Neural Coding: Integrate-and-Fire Models of Single and Multi-Neuron Responses

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

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

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

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

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

7. K h Generalized Integrate-and-Fire Model x(t) y(t) Inoise Istim Ispike related: “Spike Response Model”, Gerstner & Kistler ‘02

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

9. Model behaviors: adaptation

10. Model behaviors: bursting

11. 0 0 Model behaviors: bistability

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

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

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

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

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

17. Main Theorem: The log likelihood is concave in the parameters {K, t, s, h, VL} , for any data {x(t), ti} Maximizing the likelihood • parameter space is large (  20 to 100 dimensions) • parameters interact nonlinearly  gradient ascent guaranteed to converge to global maximum! [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 t (Data: Valerie Uzzell & E.J. Chichilnisky)

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

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

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

22. 0 time (ms) 200 P(spike) Accounting for spike timing precision

23. Accounting for reliability

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

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

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

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

28. Decoding the neural response IF model % correct LNP 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. cell 1 cell 2 2 types of correlation: • stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r1,r2| stim) = P(r1|stim)P(r2|stim) stimuli responses

31. cell 1 cell 2 2 types of correlation: • stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r1,r2| stim) = P(r1|stim)P(r2|stim) 2. “noise” correlation: arises if responses are not conditionally independent given the stimulus, i.e. P(r1,r2| stim)  P(r1|stim)P(r2|stim) Noise stimuli responses

32. y1 y2 Modeling multi-neuron responses K x h11 h12 coupling h currents: h21 K x h22

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

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

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

37. spikes s s t t i i m m u u l l u u s s f f i i l l t t e e r r IF IF novel stim spikes novel stim hij cell j spikes Pairwise coupling analysis • Compare likelihoods: • The single-cell model for cell i: • vs. • 2. The pairwise model for i with coupling from cell j

38. Pairwise coupling analysis Coupling Matrix likelihood ratio Functional Coupling

39. Accounting for the autocorrelation RGC simulated model post-spike current O F F c e l l s 1 2

40. RGC coupled model RGC, shuffled uncoupled model Accounting for cross-correlations ON-ON correlations raw (stimulus + noise) stimulus-induced 2 1 5 1 0 1 5 0 0 - 1 - 5 - 1 0 0 - 5 0 0 5 0 1 0 0 - 1 0 0 - 5 0 0 5 0 1 0 0 t i m e ( m s ) t i m e ( m s ) 4 to 5 5 to 4

41. RGC coupled model RGC, shuffled uncoupled model raw (stimulus + noise) stimulus-induced OFF-OFF cell correlations 1 6 4 0 1 vs 3 2 - 1 0 - 2 - 2 1 6 4 2 vs 3 0 2 1 vs 3 0 - 1 - 2 - 1 0 0 - 5 0 0 5 0 1 0 0 t i m e ( m s ) 3 to 2 2 to 3

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

43. 5 0 - 5 2 0 - 2 2 0 - 2 - 4 4 2 0 - 2 - 4 1 5 1 0 5 0 - 5 - 1 0 0 - 5 0 0 5 0 1 t i m e ( m OFF-ON cell correlations raw (stimulus + noise) stimulus-induced 1 vs 5 2 vs 4 2 vs 5 3 vs 4 3 vs 5 - 1 0 0 - 5 0 0 5 0 1 0 0 0 0 s ) t i m e ( m s )

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 Likelihood ratio for fully connected model Functional Coupling

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