1. Scaling-up Cortical Representations David W. McLaughlin Courant Institute & Center for Neural Science New York University http://www.cims.nyu.edu/faculty/dmac/ MSRI -- March ‘04

2. In collaboration with:   David Cai Louis Tao Michael Shelley Aaditya Rangan

4. Coarse-Grained Asymptotic Representations Needed for “Scale-up”

5. Lateral Connections and Orientation -- Tree Shrew Bosking, Zhang, Schofield & Fitzpatrick J. Neuroscience, 1997

10. Coarse-Grained Reductions for V1 Average firing rate models (Cowan & Wilson; ….; Shelley & McLaughlin) m  (x,t),  = E,I

11. But realistic networks have a very noisy dynamics Strong temporal fluctuations On synaptic timescale Fluctuation driven spiking

12. Experiment Observation Fluctuations in Orientation Tuning (Cat data from Ferster’s Lab) Ref: Anderson, Lampl, Gillespie, Ferster Science, 1968-72 (2000) threshold (-65 mV)

13. Fluctuation-driven spiking Solid: average ( over 72 cycles) Dashed: 10 temporal trajectories (very noisy dynamics, on the synaptic time scale)

14. To accurately and efficiently describe these fluctuations, the scale-up method will require pdf representations –   (v,g; x,t),  = E,I To “benchmark” these, we will numerically simulate I&F neurons within one CG “patch”

15. Coarse-Grained Reductions for V1 PDF representations (Knight & Sirovich; Tranchina, Nykamp & Haskell; Cai, Tao, Shelley & McLaughlin)   (v,g; x,t),  = E,I Sub-network of embedded point neurons -- in a coarse- grained, dynamical background (Cai,Tao & McLaughlin)

17. 1 - p: Synaptic Failure rate

20. Under ASSUMPTIONS: 1) 2) Summed intra-cortical low rate spike events become Poisson : 2) Summed intra-cortical low rate spike events become Poisson :

23. Closures:

25. PDF of v Theory→ ←I&F (solid) Fokker-Planck→ Theory→ ←I&F ←Mean-driven limit ( ): Hard thresholding Fluctuation-Driven Dynamics N=75 σ=5msec S=0.05 f=0.01 Experiment firing rate (Hz)

26. Mean­Driven: Bistability and Hysteresis   Network of Simple, Excitatory only Fluctuation­Driven: N=16 Relatively Strong Cortical Coupling: N=16!

27. Mean­Driven: N=16! Bistability and Hysteresis   Network of Simple, Excitatory only Relatively Strong Cortical Coupling:

28. Simple & Complex Cells; Multiple interacting CG Patches

29. Recall:

35. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory

36. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Complex Excitatory Cells Mean-Driven

37. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Complex Excitatory Cells Mean-Driven

38. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Complex Excitatory Cells Mean-Driven

39. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Complex Excitatory Cells Mean-Driven

40. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Simple Excitatory Cells Mean-Driven

41. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Simple Excitatory Cells Mean-Driven

42. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Simple Excitatory Cells Mean-Driven

43. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Simple Excitatory Cells Mean-Driven

44. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Simple Excitatory Cells Mean-Driven

45. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Complex Excitatory Cells Fluctuation-Driven

46. Incorporation of Inhibitory Cells 4 Population Dynamics Simple:   Excitatory   Inhibitory Complex:   Excitatory   Inhibitory Simple Excitatory Cells Fluctuation-Driven

47. Three Dynamic Regimes of Cortical Amplification: 1) Weak Cortical Amplification No Bistability/Hysteresis 2) Near Critical Cortical Amplification 3) Strong Cortical Amplification Bistability/Hysteresis (2) (1) (3) I&F Excitatory Complex Cells Shown (2) (1)

48. Embedded Point Neurons Firing rate codes Vs Spike timing codes

50. Embedded Point Neurons For “scale-up” – computer efficiency Yet maintaining firing properties of individual neurons -- for spike coding, coincidence detection, etc. Model relevant for biologically distinguished sparse, strong sub- networks – perhaps such as long-range connections Point neurons -- embedded in, and fully interacting with, coarse- grained kinetic theory, Or, when kinetic theory accurate by itself, embedded as “test neurons”

53. I&F vs. Embedded Network Spike Rasters a) I&F Network: 50 “Simple” cells, 50 “Complex” cells. “Simple” cells driven at 10 Hz b)-d) Embedded I&F Networks: b) 25 “Complex” cells replaced by single kinetic equation; c) 25 “Simple” cells replaced by single kinetic equation; d) 25 “Simple” and 25 “Complex” cells replaced by kinetic equations. In all panels, cells 1-50 are “Simple” and cells 51-100 are “Complex”. Rasters shown for 5 stimulus periods.

