1. Scaling-up Cortical Representations in Fluctuation-Driven Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University http://www.cims.nyu.edu/faculty/dmac/ Cold Spring Harbor -- July ‘04

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

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

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

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

11. Cortical 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 networks requires that fluctuations be retained in a coarse-grained representation. “Pdf ” representations –   (v,g; x,t),  = E,I will retain fluctuations. But will not be very efficient numerically Needed – a reduction of the pdf representations which retains 1.Means & 2.Variances PT #1: Kinetic Theory provides this representation Ref: Cai, Tao, Shelley & McLaughlin, PNAS, pp 7757-7762 (2004)

15. First, tile the cortical layer with coarse-grained (CG) patches

16. Kinetic Theory begins from PDF representations   (v,g; x,t),  = E,I Knight & Sirovich; Tranchina, Nykamp & Haskell;

17. 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)

19. We’ll replace the 200 neurons in this CG cell by an effective pdf representation

20. First, replace the 200 neurons in this CG cell by an effective pdf representation Then derive from the pdf rep, kinetic thry For convenience of presentation, I’ll sketch the derivation a single CG cell, with 200 excitatory Integrate & Fire neurons The results extend to interacting CG cells which include inhibition – as well as “simple” & “complex” cells.

21. 1 - p: Synaptic Failure rate

23. N excitatory neurons (within one CG cell) Random coupling throughout the CG cell; AMPA synapses (with time scale  )   t v i = -(v – V R ) – g i (v-V E )   t g i = - g i +  l f  (t – t l ) + (Sa/N)  l,k  (t – t l k )

24. N excitatory neurons (within one CG cell) All-to-all coupling; AMPA synapses (with time scale  )   t v i = -(v – V R ) – g i (v-V E )   t g i = - g i +  l f  (t – t l ) + (Sa/N)  l,k  (t – t l k )  (g,v,t)  N -1  i=1,N E{  [v – v i (t)]  [g – g i (t)]}, Expectation “E” over Poisson spike train

25.   t v i = -(v – V R ) – g i (v-V E )   t g i = - g i +  l f  (t – t l ) + (Sa/N)  l,k  (t – t l k ) Evolution of pdf --  (g,v,t): (i) N>1; (ii) the total input to each neuron is (modulated) Poisson spike trains.  t  =  -1  v {[(v – V R ) + g (v-V E )]  } +  g {(g/  )  } + 0 (t) [  (v, g-f/ , t) -  (v,g,t)] + N m(t) [  (v, g-Sa/N , t) -  (v,g,t)], 0 (t) = modulated rate of Poisson spike train from LGN; m(t) = average firing rate of the neurons in the CG cell =  J (v) (v,g;  )| (v= 1) dg, and where J (v) (v,g;  ) = -{[(v – V R ) + g (v-V E )]  }

26.  t  =  -1  v {[(v – V R ) + g (v-V E )]  } +  g {(g/  )  } + 0 (t) [  (v, g-f/ , t) -  (v,g,t)] + N m(t) [  (v, g-Sa/N , t) -  (v,g,t)], N>>1; f << 1; 0 f = O(1);  t  =  -1  v {[(v – V R ) + g (v-V E )]  } +  g {[g – G(t)]/  )  } +  g 2 /   gg  + … where  g 2 = 0 (t) f 2 /(2  ) + m(t) (Sa) 2 /(2N  ) G(t) = 0 (t) f + m(t) Sa

27. Kinetic Theory Begins from Moments  (g,v,t)  (g) (g,t) =   (g,v,t) dv  (v) (v,t) =   (g,v,t) dg  1 (v) (v,t) =  g  (g,t  v) dg where  (g,v,t) =  (g,t  v)  (v) (v,t).  t  =  -1  v {[(v – V R ) + g (v-V E )]  } +  g {(g/  )  } + 0 (t) [  (v, g-f/ , t) -  (v,g,t)] + N m(t) [  (v, g-Sa/N , t) -  (v,g,t)],

28.  t  =  -1  v {[(v – V R ) + g (v-V E )]  } +  g {[g – G(t)]/  )  } +  g 2 /   gg  + …

29. Moments  (g,v,t)  (g) (g,t) =   (g,v,t) dv  (v) (v,t) =   (g,v,t) dg  1 (v) (v,t) =  g  (g,t  v) dg where  (g,v,t) =  (g,t  v)  (v) (v,t) Integrating  (g,v,t) eq over v yields:   t  (g) =  g {[g – G(t)])  (g) } +  g 2  gg  (g)

30. Integrating  (g,v,t) eq over g yields:  t  (v) =  -1  v [(v – V R )  (v) +  1 (v) (v-V E )  (v) ] Integrating [g  (g,v,t)] eq over g yields an equation for  1 (v) (v,t) =  g  (g,t  v) dg, where  (g,v,t) =  (g,t  v)  (v) (v,t)

31.  t  =  -1  v {[(v – V R ) + g (v-V E )]  } +  g {[g – G(t)]/  )  } +  g 2 /   gg  + …

