Reading population codes: a neural implementation of ideal observers

1. Reading population codes: a neural implementation of ideal observers Sophie Deneve, Peter Latham, and Alexandre Pouget

2. encode Stimulus (s) neurons Response (r) decode

3. Tuning curves • sensory and motor info often encoded in “tuning curves” • neurons give a characteristic “bell shaped” response

4. Difficulty of decoding • noisy neurons create variable responses to same stimuli • brain must estimate encoded variables from the “noisy hill” of a population response

5. Population vector estimator • assign each neuron a vector • vector length is proportional to activity • vector direction corresponds to preferred direction Sum vectors

6. Population vector estimator • Vector summation is equivalent to fitting a cosine function • peak of cosine is estimate of direction

7. How good is an estimator? • need to compare variance of estimator after repeated presentations to a lower bound • the maximum likelihood estimate gives the lower variance bound for a given amount of independent noise VS

8. encode Stimulus (s) neurons Response (r) decode

9. Maximum Likelihood Decoding Maximum likelihood estimator Decoding Encoding

10. Goal: biological ML estimator • recurrent neural network with broadly tuned units • can achieve ML estimate with noise independent of firing rate • can approximate ML estimate with activity-dependent noise

11. General Architecture Pλ Preferred Frequency • units are fully connected and are arranged in frequency columns and orientation rows • weights implement a 2-D Gaussian filter: 20 Preferred orientation PΘ 20

12. Input tuning curves • circular normal functions with some spontaneous activity: • Gaussian noise is added to inputs:

13. Unit updates & normalization • units are convolved with filter (local excitation) • responses are normalized divisively (global inhibition)

14. Results • Rapidly converges • strongly dependent on contrast

15. Results • sigmoidal response curve after 3 iterations, becomes a step after 20 • actual neuron

16. Noise Effects Flat Noise • Width of input tuning curve held constant • width of output tuning curve varied by adjusting spatial extent of the weights Proportional Noise

17. Analysis Flat Noise Q1: Why does the optimal width depend on noise? Q2: Why does the network perform better for flat noise? Proportional Noise

18. Analysis Smallest achievable variance: = inverse of the covariance matrix of the noise Θ = vector of the derivative of the input tuning curve with respect to For Gaussian noise: Trace term is 0 when R is independent of Θ (flat noise)

19. Summary • network gives a good approximation of the optimal tuning curve determined by ML • type of noise (flat vs proportional) affected variance and optimal tuning width