55. I&F vs. Embedded Network Spike Rasters a) I&F Network: 40 “Simple” cells, 40 “Complex” cells. “Simple” cells driven at 10 Hz; b)-d) Embedded I&F Networks: b) 20 “Complex” cells replaced by log-form rate equation; c) 20 “Simple” cells replaced by log-form rate equation; d) 20 “Simple” and 20 “Complex” cells replaced by log-form rate equations. In all panels, cells 1-40 are “Simple” and cells 41-80 are “Complex”. Rasters shown for 5 stimulus periods.

56. a) I&F Network: 50 Simple cells, 50 Complex”cells. “Simple”cells driven at 10 Hz; b) Embedded I&F Networks, with Complex I&F neurons receiving input but not outputing to the network. 25 Complex”cells replaced by kinetic equations. In all panels, cells 1-50 are Simple and cells 51-75 are Complex”. Rasters shown for 5 stimulus periods.

57. Dynamic Firing Rates: (Shown: Exc Simple Pop in a 4 Pop Model) Forcing--time-dependent Poisson process, sinusoidal driving at 10 Hz. In the embedded models, excitatory neurons are I&F and inhibitory neurons are replaced by Kinetic Theory.

58. Cycle-averaged Firing Rate Curves [Shown: Exc Cmplx Pop in a 4 population model): Full I&F network (solid), Full I&F + KT (dotted); Full I&F coupled to Full KT but with mean only coupling (dashed).] In both embedded cases (where the I&F units are coupled to KT), half the simple cells are represented by Kinetic Theory The Importance of Fluctuations

59. Steady State Firing Rate Curves (for the excitatory population): I&F, Exc. & Inh. (Magenta), Kinetic Theory, Exc. & Inh. (Red), Embedded Network (Blue), Log-form (Green). In the Embedded model, Excitatory neurons are I&F and Inhibitory neurons are modelled by Kinetic Theory.

60. Steady State Firing Rate Curves for Embedded Sub-networks: (Shown, Exc Cmplx Pop): In the embedded models, half of the simple cells are replaced by Kinetic Theory. However, in the “I&F + KT (mean only), they are coupled back to the I&F neurons as mean conductance drives.

61. (From left to right) Rasters, Cross-correlation and ISI distributions for two simulations: (Upper panels) Kinetic Theory of a neuronal patch driving test neurons, which are not coupled to each other; (Lower panels) KT of a neuronal patch driving strongly coupled neurons. In both cases, the CGed patch is being driven at 1 Hz. Neurons 1-6 are excitatory; Neurons 7-8 are inhibitory; the S-matrix is (S ee, S ei, S ie, S ii ) = (0.3, 0.4, 0.6, 0.4). EPSP time constant 3 ms; IPSP time constant 10 ms.

62. (From left to right) Rasters, Cross-correlation and ISI distributions for two simulations: (Upper panels) Kinetic Theory of a neuronal patch driving test neurons, which are not coupled to each other; (Lower panels) KT of a neuronal patch driving strongly coupled neurons. In both cases, the CGed patch is being driven “asynchronously,” and are firing at a fixed rate. I&F Neurons 1-6 are excitatory; Neurons 7-8 are inhibitory; the S-matrix is (S ee, S ei, S ie, S ii ) = (0.4, 0.4, 0.8, 0.4). EPSP time constant 3 ms; IPSP time constant 10 ms.

63. Raster Plots, Cross-correlation and ISI distributions. (Upper panels) KT of a neuronal patch with strongly coupled embedded neurons; (Lower panels) Full I&F Network. Shown is the sub-network, with neurons 1-6 excitatory; neurons 7-8 inhibitory; EPSP time constant 3 ms; IPSP time constant 10 ms. Embedded Network Full I & F Network

64. ISI distributions for two simulations: (Left) Test Neuron driven by a CG neuronal patch; (Right) Sample Neuron in the I&F Network. “Test neuron” within a CG Kinetic Theory

65. Reverse Time Correlations Correlates spikes against driving signal Triggered by spiking neuron Frequently used experimental technique to get a handle on one description of the system P( ,  ) – probability of a grating of orientation , at a time  before a spike -- or an estimate of the system’s linear response kernel as a function of ( ,  )

66. Time → Reverse-Time Correlation (RTC)   System analysis   Probing network dynamics

67. Reverse Correlation Left: I&F Network of 128 “Simple” and 128 “Complex” cells at pinwheel center. RTC P(  ) for single Simple cell. Below: Embedded Network of 128 “Simple” cells, with 128 “Complex” cells replaced by single kinetic equation. RTC P(  ) for single Simple cell.

