Decoding

1. Decoding • How well can we learn what the stimulus is by looking • at the neural responses? • Two approaches: • devise explicit algorithms for extracting • a stimulus estimate • directly quantify the relationship between • stimulus and response using information theory

2. Solving for K(t), Predicting the firing rate Starting with a rate response, r(t) and a stimulus, s(t), the optimal linear estimator finds the best kernel K such that: is close to r(t), in the least squares sense.

3. Stimulus reconstruction K t

4. Stimulus reconstruction

5. Stimulus reconstruction

6. Reading minds: the LGN Yang Dan, UC Berkeley

7. Motion detection task: two-state forced choice Britten et al. ‘92: behavioral monkey data + neural responses

8. Behavioral performance Neural data at different coherences Discriminability: d’ = ( <r>+ - <r>- )/ sr

9. z p(r|-) p(r|+) <r>+ <r>- Signal detection theory: r is the “test”. Decoding corresponds to comparing test to threshold. a(z) = P[ r ≥ z|-] false alarm rate, “size” b(z) = P[ r ≥ z|+] hit rate, “power” Find by maximizing P[correct] = (b(z) + 1 – a(z))/2

10. ROC curves: summarize performance of test for different thresholds z Want b 1, a  0.

11. The area under the ROC curve corresponds to P[correct] for a two-alternative forced choice task: Response on one presentation acts as threshold for the other. Say present + then -; procedure will be correct if r1 > z where z = r2, true with probability P[r1 > z | +] = b(z), z = r2. total probability correct is this times probability of obtaining a particular value r2 = z Probability of r2 being in range z to z+dz, p[z|-]dz Then P[correct] = ∫∞0 dz p[z|-] b(z).

12. P[correct] = ∫∞0 dz p[z|-] b(z). As a = probability that r > z for a - stimulus, da = -p[z|-] dz, P[correct] = ∫ dab.

13. Appearance of the logistic function: If p[r|+] and p[r|-] are both Gaussian, P[correct] = ½ erfc(-d’/2). To interpret results as two-alternative forced choice, need simultaneous responses from + neuron and from – neuron. Get “- neuron” responses from same neuron in response to – stimulus. Ideal observer: performs as area under ROC curve.

14. Close correspondence between neural and behaviour.. Why so many neurons? Correlations limit performance.

15. What is the best test function to use? (other than r) • The optimal test function is the likelihood ratio, • l(r) = p[r|+] / p[r|-]. • (Neyman-Pearson lemma) • Note that • l(z) = (db/dz) / (da/dz) = db/da • i.e. slope of ROC curve

16. Likelihood as loss minimisation: Penalty for incorrect answer: L+, L- Observe r; Expected loss Loss+ = L+P[-|r] Loss- = L-P[+|r] Cut your losses: answer + when Loss+ < Loss- i.e. L+P[-|r] > L-P[+|r]. Using Bayes’, P[+|r] = p[r|+]P[+]/p(r); P[-|r] = p[r|-]P[-]/p(r);  l(r) = p[r|+]/p[r|-] > L+P[-] / L-P[+] .

17. Again, if p[r|-] and p[r|+] are Gaussian, • P[+] and P[-] are equal, • P[+|r] = 1/ [1 + exp(-d’ (r - <r>)/ s)]. • d’ is the slope of the sigmoidal fitted to P[+|r]

18. “Score” Compared to threshold For small separations between stimulus values, s and s + ds