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## Decoding

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**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**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.**Reading minds: the LGN**Yang Dan, UC Berkeley**Motion detection task: two-state forced choice**Britten et al. ‘92: behavioral monkey data + neural responses**Behavioral performance**Neural data at different coherences Discriminability: d’ = ( <r>+ - <r>- )/ sr**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**ROC curves: summarize performance of test for**different thresholds z Want b 1, a 0.**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).**P[correct] = ∫∞0 dz p[z|-] b(z).**As a = probability that r > z for a - stimulus, da = -p[z|-] dz, P[correct] = ∫ dab.**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.**Close correspondence between neural and behaviour..**Why so many neurons? Correlations limit performance.**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**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[+] .**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]**“Score”**Compared to threshold For small separations between stimulus values, s and s + ds