The Bayesian Brain. Robert Jacobs Department of Brain & Cognitive Sciences University of Rochester. May 11, 2017

Transcription

1 The Bayesian Brain Robert Jacobs Department of Brain & Cognitive Sciences University of Rochester May 11, 2017

2 Bayesian Brain How do neurons represent the states of the world? How do neurons represent likelihood functions? How do neurons use these representations to calculate posterior distributions?

3 Biology of a Single Neuron Think of an individual neuron as an input-output device Neurons transmit electrical impulses (action potentials; aka spikes) away from their cell bodies along output structures (axons) When a spike reaches the end of an axon (axon terminal), it induces the release of neurotransmitter molecules that diffuse across a narrow synaptic cleft and bind to receptors on the input structure (dendrites) of the postsynaptic neuron

4 Effect of the transmitter released by a given input neuron may be inhibitory (reducing the probability that the postsynaptic neuron will fire spikes of its own) or excitatory (increasing the probability that the postsynaptic neuron will fire) depending on the transmitter released and on the receptor that binds it Each neuron receives inputs from a large number of other (presynaptic) neurons Each neuron sends outputs to a large number of other (postsynaptic) neurons

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8 Each input neuron fires a number of spikes over a relevant time interval These result in neurotransmitter release onto our neuron of interest. This postsynaptic neuron integrates these inputs and produces an output spike train of its own We will focus on the number of output spikes in a time interval, either in response to a stimulus or in response to the numbers of input spikes it receives from other neurons

9 Neural Coding Let s denote the sensory stimulus Special neurons (sensory receptors) convert the stimulus into a pattern of spikes For example, photoreceptors are neurons that convert a pattern of light falling on the back of the retina into a pattern of spikes. Each individual photoreceptor responds to the amount of light falling on a very small region of the retina

10 Neural Coding Important: Even when the stimulus s is held constant, the pattern of spiking activities in any set of neurons will be variable Goal of neural coding: Characterize the (stochastic) mapping from stimulus s to neural responses r Because of the stochastic nature of this mapping, it is characterized as a probability distribution, denoted p(r s) We do not currently understand why this mapping is stochastic That is, we do not understand the noise factors contributing to the stochastic nature of neural responses

11 Neural Tuning Curve The tuning curve of a neuron describes its mean response to stimulus s For example, let the stimulus be an oriented Gabor: If the orientation is 90 (s = 90 ), the neuron may emit (on average) 10 spikes (r = 10) If s = 100, then r = 40 If s = 110, then r = 80 If s = 120, then r = 40 If s = 130, then r = 10 On average, this neuron is most activated by a Gabor whose orientation is 110 (i.e., s pref = 110 )

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17 Example: Bell-Shaped Tuning Curve In primary visual cortex (area V1), neurons are tuned to the orientation of a visual stimulus. The tuning curve is typically unimodal and symmetric around the preferred orientation. Von Mises (circular Gaussian) function: f i (s) = g e κ(cos(s s i) 1) + b where b is an offset (baseline level of activity), g is a gain, s i is neuron i s preferred orientation, and κ is a concentration parameter (the higher κ, the more narrowly tuned the neuron)

18 Example: Monotonic Tuning Curve Simplest form is a rectified linear function: f i (s) = [g s + b] + where g is positive for monotonically increasing curves and negative for monotonically decreasing curves Problem: function is unbounded (may need to place a ceiling on function value)

19 Neural Variability

20 Example: Poisson Variability Need to characterize variability in neuron s spike count (non-negative integer) Mean spike count is f i (s) (not necessarily an integer) On any given trial, actual spike count will vary around f i (s). For every possible count r, we seek its probability: p(r i s) = 1 r i! e f i(s) f i (s) r i Note: mean = variance = f i (s)

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22 Population Codes Neural populations are groups of neurons that are all selective for a particular stimulus feature (e.g., orientation) Typically have similarly-shaped tuning curves but with different preferred stimulus values Population code is the vector of stimulus-evoked firing rates of the neurons in a neural population (denoted r = [r 1,..., r N ] T ) Neural coding: need to characterize p(r s)

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24 Independent Poisson Variability in a Population Assume: For a given stimulus, neuron responses are independent: p(r s) = N p(r i s) i=1 If we also assume Poisson variability, then: p(r s) = N i=1 1 r i! e f i(s) f i (s) r i

25 Problem (Problem 10.3 from draft of textbook by Ma, Körding, and Goldreich) We assume a population of 9 independent Poisson neurons with Gaussian tuning curves and preferred orientations from -40 to 40 in steps of 10. The tuning curve parameters have values g = 10, b = 0, and σ tc = 20. A stimulus s = 0 is presented to this population. What is the probability that all neurons stay silent?

