1. Neural Coding: Integrate-and-Fire Models of Single and Multi-Neuron Responses Jonathan Pillow HHMI and NYU http://www.cns.nyu.edu/~pillow Oct 5, Course lecture: “Computational Modeling of Neuronal Systems” Fall 2005, New York University

2. General Goal: understand the mapping from stimuli to spike responses with the use of a model model x stimulus y spike response Model criteria: flexibility (captures realistic neural properties) tractability (for fitting to data)

3. Example 1: Hodgkin-Huxley stimulus spike response Na + activation (fast) Na + inactivation (slow) K + activation (slow) + flexible, biophysically realistic - not easy to fit

4. Example 2: LNP + easy to fit (spike-triggered averaging) + not biologically plausible K (receptive field) f y x

5. LNP model stimulus filter output spike rate spikes time (sec) filter K

6. “cascade” models model x stimulus y spike response Linear Filtering Nonlinear Probabilistic Spiking more realistic models of spike generation

7. Generalized Integrate-and-Fire Model K h I stim I spike I noise related: “Spike Response Model”, Gerstner & Kistler ‘02 x(t)y(t)

8. Generalized Integrate-and-Fire Model + powerful, flexible + tractable for fitting K h

9. Model behaviors: adaptation

10. Model behaviors: bursting

11. Model behaviors: bistability 0 0

12. K = stimulus filter g = leak conductance  2 = noise variance h = response current V L = reversal potential The Estimation Problem Solution: Maximum Likelihood - need an algorithm to compute P  (y|x) Learn the model parameters: From: stimulus train x(t) spike times t i K h

13. Likelihood function P(spike at t i ) = fraction of paths crossing threshold at t i titi hidden variable:

14. Likelihood function P(spike at t i ) = fraction of paths crossing threshold at t i hidden variable: titi

15. Computing Likelihood linear dynamics additive Gaussian noise titi P(V,t) fast methods for solving linear PDE efficient procedure for computing likelihood  P(V,t+  t) Diffusion Equation: 11

16. Computing Likelihood ISIs are conditionally independent  likelihood is product over ISIs linear dynamics additive Gaussian noise fast methods for solving linear PDE efficient procedure for computing likelihood  reset Diffusion Equation: (Fokker-Planck) 11

17. Maximizing the likelihood The log likelihood is concave in the parameters {K, , , h, V L }, for any data {x(t), t i } Main Theorem:  gradient ascent guaranteed to converge to global maximum! parameter space is large (  20 to 100 dimensions) parameters interact nonlinearly [Paninski, Pillow & Simoncelli. Neural Comp. ‘04

18. Application to Macaque Retina isolated retinal ganglion cell (RGC) stimulated with full-field random stimulus (flicker) fit using 1-minute period of response (Data: Valerie Uzzell & E.J. Chichilnisky) t

19. time (ms) IF model simulation Stimulus filter K I inj V

20. time (ms) IF model simulation Stimulus filter K I inj V h Noise

21. RGC IF ON cell LNP 74% of var 92 % of var

22. Accounting for spike timing precision P(spike) 0200time (ms)

23. Accounting for reliability

24. Decoding the neural response Resp 1Resp 2 ? Stim 1Stim 2

25. Solution: use P(resp|stim) Resp 1Resp 2 ? Stim 1Stim 2 a P(R1|S1)P(R2|S2)P(R1|S2)P(R2|S1)

26. Discriminate each repeat using P(Resp|Stim) Resp 1Resp 2 ? Stim 1Stim 2 P(R1|S1)P(R2|S2)P(R1|S2)P(R2|S1)

27. Discriminate each repeat using P(Resp|Stim) Resp 1Resp 2 ? Stim 1Stim 2 94 % correct LNP: 68 % correct Compare to LNP model P(Resp|Stim)

28. Decoding the neural response LNP model % correct IF model % correct

29. Part 2: how to characterize the responses of multiple neurons? Want to capture: the stimulus dependence of each neuron’s response the response dependencies between neurons.

30. 2 types of correlation: 1.stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r 1,r 2 | stim) = P(r 1 |stim)P(r 2 |stim) cell 1cell 2 stimuliresponses

31. 2 types of correlation: cell 2cell 1 Noise stimuliresponses 1.stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r 1,r 2 | stim) = P(r 1 |stim)P(r 2 |stim) 2. “noise” correlation: arises if responses are not conditionally independent given the stimulus, i.e. P(r 1,r 2 | stim)  P(r 1 |stim)P(r 2 |stim)

32. Modeling multi-neuron responses y1y1 y2y2 K K h 11 h 22 h 12 h 21 x x coupling h currents:

33. Methods spatiotemporal binary white noise (24 x 24 pixels, 120Hz frame rate) simultaneous multi-electrode recordings of macaque RGCs Model parameters fit to five RGCs using 10 minutes of response to a non-repeating binary white noise stimulus

34. Fits OFF cells ON cells cell 1 cell 2 cell 5 cell 3 cell 4

35. Fits ON + OFF cells cell 1 cell 2 cell 5 cell 3 cell 4

36. Fits cell 1 cell 2 cell 5 cell 3 cell 4

37. Compare likelihoods: 1.The single-cell model for cell i : vs. 2. The pairwise model for i with coupling from cell j Pairwise coupling analysis stimulusfilter IF novel stim spikes novel stim cell j spikes h ij spikes stimulusfilter IF

38. Pairwise coupling analysis Coupling Matrix likelihood ratio Functional Coupling

39. Accounting for the autocorrelation 1 2 OFFcells RGC simulated model post-spike current

40. Accounting for cross-correlations RGC coupled model -5 0 5 10 15 -100-50050100 time(ms) -100-50050100 time(ms) -1 0 1 2 raw (stimulus + noise)stimulus-induced RGC, shuffled uncoupled model ON-ON correlations 4 to 5 5 to 4

41. OFF-OFF cell correlations RGC coupled model raw (stimulus + noise)stimulus-induced RGC, shuffled uncoupled model 3 to 2 2 to 3 -2 0 2 4 6 -2 0 2 4 6 -100-50050100 time(ms) -2 -1 0 1 -1 0 1 1 vs 3 2 vs 3

42. OFF-ON cell correlations RGC coupled model raw (stimulus + noise)stimulus-induced RGC, shuffled uncoupled model 1 to 4 4 to 1 1 vs 4 -6 -4 -2 0 2 4

43. OFF-ON cell correlations stimulus-induced 1 vs 5 s) -100-50050100 raw (stimulus + noise) 00 time(ms) 2 vs 4 2 vs 5 3 vs 4 3 vs 5

44. Conclusions 1. generalized-IF model: flexible, tractable tool for modeling neural responses 2. fitting with maximum likelihood 3. probabilistic framework: useful for encoding (precision, response variability) and decoding 4. easily extended to multi-neuron responses 5. likelihood test of functional connectivity between cells 6. explains auto- and cross-correlations 7. resolves cross-correlations into “stimulus-induced” and “noise-induced”

45. My collaborators: E.J. Chichilnisky Valerie Uzzell - The Salk Institute Jonathon Shlens Eero Simoncelli - HHMI & NYU Liam Paninski- Columbia U.

46. Basis used for coupling currents

47. Extra slides:

48. 5-way coupling analysis Functional Coupling Likelihood ratio for fully connected model

49. 5-way coupling analysis Conclusion: the fully connected model gives an improved description of multi-cell responses to white noise stimuli. Likelihood ratio for fully connected model