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Signal is carried chemically across the synaptic cleft

Synapses. Signal is carried chemically across the synaptic cleft. Post-synaptic conductances. Requires pre- and post-synaptic depolarization Coincidence detection, Hebbian. Synaptic plasticity. LTP, LTD Spike-timing dependent plasticity. Short-term synaptic plasticity. Facilitation.

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Signal is carried chemically across the synaptic cleft

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  1. Synapses Signal is carried chemically across the synaptic cleft

  2. Post-synaptic conductances Requires pre- and post-synaptic depolarization Coincidence detection, Hebbian

  3. Synaptic plasticity • LTP, LTD • Spike-timing dependent plasticity

  4. Short-term synaptic plasticity Facilitation Depression

  5. I = 0 phase portrait A simple model neuron: FitzHugh-Nagumo

  6. Phase portrait of the FitzHugh-Nagumo neuron model W V

  7. Reduced dynamical model for neurons

  8. Population coding • Population code formulation • Methods for decoding: • population vector • Bayesian inference • maximum a posteriori • maximum likelihood • Fisher information

  9. Cricket cercal cells coding wind velocity

  10. Population vector RMS error in estimate Theunissen & Miller, 1991

  11. Population coding in M1 Cosine tuning: Pop. vector: For sufficiently large N, is parallel to the direction of arm movement

  12. The population vector is neither general nor optimal. “Optimal”: Bayesian inference and MAP

  13. Bayesian inference By Bayes’ law, Introduce a cost function, L(s,sBayes); minimise mean cost. For least squares, L(s,sBayes) = (s – sBayes)2 ; solution is the conditional mean.

  14. MAP and ML MAP: s* which maximizes p[s|r] ML: s* which maximizes p[r|s] Difference is the role of the prior: differ by factor p[s]/p[r] For cercal data:

  15. Decoding an arbitrary continuous stimulus E.g. Gaussian tuning curves

  16. Need to know full P[r|s] Assume Poisson: Assume independent: Population response of 11 cells with Gaussian tuning curves

  17. Apply ML: maximise P[r|s] with respect to s Set derivative to zero, use sum = constant From Gaussianity of tuning curves, If all s same

  18. Apply MAP: maximise p[s|r] with respect to s Set derivative to zero, use sum = constant From Gaussianity of tuning curves,

  19. Given this data: Prior with mean -2, variance 1 Constant prior MAP:

  20. How good is our estimate? For stimulus s, have estimated sest Bias: Variance: Mean square error: Cramer-Rao bound: Fisher information

  21. Fisher information Alternatively: For the Gaussian tuning curves w/Poisson statistics:

  22. Fisher information for Gaussian tuning curves Quantifies local stimulus discriminability

  23. Do narrow or broad tuning curves produce better encodings? Approximate: Thus,  Narrow tuning curves are better But not in higher dimensions!

  24. Fisher information and discrimination Recall d' = mean difference/standard deviation Can also decode and discriminate using decoded values. Trying to discriminate s and s+Ds: Difference in estimate is Ds (unbiased) variance in estimate is 1/IF(s). 

  25. Comparison of Fisher information and human discrimination thresholds for orientation tuning Minimum STD of estimate of orientation angle from Cramer-Rao bound data from discrimination thresholds for orientation of objects as a function of size and eccentricity

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