26 Solution Gaussian tuning curve: [ 1 f i (s) = g exp 1 ] (s s 2πσtc 2σtc 2 i ) 2 + b Thus p(r = 0 s = 0 ) = 9 i=1 e f i(s) = e f 1(s) f 2 (s) f 9 (s)

27 Problem (modified from Problem 10.5 from draft of textbook by Ma, Körding, and Goldreich) In a population of 1000 independent neurons with Gaussian tuning curves (let g = 10, b = 0, and σ tc = 20), examine the claim that Σ i f i (s) is more or less independent of s. Given certain assumptions, Σ i f i (s) is, in fact, independent of s. We will use this important fact later in this lecture.

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29 Sum of Neurons Mean Responses Stimulus

30 Problem (modified from Problem from draft of textbook by Ma, Körding, and Goldreich) Define 1000 time points. At each time point, determine whether a spike is fired by generating a random number according to a Poisson distribution (use poissrnd function in Matlab) that leads to a yes probability of (this corresponds to a mean of 3.2 spikes over all 1000 time points). Count the total number of spikes generated. Repeat for 10,000 trials. Plot a histogram of spike counts and compare to Figure 10.3a.

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32 Repeat for a mean of 9.5 spikes. Compare the resulting histogram to Figure 10.3b.

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34 A property of a Poisson process is that the time between two subsequent spikes (interspike interval, denoted t) follows an exponential distribution: p( t) = exp( t/λ)/λ, where λ is the mean of the Poisson process. Verify this by plotting the histogram of interspike intervals across all Poisson spike trains generated above.

35 Mean = 3.2 spikes

36 Mean = 9.5 spikes 6 x

37 Neural Decoding Neural Decoding: Given neural responses r, what can we say about the external stimulus s? Goal: Infer posterior distribution p(s r) via Bayes rule Is this like mind reading? For example, given a person s neural responses, we want to infer what the person is looking at

38 Neural Likelihood Function for a Single Neuron Recall that the probability that a neuron fires r spikes in a given time interval (based on a Poisson noise distribution) is: p(r s) = 1 r! e f(s) f(s) r Suppose that 11 spikes are observed (r = 11). Given r, the neural likelihood of a hypothesized stimulus value s is the probability that r spikes were produced by that value of s. That is, the likelihood of s is: L(s r) = p(r s)

39 Example: Gaussian Tuning Curve Suppose that a neuron has a Gaussian tuning curve (with mean s pref = 0, width σ tc = 10, and gain g = 5: [ f(s) = g exp 1 ] (s s 2σtc 2 pref ) 2 + b 1 For the purposes of this slide, the constant 2πσtc (normally appearing in a Gaussian distribution) has been absorbed into the constant gain g

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41 Intuition Behind Likelihood Suppose that neuron fires 4 spikes. What can we conclude about stimulus s? r = 4 could have been produced by, for example, s = 10...or, with equal probability, it could have been produced by s = 10...or, with much lower probability, it could have been produced by s = 21

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43 Unlike the likelihood functions frequently encountered in previous classes, this likelihood function is not Gaussian This likelihood function is not normalized In general, likelihood functions do not need to be normalized. Recall that they are not probability distributions. L(s r) (which equals p(r s)) is a function of the unknown stimulus value s, not the known rate count r

44 Likelihood functions for several different values of r

45 Example: Britten et al. (1992) Monkeys viewed random-dot kinematograms:

46 For each neuron in area MT, consider two directions of motion: Direction that produced the maximum response in that neuron Opposite direction Behavioral task: Monkey s task was to discriminate between these two motion directions Q: Can we decode the neuron s responses in order to discriminate between these two directions?

