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Neural Computation Chapter 3

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Neural Computation Outline Comparison of behavioral and neural response on a discrimination task –Bayes rule –ROC curves –Neyman Pearson Lemma Population decoding –Cricket cercal system –Monkey M1 motoneurons Optimal decoding –MAP and Bayesian estimates –Relation to population vector –Fisher information

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Neural Computation Bayes’ rule Let s denote a stimulus and r=(r 1,…,r N ) denote the response of one or more neurons. We define –The stimulus probability p(s) –The response probability p(r) –The joint probability p(r,s) and conditional probabilities p(r|s) and p(s|r) Bayes rule:

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Neural Computation Discrimination of movements Stimulus is moving dot pattern with variable % of coherently moving dots. Monkey behavioral forced choice task to report direction of motion (+ or -) as a function of coherence in the stimulus (filled circles) Monkey decides on basis of neural response. Open circles are optimal discrimination performance given the neural response data

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Neural Computation Two alternative forced choice

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Neural Computation

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Optimal decision: ROC curve The classification requires the definition of a threshold. The threshold z affects the classification performance: –Define the false alarm rate (size) =p(r>z|s=-) and hit rate (power) =p(r>z|s=+) –ROC (‘receiver operating characteristic’) plot (z) vs. (z) Area under curve s d / classification performance: –½ is random guessing –1 is perfect classification

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Neural Computation Two alternative forced choice Given a stimulus s=+ and the response in two neurons p+ and p-, That give rates r+ and r- respectively. Stimulus is classified by the highest rate. What is the probability of correct classification?

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Neural Computation Two alternative forced choice White circles are p(correct) as a function of stimulus coherence Monkeys response is as if based on two alternative forced choice

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Neural Computation Discriminability

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Neural Computation Discriminability (details) Discuss ex. 3.1

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Neural Computation Likelihood ratio

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Neural Computation Neyman-Pearson Lemma

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Neural Computation Neyman-Pearson Lemma

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Neural Computation Neyman-Pearson Lemma

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Neural Computation Cricket cercal system Consider response of a population of neurons p(r 1,…,r N |s) Consider a stimulus that is parametrized by a continuous value Cricket cercal system –Hair cells send spike when deflected by wind –4 inter-neurons receive input from thousands of hair cells

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Neural Computation Cricket cercal system

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Neural Computation Cricket cercal system

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Neural Computation Monkey primary motor cortex (M1)

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Neural Computation Monkey primary motor cortex (M1)

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Neural Computation Optimal decoding

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Neural Computation Optimal decoding

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Optimal decoding details log p(r|s) \propto \sum_{a=1}^4 (r_a-f_a(s))^2 Assume f_a(s)=c_{ai} v_i Then log p(r|s) as a function of v has maximum at Sum_{ja} c_{ia} c_{ja} v_j = sum_a c_{ia} r_a This is the MAP estimate Bayesian estimate requires p(s|r) which is a normalized version of p(r|s) as a function of s. Neural Computation

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Bayesian vs. Population vector decoding

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Neural Computation Bayesian vs. Population vector decoding

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Neural Computation Bayesian vs. Population vector decoding

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Neural Computation Bias and variance

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Neural Computation Bias and variance

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Neural Computation Fisher information

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Neural Computation

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Fisher information

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Neural Computation Fisher information

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Neural Computation Fisher information

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Neural Computation

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S_est=sum_i a_i r_i sum_i a_i=1 r_i is unbiased estimator of s = sum_i a_i s= s Sigma^2_est=sum_i a_i^2 sigma^2 A_i =1/n -> Sigma^2_est = sigma^2/n optimal A_i different is suboptimal.

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Neural Computation Fisher information

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Neural Computation Fisher information

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Neural Computation

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Spike Train decoding

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Neural Computation Spike Train decoding

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Neural Computation Spike Train decoding

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Neural Computation Summary Decoding of stimulus from response –Two choice case Discrimination ROC curves –Population decoding MAP and ML estimators Bias and variance Fisher information, Cramer-Rao bound –Spike train decoding

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Neural Computation

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