorded neurons 3 X cells Ycells Information gain (bits/spk) 48

49 Precision computation Stimulus Linear Filter Suppressive filter ** RF 1 RF 2 Spike generation (Rectified) Suppression firing rate Response (Spike Train) - sup linear

50 long-range excitatory project ion General role of local inhibition? Retina Exc: bipolar cell Inh: spiking amacrine LGN V1 local inhibitory interneuron Exc: LGN input Inh: interneurons Exc: RGC input Inh: interneurons 50

51 Circuitry of the retina OUTER PLEXIFORM LAYER Role in spatial processing? INNER PLEXIFORM LAYER Role in temporal processing? 51

52 Conclusions/Parting Thoughts 1. Visual neuron modeling Don t forget the LN model -- it is everywhere Basis for sensory models in neuroscience ( minimal model ) 2. Neuroscience has been (but no longer is?) stuck with standard statistics Brought field to where it is (VERY USEFUL) but... could not go much further Neuroscience-statisticians are having large impact on basic science (Emory Brown, Liam Paninski, Rob Kass, Valerie Ventura, Han Amarsingham,... ) 3. Maximum likelihood modeling System of models that have smooth likelihood surface Ability to solve higher-order models with limited data 52

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