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48 Percent correct of monkey is about the same as the percent correct of the single neuron (!!!) But, but, but...monkey has lots of neurons. Shouldn t its performance be better than that of an individual neuron?

49 Likelihood Function for a Population of Neurons Assuming neurons are independent with Poisson noise: L(s r) = p(r s) = = N p(r i s) i=1 N i=1 1 r i! e f i(s) f i (s) r i

50 Log likelihood function: log L(s r) = N i=1 ( ) 1 log r i! e f i(s) f i (s) r i N = log r! i=1 N f i (s) + i=1 N r i log f i (s) i=1

51 In General... Likelihood values can be very small. These values are products of probabilities (numbers less than one) Likelihood functions can be very far from Gaussian. Some functions have multiple maxima, or a flat top, or are skew (i.e., long tails). If the number of neurons is large, the likelihood function is likely to have a dominant peak and may look (roughly) Gaussian The likelihood function is different on every trial, even if the stimulus is kept the same. This is because the likelihood function is determined by the pattern of neural activity on a trial, and this pattern varies stochastically from trial to trial

52 Neural Likelihood Function and Sensory Integration At a conceptual level, the computation of a likelihood function based on a set of neurons with independent noise is similar to sensory integration (or cue combination) as discussed earlier in the semester A neuron s spike count is analogous to a sensory or cue measurement The likelihood is obtained by multiplying the likelihoods from the individual neurons, just as the likelihoods from individual cues are multiplied together in cue combination

53 Example: Cercal System of Crickets Cercal system to sense the direction of incoming air currents. Four neurons represent this direction at low velocities.

54 Neural tuning curves are well-approximated by half-wave rectified cosine functions: Change of notation: f(s) = r max [cos(s s pref )] + Let spatial vector v point parallel to the wind velocity and have unit length (i.e., v = 1) Let c i (of unit length) denote the preferred direction of neuron i

55 In this case: f i (s) = r max [ v c i ] + Neural decoding: The wind direction represented by the four neurons is: 4 ( ) ri v pop = c i i=1 r max Warning: In this example, neural decoding is performed in a heuristic (non-probabilistic) manner. (Nonetheless, it illustrates the concept of neural decoding based on a population of neurons.)

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57 fmri Example: Harrison and Tong (2009) Decoding can also be performed using fmri data Stimuli: Two orthogonal gratings (25 and 115 ) Two experiments on visual working memory Q: Can we decode fmri signals in order to estimate the contents of a person s visual working memory?

58 Experiment 1: On each trial: 1. Display 1 grating (picked at random) 2. Display other grating 3. Display cue (either 1 or 2) indicating which grating to remember second delay period 5. Display test grating 6. Response: Is test grating rotated clockwise or anticlockwise relative to the cued grating?

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61 Experiment 2: On each trial: 1. Display sequence of letters at center of screen with low-contrast gratings flashing in the surround 2. Response: Identify if sequence contained a J or a K (grating is task-irrelevant)

62 Experiment 1: green curve Experiment 2: red curve

63 Yes, we can decode fmri signals in order to estimate the contents of a person s visual working memory Decoding works well for both attended and unattended stimuli Lower-level areas (e.g., V1, V2) respond similarli regardless of whether a stimulus is attended. In contrast, higher-level areas (e.g., V3A-V4) are sensitive to whether a stimulus is attended. Warning: Decoding was performed using a non-probabilistic (linear support vector machine) method. (Nonetheless, this example illustrates the concept of neural decoding based on fmri signals from populations of neurons.)

64 Toy Model Next, make assumptions underlying a toy model that is unrealistic, but useful nonetheless Goal: Show that, given assumptions, neural likelihood function is Gaussian Assume neuron tuning curves are Gaussian (with b = 0), and are translated versions of each other Assume preferred stimuli of the neurons are equally and densely spaced across entire real line (there are thus infinitely many neurons)

65 Thus, neuron i s tuning function is: [ f i (s) = g exp 1 ] (s s 2σtc s pref ) 2

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67 Log Likelihood Function Recall the log likelihood function: N log L(s r) = log r! i=1 N f i (s) + i=1 N r i log f i (s) i=1 On a given trial, first term is a constant (i.e., it is independent of stimulus s) Second term is also (approximately) a constant (i.e., independent of s) This was demonstrated (via simulation) on an earlier slide Known as the constant sum approximation

