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Kinetic Theory for the Dynamics of Fluctuation-Driven Neural Systems David W. McLaughlin Courant Institute & Center for Neural Science New York University Toledo – June ‘06

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Happy Birthday, Peter & Louis

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Kinetic Theory for the Dynamics of Fluctuation-Driven Neural Systems In collaboration with: David Cai Louis Tao Michael Shelley Aaditya Rangan

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Visual Pathway: Retina --> LGN --> V1 --> Beyond

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Integrate and Fire Representation t v = -(v – V R ) – g (v-V E ) t g = - g + l f (t – t l ) + (Sa/N) l,k (t – t l k ) plus spike firing and reset v (t k ) = 1; v (t = t k + ) = 0

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Nonlinearity from spike-threshold: Whenever V(x,t) = 1, the neuron "fires", spike-time recorded, and V(x,t) is reset to 0,

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The “primary visual cortex (V1)” is a “layered structure”, with O(10,000) neurons per square mm, per layer.

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O(10 4 ) neuons per mm 2 per mm 2 Map of OrientationPreference With both regular & random patterns of neurons’ preferences

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

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Line-Motion-Illusion LMI

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Coarse-Grained Asymptotic Representations Needed for “Scale-up” Larger lateral area Multiple layers

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

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Coarse-Grained Reductions for V1 Average firing rate models [Cowan & Wilson (’72); ….; Shelley & McLaughlin(’02)] Average firing rate of an excitatory (inhibitory) neuron, within coarse-grained patch located at location x in the cortical layer: m (x,t), = E,I

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Cortical networks have a very “noisy” dynamics Strong temporal fluctuations On synaptic timescale Fluctuation driven spiking

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Experiment Observation Fluctuations in Orientation Tuning (Cat data from Ferster’s Lab) Ref: Anderson, Lampl, Gillespie, Ferster Science, (2000)

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

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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 Kinetic Theory provides this representation Ref: Cai, Tao, Shelley & McLaughlin, PNAS, pp (2004)

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Kinetic Theory begins from PDF representations (v,g; x,t), = E,I Knight & Sirovich; Nykamp & Tranchina, Neural Comp (2001) Haskell, Nykamp & Tranchina, Network (2001) ;

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

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N excitatory neurons (within one CG cell) Random coupling throughout the CG cell; AMPA synapses (with a short time scale ) t v i = -(v i – V R ) – g i (v i -V E ) t g i = - g i + l f (t – t l ) + (Sa/N) l,k (t – t l k ) plus spike firing and reset v i (t i k ) = 1; v i (t = t i k + ) = 0

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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 ) (g,v,t) N -1 i=1,N E{ [v – v i (t)] [g – g i (t)]}, Expectation “E” over Poisson spike train { t l }

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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 incoming Poisson spike train; 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 )] }

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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

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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 – G(t)]/ ) } + g 2 / gg + … First, integrating (g,v,t) eq over v yields: t (g) = g {[g – G(t)]) (g) } + g 2 gg (g)

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Fluctuations in g are Gaussian t (g) = g {[g – G(t)]) (g) } + g 2 gg (g)

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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)

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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:

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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)

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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)

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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)

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

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

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Computational Efficiency For statistical accuracy in these CG patch settings, Kinetic Theory is more efficient than I&F;

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Realistic Extensions Extensions to coarse-grained local patches, to excitatory and inhibitory neurons, and to neurons of different types (simple & complex). The pdf then takes the form , (v,g; x,t), where x is the coarse-grained label, = E,I and labels cell type

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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) Excitatory Cells Shown

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Firing rate vs. input conductance for 4 networks with varying pN : 25 (blue), 50 (magneta), 100 (black), 200 (red). Hysteresis occurs for pN =100 and 200. Fixed synaptic coupling S exc /pN

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Summary Kinetic Theory is a numerically efficient ( more efficient than I&F), and remarkably accurate, method for “scale-up” Ref: PNAS, pp (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. 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: Tao, Cai, McLaughlin, PNAS, (2004)

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Too good to be true? What’s missing? First, the zeroth moment is more accurate than the first moment, as in many moment closures

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Too good to be true? What’s missing? Second, again as in many moment closures, existence can fail -- (Tranchina, et al – 2006). That is, at low but realistic firing rates, equations too rigid to have steady state solutions which satisfy the boundary conditions. Diffusion (in v) fixes this existence problem – by introducing boundary layers

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Too good to be true? What’s missing? But a far more serious problem Kinetic Theory does not capture detailed “spike-timing” information

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Why does the kinetic theory (Boltzman-type approach in general) not work? Note

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Too good to be true? What’s missing? But a far more serious problem Kinetic Theory does not capture detailed “spike-timing” statistics

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Too good to be true? What’s missing? But a far more serious problem Kinetic Theory does not capture detailed “spike-timing” statistics And most likely the cortex works, on very short time time scales, through neurons correlated by detailed spike timing. Take, for example, the line-motion illusion

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Line-Motion-Illusion LMI

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Model Voltage Model NMDA time space Trials 40% ‘coarse’ 0% ‘coarse’ Direct ‘naïve’ coarse graining may not suffice: Priming mechanism relies on Recruitment Recruitment relies on locally correlated cortical firing events Naïve ensemble average destroys locally correlated events Stimulus

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Conclusion Kinetic Theory is a numerically efficient ( more efficient than I&F), and remarkably accurate. Kinetic Theory accurately captures firing rates in fluctuation dominated systems Kinetic Theory does not capture detailed spike- timed correlations – which may be how the cortex works, as it has no time to average. So we’ve returned to integrate & fire networks, and have developed fast “multipole” algorithms for integrate & fire systems (Cai and Rangan, 2005).

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