1. 6. Population Codes Presented by Rhee, Je-Keun © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/1

2. Preview The standard view of population codes  How populations of neurons encode information about single variables  How this information can be decoded from the population activity An alternative view - The information conveyed by the population has associated uncertainty due to noisy neural activity.  How neuronal populations may offer a rich representation of such things as uncertainty or probability distribution over the encoded stimulus © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 2

3. Homogeneous population code An (idealized ) example of the tuning curves of a population of orientation-tuned cells © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 3

4. Coding and Decoding On any give trial, the actual number of spikes r i is not exactly equal to what the tuning curve, f i (s), predicts because neural activity in the cortex is variable. In other words, r i is a random quantity (called a random variable) with mean =f i (s). Consider the simplest model for which the number of spikes r i has a Poisson distribution. The equation is called an encoding model of orientation, specifying how orientation information is encoded in the population response. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 4

5. Population Vector A simple heuristic method for decoding orientation is to say that cell i “votes” for a vector, c i, pointing in the direction of its preferred orientation s i with a strength determined by its activity r i. Then, the population vector,, is computed by pooling all votes, and an estimate can be derived from the direction of. The main problem with the population vector method is that it is not sensitive to the noise process that generates the actual rates r i from the mean rates f i (s). As a results, it tends to produce large errors. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 5

6. Error Quantification – bias and variance One popular approach entails computing the mean and standard deviation of the estimator over many trials in which the value of s was maintained constant. Ideally, the mean should be as close as possible to the true value of s on these trials. The difference between the mean of the estimate and the true value is called the bias, and the estimator is said to be unbiased when the bias is equal to zero. Likewise, the variance of the estimator should be as small as possible, that is, if the same stimulus s is presented over many trials, it is best to have as little variability as possible in the estimates. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 6

7. Maximum Likelihood Estimator The maximum likelihood estimator starts from the full probabilistic encoding model, which, by taking into account the noise corrupting the activities of the neurons, specifies the probability p(r|s) of observing activities r if the stimulus is s. In general, maximum likelihood outperforms the population vector estimator. However, estimating s i (the population vector method) can be done with a lot less data than are needed to estimate the full tuning curves. Also, if the estimate of f i (s) is poor, the performance of the estimator will suffer. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 7

8. Bayesian Estimates Bayesian estimates combine the likelihood p(r|s) with any prior information about the stimulus s to produce a posterior distribution p(s|r). Bayesian inference proceeds using a loss function L(s’,s), which indicates the cost of reporting s’ when the true value is s; it is optimal to decode to the value that minimizes the cost, averaged over the posterior distribution In most situations, the Bayesian and maximum likelihood decoding typically outperform the population vector approach by a rather large margin. In general, the greater the number of cells, the greater the accuracy with which the stimulus can be decoded by any method, since more cells can provide more information about s. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 8

9. Assessing the Quality of an Estimator : Fisher information The quality of an estimator can be assessed by computing the absolute value of the bias and the variance of the estimator. The best estimators are the ones minimizing these two measures. Assume that we are dealing with unbiased estimators Derive a lower bound related to a quantity known as Fisher information, I F (s) If Fisher information is large, the best decoder will recover s with a small variance, that is to say with a high accuracy. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 9

10. Assessing the Quality of an Estimator : Fisher information Another approach to quantify the quality of an estimator would be to compute the amount by which s must be changed in order to detect the change reliably (say 80% of the time) from observation of the spike counts, r. If this change,, is small, one can conclude that the neural code contains a lot of information about s. In psychophysics, is known as the discrimination threshold of an ideal observer of r. Interestingly, is inversely proportional to Fisher information. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 10

11. Representing Uncertainty with Population Codes The information conveyed by the population has associated uncertainty due to noisy neural activities, and this inherent uncertainty can be seen as a nuisance with respect to decoding the responses. The uncertain information can be described in terms of probability distributions. The term distributional population codes for population code representations of such probability distributions are used. The task of representing distributions can be seen as involving two spaces:  an explicit space of neural activities, and an implicit space in which a probability distribution over the random variable is associated with the neurons. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 11

12. Direct Encoding Perhaps the simplest method of representing a distribution is when the neural activity corresponds directly to the probability. For example, if the random variable s can take on two values, with probability p = P(s l ) and 1-p = P(s 2 ), then two neurons a and b could represent p by some simple function of their activities: p could be the difference in the spike counts of neurons a and b, or log p could be the ratio of spike counts Alternatively, the spike counts of neurons a and b could be seen as samples drawn from P(s l ) and P(s 2 ), respectively © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 12

13. Probabilistic Population Codes A second distributional population coding scheme is closely related to the Bayesian decoder. If the noise in the response of neurons in a large population is assumed to be independent, the law of large numbers dictates that this posterior distribution converges to a Gaussian. The mean of this posterior distribution is controlled by the position of the noisy hill of activity. When the noise is independent and follows a Poisson distribution, the standard deviation of the posterior distribution is controlled by the amplitude of the hill. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 13 : some arbitrary function g : the gain of the population activity r

14. Probabilistic Population Codes © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 14

15. Convolution Encoding An alternative form of distributional population coding involves a Bayesian formulation of the representation of any form of posterior distribution p(s|I) over the sensory input s given the encoded variable I. These coding schemes therefore are particularly suitable for a non- Gaussian distribution p(s|I), which cannot be characterized by a few parameters such as its mean and variance. One possibility inspired by the encoding of nonlinear functions is to represent the distribution using a convolution code, obtained by convolving the distribution with a particular set of kernel functions. The activity of neuron i by taking the dot product between a kernel function assigned to that neuron and the function being encoded. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 15

16. Convolution Encoding With the convolution code, one solution to decoding is to use deconvolution, a linear filtering operation which reverses the application of the kernel functions. An alternative to this linear decoding scheme for convolution codes is to adopt a probabilistic approach. one should not try to recover the most likely value of s, but rather the most likely distribution overs, p(s|I). This can be achieved using a nonlinear regression method such as the expectation-maximization (EM) algorithm. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 16

17. Convolution Decoding The activity r i of neuron i is considered to be a vote for a particular (usually probabilistic) decoding basis function p i (s). The advantage of this scheme is the straightforward decoding model One disadvantage is the difficulty of formulating an encoding model for which this decoder is the appropriate Bayes optimal approach. A second disadvantage of this scheme is shared with the linear deconvolution approach: it cannot readily recover the high frequencies that are important for sharply peaked distributions p(s|I), which arise in the case of ample information in I. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 17

18. Convolution Decoding Another form of convolutional decoding is suggested by maximum entropy, or random field approaches, such as the product of experts model, in which the decoded distribution is a product of contributions of each neuron (the experts) in the population. The decoded distribution is not directly given by this weighted combination, but instead is a normalized exponential of the sum. This scheme can be extended to allow spikes to form a representation of a temporally changing probability distribution © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 18 is 1 iff neuron i spiked at time is a decoding temporal kernel function