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
Depression
Facilitation

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

6. Phase portrait of the FitzHugh-Nagumo neuron modelW 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: s* which maximizes p[s|r] ML: s* which maximizes p[r|s]
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