32. Under the conditions, N>1; f < 1; 0 f = O(1), And the Closure: (i)  v  2 (v) = 0; (ii)  2 (v) =  g 2 where  2 (v) =  2 (v) – (  1 (v) ) 2,  g 2 = 0 (t) f 2 /(2  ) + m(t) (Sa) 2 /(2N  ) G(t) = 0 (t) f + m(t) Sa One obtains:

33.  t  1 (v) = -  -1 [  1 (v) – G(t)] +  -1 {[(v – V R ) +  1 (v) (v-V E )]  v  1 (v) } +  2 (v)/ (  (v) )  v [(v-V E )  (v) ] +  -1 (v-V E )  v  2 (v) where  2 (v) =  2 (v) – (  1 (v) ) 2. Closure: (i)  v  2 (v) = 0; (ii)  2 (v) =  g 2 One obtains:

34.  t  (v) =  -1  v [(v – V R )  (v) +  1 (v) (v-V E )  (v) ]  t  1 (v) = -  -1 [  1 (v) – G(t)] +  -1 {[(v – V R ) +  1 (v) (v-V E )]  v  1 (v) } +  g 2 / (  (v) )  v [(v-V E )  (v) ] Together with a diffusion eq for  (g) (g,t):   t  (g) =  g {[g – G(t)])  (g) } +  g 2  gg  (g)

35. Fluctuations in g are Gaussian   t  (g) =  g {[g – G(t)])  (g) } +  g 2  gg  (g)

36. 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 firing rate (Hz)

37. 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)

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

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

40. But we can go further for AMPA (   0):  t  1 (v) = - [  1 (v) – G(t)] +  -1 {[(v – V R ) +  1 (v) (v-V E )]  v  1 (v) } +  g 2 / (  (v) )  v [(v-V E )  (v) ] Recall  g 2 = f 2 /(2  ) 0 (t) + m(t) (Sa) 2 /(2N  ); Thus, as   0,  g 2 = O(1). Let   0: Algebraically solve for  1 (v) :  1 (v) = G(t) +  g 2 / (  (v) )  v [(v-V E )  (v) ]

41. Result: A Fokker-Planck eq for  (v) (v,t):   t  (v) =  v {[ (1 + G(t) +  g 2 /  ) v – ( V R + V E (G(t) +  g 2 /  ) ) ]  (v) +  g 2 /  (v- V E ) 2  v  (v) }  g 2 /  -- Fluctuations in g

42. Remarks: (i) Boundary Conditions; (ii) Inhibition, spatial coupling of CG cells, simple & complex cells have been added; (iii) N   yields “mean field” representation.

43. New Pdf Representation  (g,v,t) -- (i) Evolution eq, with jumps from incoming spikes; (ii) Jumps smoothed to diffusion in g by a “large N expansion”  (g) (g,t) =   (g,v,t) dv -- diffuses as a Gaussian  (v) (v,t) =   (g,v,t) dg ;  1 (v) (v,t) =  g  (g,t  v) dg Coupled (moment) eqs for  (v) (v,t) &  1 (v) (v,t), which are not closed [but depend upon  2 (v) (v,t) ] Closure -- (i)  v  2 (v) = 0; (ii)  2 (v) =  g 2, where  2 (v) =  2 (v) – (  1 (v) ) 2.   0  eq for  1 (v) (v,t) solved alge…y in terms of  (v) (v,t), resulting in a Fokker-Planck eq for  (v) (v,t)

44. Local temporal asynchony enhanced by synaptic failure – permitting better amplification

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

48. Closures:

51. Simple & Complex Cells; Multiple interacting CG Patches

52. Recall:

56. 1 - p: Synaptic Failure rate

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

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

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

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

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

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

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

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

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

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

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

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

71. 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)

72. Simple & Complex Cells; Multiple interacting CG Patches

73. 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

74. Computational Efficiency For statistical accuracy in these CG patch settings, Kinetic Theory is 10 3 -- 10 5 more efficient than I&F;

75. Average firing rates Vs Spike-time statistics

77. Coarse-grained theories involve local averaging in both space and time. Hence, coarse-grained theories average out detailed spike timing information. Ok for “rate codes”, but if spike-timing statistics is to be studied, must modify the coarse-grained approach

78. PT #2: Embedded point neurons will capture these statistical firing properties [ Ref: Cai, Tao & McLaughlin, PNAS (to appear)] For “scale-up” – computer efficiency Yet maintaining statistical firing properties of multiple neurons Model especially 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”

82. 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.

83. 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

84. 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

86. 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.

87. 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.

88. 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.

89. 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

90. 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.

91. 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.

92. (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.

93. (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.

94. 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 ( ,  )

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

96. 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.

97. 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)

98. Conclusions Kinetic Theory is a numerically efficient, and remarkably accurate, method for “scale-up” – Ref: PNAS, pp 7757-7762 (2004) 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” statistics Sub-networks of point neurons can be embedded within kinetic theory to capture spike timing statistics, with a range from test neurons to fully interacting sub-networks. Ref: PNAS, to appear (2004)

101. 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

102. 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)

103. Summary & Conclusion

104. 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.

105. 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

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

108. Closures:

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

117. 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 :

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

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

120.   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: