Neural Computation Chapter 3
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
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:
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
Neural Computation Two alternative forced choice
Neural Computation
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
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?
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
Neural Computation Discriminability
Neural Computation Discriminability (details) Discuss ex. 3.1
Neural Computation Likelihood ratio
Neural Computation Neyman-Pearson Lemma
Neural Computation Neyman-Pearson Lemma
Neural Computation Neyman-Pearson Lemma
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
Neural Computation Cricket cercal system
Neural Computation Cricket cercal system
Neural Computation Monkey primary motor cortex (M1)
Neural Computation Monkey primary motor cortex (M1)
Neural Computation Optimal decoding
Neural Computation Optimal decoding
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
Bayesian vs. Population vector decoding
Neural Computation Bayesian vs. Population vector decoding
Neural Computation Bayesian vs. Population vector decoding
Neural Computation Bias and variance
Neural Computation Bias and variance
Neural Computation Fisher information
Neural Computation
Fisher information
Neural Computation Fisher information
Neural Computation Fisher information
Neural Computation
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.
Neural Computation Fisher information
Neural Computation Fisher information
Neural Computation
Spike Train decoding
Neural Computation Spike Train decoding
Neural Computation Spike Train decoding
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
Neural Computation