68. Computational Efficiency For statistical accuracy in these CG patch settings, Kinetic Theory is 10 3 -- 10 5 more efficient than I&F; The efficiency of the embedded sub-network scales as N 2, where N = # of embedded point neurons; (i.e. 100  20 yields 10,000  400)

69. Conclusions Kinetic Theory is a numerically efficient, and remarkably accurate, method for “scale-up”. Kinetic Theory introduces no new free parameters into the model, and has a large dynamic range from the rapid firing “mean-driven” regime to a fluctuation driven regime. Kinetic Theory does not capture detailed “spike-timing” Sub-networks of point neurons can be embedded within kinetic theory to capture spike timing, with a range from test neurons to fully interacting sub-networks.

72. Conclusions and Directions Constructing ideal network models to discern and extract possible principles of neuronal computation and functions Mathematical methods for analytical understanding Search for signatures of identified mechanisms Mean-driven vs. fluctuation-driven kinetic theories New closure, Fluctuation and correlation effects Excellent agreement with the full numerical simulations Large-scale numerical simulations of structured networks constrained by anatomy and other physiological observations to compare with experiments Structural understanding vs. data modeling New numerical methods for scale-up --- Kinetic theory

73. Three Dynamic Regimes of Cortical Amplification: 1) Weak Cortical Amplification No Bistability/Hysteresis 2) Near Critical Cortical Amplification 3) Strong Cortical Amplification Bistability/Hysteresis (2) (1) (3) I&F Excitatory Cells Shown  Possible Mechanism for Orientation Tuning of Complex Cells Regime 2 for far-field/well-tuned Complex Cells Regime 1 for near-pinwheel/less-tuned Summed Effects (2) (1)

74. Fluctuation­Driven Tuning Dynamics Near Critical Amplification vs. Weak Cortical Amplification Sensitivity to Contrast Ring Model A less-tuned complex cell Ring Model — far field A well-tuned complex cell Large V1 Model A complex cell in the far-field Ring Model of Orientation Tuning: Near Pinwheel Far Field A Cell in Large V1 Model

75. Summary & Conclusion

76. Summary Points for Coarse-Grained Reductions needed for Scale-up 1.Neuronal networks are very noisy, with fluctuation driven effects. 2.Temporal scale-separation emerges from network activity. 3.Local temporal asynchony needed for the asymptotic reduction, and it results from synaptic failure. 4.Cortical maps -- both spatially regular and spatially random -- tile the cortex; asymptotic reductions must handle both. 5.Embedded neuron representations may be needed to capture spike-timing codes and coincidence detection. 6.PDF representations may be needed to capture synchronized fluctuations.

77. Scale-up & Dynamical Issues for Cortical Modeling of V1 Temporal emergence of visual perception Role of spatial & temporal feedback -- within and between cortical layers and regions Synchrony & asynchrony Presence (or absence) and role of oscillations Spike-timing vs firing rate codes Very noisy, fluctuation driven system Emergence of an activity dependent, separation of time scales But often no (or little) temporal scale separation

78. Under ASSUMPTIONS: 1) 2) Summed intra-cortical low rate spike events become Poisson : 2) Summed intra-cortical low rate spike events become Poisson :

80. Closures:

88. Kinetic Theory for Population Dynamics Population of interacting neurons: 1-p: Synaptic Failure rate

89. Kinetic Equation: Under ASSUMPTIONS: 1) 2) Summed intra-cortical low rate spike events become Poisson : 2) Summed intra-cortical low rate spike events become Poisson :

90. Fluctuation-Driven Dynamics Physical Intuition: Fluctuation-driven/Correlation between g and V Hierarchy of Conditional Moments

91. Closure Assumptions: Closed Equations — Reduced Kinetic Equations Closed Equations — Reduced Kinetic Equations :

92.   Fluctuation Effects   Correlation Effects Fokker-Planck Equation: Flux: Determination of Firing Rate: For a steady state, m can be determined implicitly Coarse-Graining in Time: