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

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Stimulus (s) neurons encode Response (r) decode

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

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

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

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

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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

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Stimulus (s) neurons encode Response (r) decode

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Maximum Likelihood Decoding Maximum likelihood estimator Decoding Encoding

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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

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General Architecture units are fully connected and are arranged in frequency columns and orientation rows weights implement a 2-D Gaussian filter: 20 Preferred Frequency Preferred orientation PΘPΘ PλPλ

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

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

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Results Rapidly converges strongly dependent on contrast

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

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

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

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Analysis Smallest achievable variance: = inverse of the covariance matrix of the noise = 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) Θ

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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

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