68 log L(s r) = = = N i=1 N i=1 N i=1 [ r i log g exp 1 ] (s s 2σtc s pref ) 2 + constant ( r i log g 1 ) (s s 2σtc s pref ) 2 + constant r i log g 1 2σ s tc N r i (s s pref ) 2 + constant i=1 = 1 2σ s tc N r i (s s pref ) 2 + constant i=1 In last step, N i=1 r i log g is a constant (independent of s), and thus absorbed into additive constant

69 After re-arranging terms, it is easy to show: [ ] 1 L(s r) exp (s µ likelihood ) 2 2σ 2 likelihood µ likelihood = N i=1 r i s pref,i N i=1 r i σ s likelihood = σ 2 tc N i=1 r i Likelihood is an (unnormalized) Gaussian (!!!)

70 Maximum Likelihood Estimate In other words, the maximum likelihood estimate of s is: ŝ MLE = N i=1 r i s pref,i N i=1 r i Width of this estimate ( sensory uncertainty ) is: σ likelihood = σ tc N i=1 r i

71 Higher the total spike count in the population, narrower the likelihood function (i.e., lower the sensory uncertainty)

72 MLE estimate and width varies from trial to trial (even when the stimulus value s is constant) What is the distribution of errors (over many trials) in the MLE estimate?

73 What is the distribution of widths (over many trials) in the MLE estimate?

74 Relationships between Behavioral and Neural Models

75 Think of the neural model as generating quantities that are used by the behavioral model The neural quantity µ likelihood is used as the measurement x in the behavioral model The neural quantity σlikelihood 2 is used as the error σ in the measurement x in the behavioral model

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77 Problem (modified from Problem 10.3 [in Chapter 10, not 9] from draft of textbook by Ma, Kording, and Goldreich) Simulate a population of 1000 independent Poisson neurons with Gaussian tuning curves (let g = 1 [also try g = 10], b = 0, and σ tc = 20). Simulate the toy model of this chapter for 10,000 trials (all at s=0) and create a scatter plot of the squared error of the maximum likelihood estimate versus the squared width (variance) of the likelihood function. What is the correlation coefficient?

78 gain: g = Variance of Likelihood Function Squared Error of ML Estimate

79 gain: g = Variance of Likelihood Function Squared Error of ML Estimate

80 Last Topic...Cue Combination Suppose that we want to combine visual and auditory information to the location of an event One neural population encodes location based on the visual information. The activities of these neurons are denoted r V Another neural population encodes location based on the auditory information. The activities of these neurons are denoted r A

81 Assumptions: Each modality-specific population has the same number of neurons Neurons in each area have the same tuning curves f i (s) Neural activities in each area are corrupted by Poisson noise The two populations may have different gains. The mean activities of the visual neurons are g V f i (s) and the mean activities of the auditory neurons are g A f i (s) The population with the higher gain will produce more spikes, and thus should be more reliable

82 Assuming that the two populations are conditionally independent given the stimulus value s, then p(s r V, r A ) p(r V, r A s) = p(r V s) p(r A s)

83 Recalling that neural responses follow a Poisson distribution, and ignoring all factors that do not depend on s: ( N ) ( N ) p(s r V, r A ) e g V f i (s) f i (s) r V,i e g Af i (s) f i (s) r A,i i=1 i=1 [ N ] = exp ( (g V + g A )f i (s) + (r V,i + r A,i ) log f i (s) i=1 = N e (g V +g A )f i (s) f i (s) r V,i+r A,i i=1

84 Q: What is this last equation? Suppose a new population of neurons sums the activities of corresponding pairs of neurons in the visual and auditory populations r V A = r V + r A The new population activities r V A also obey independent Poisson variability (because the sum of two Poisson random variables is also a Poisson random variable) This new population encodes p(s r V, r A ) (!!!)

85 Fetsch, C. R., DeAngelis, G. C., & Angelaki, D. E. (2013). Bridging the gap between theories of sensory cue integration and the physiology of multisensory neurons. Nature Review Neuroscience, 14,

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