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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 http://www.cims.nyu.edu/faculty/dmac/ Toledo – June ‘06 Happy Birthday, Peter & Louis Kinetic Theory for the Dynamics of Fluctuation-Driven Neural Systems In collaboration with: David Cai Louis Tao Michael Shelley Aaditya Rangan Visual Pathway: Retina --> LGN --> V1 --> Beyond Integrate and Fire Representation t v = -(v – VR) – g (v-VE) t g = - g + l f (t – tl) + (Sa/N) l,k (t – tlk) plus spike firing and reset v (tk) = 1; v (t = tk + ) = 0 Nonlinearity from spike-threshold: Whenever V(x,t) = 1, the neuron "fires", spike-time recorded, and V(x,t) is reset to 0 , The “primary visual cortex (V1)” is a “layered structure”, with O(10,000) neurons per square mm, per layer. Map of Orientation Preference O(104) neuons per mm2 With both regular & random patterns of neurons’ preferences Lateral Connections and Orientation -- Tree Shrew Bosking, Zhang, Schofield & Fitzpatrick J. Neuroscience, 1997 Line-Motion-Illusion LMI Coarse-Grained Asymptotic Representations Needed for “Scale-up” • Larger lateral area • Multiple layers First, tile the cortical layer with coarse-grained (CG) patches 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 Cortical networks have a very “noisy” dynamics • Strong temporal fluctuations • On synaptic timescale • Fluctuation driven spiking Experiment Observation Fluctuations in Orientation Tuning (Cat data from Ferster’s Lab) Ref: Anderson, Lampl, Gillespie, Ferster Science, 1968-72 (2000) Fluctuation-driven spiking (very noisy dynamics, on the synaptic time scale) Solid: average ( over 72 cycles) Dashed: 10 temporal trajectories • 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. 2. • Means & Variances Kinetic Theory provides this representation Ref: Cai, Tao, Shelley & McLaughlin, PNAS, pp 7757-7762 (2004) Kinetic Theory begins from PDF representations (v,g; x,t), = E,I • Knight & Sirovich; • Nykamp & Tranchina, Neural Comp (2001) • Haskell, Nykamp & Tranchina, Network (2001) ; • 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. • N excitatory neurons (within one CG cell) • Random coupling throughout the CG cell; • AMPA synapses (with a short time scale ) t vi = -(vi – VR) – gi (vi -VE) t gi = - gi + l f (t – tl) + (Sa/N) l,k (t – tlk) plus spike firing and reset vi (tik) = 1; vi (t = tik + ) = 0 • N excitatory neurons (within one CG cell) • Random coupling throughout the CG cell; • AMPA synapses (with time scale ) t vi = -(v – VR) – gi (v-VE) t gi = - gi + l f (t – tl) + (Sa/N) l,k (t – tlk) (g,v,t) N-1 i=1,N E{[v – vi(t)] [g – gi(t)]}, Expectation “E” over Poisson spike train { tl } t vi = -(v – VR) – gi (v-VE) t gi = - gi + l f (t – tl) + (Sa/N) l,k (t – tlk) Evolution of pdf -- (g,v,t): (i) N>1; (ii) the total input to each neuron is (modulated) Poisson spike trains. t = -1v {[(v – VR) + g (v-VE)] } + 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 – VR) + g (v-VE)] } t = -1v {[(v – VR) + g (v-VE)] } + 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 = -1v {[(v – VR) + g (v-VE)] } + g {[g – G(t)]/) } + g2 / gg + … where g2 = 0(t) f2 /(2) + m(t) (Sa)2 /(2N) G(t) = 0(t) f + m(t) Sa 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,tv) dg where (g,v,t) = (g,tv) (v)(v,t). t = -1v {[(v – VR) + g (v-VE)] } + g {[g – G(t)]/) } + g2 / gg + … First, integrating (g,v,t) eq over v yields: t (g) = g {[g – G(t)]) (g)} + g2 gg (g) Fluctuations in g are Gaussian t (g) = g {[g – G(t)]) (g)} + g2 gg (g) Integrating (g,v,t) eq over g yields: t (v) = -1v [(v – VR) (v) + 1(v) (v-VE) (v)] Integrating [g (g,v,t)] eq over g yields an equation for 1(v)(v,t) = g (g,tv) dg, where (g,v,t) = (g,tv) (v)(v,t) t 1(v) = - -1[1(v) – G(t)] + -1{[(v – VR) + 1(v)(v-VE)] v 1(v)} + 2(v)/ ((v)) v [(v-VE) (v)] + -1(v-VE) v2(v) where 2(v) = 2(v) – (1(v))2 . Closure: One obtains: (i) v2(v) = 0; (ii) 2(v) = g2 t (v) = -1v [(v – VR) (v) + 1(v)(v-VE) (v)] t 1(v) = - -1[1(v) – G(t)] + -1{[(v – VR) + 1(v)(v-VE)] v 1(v)} + g2 / ((v)) v [(v-VE) (v)] Together with a diffusion eq for (g)(g,t): t (g) = g {[g – G(t)]) (g)} + g2 gg (g) Fluctuation-Driven Dynamics PDF of v Theory→ ←I&F (solid) firing rate (Hz) N=75 N=75 σ=5msec S=0.05 f=0.01 Fokker-Planck→ Theory→ ←I&F ←Mean-driven limit ( Hard thresholding ): N Bistability and Hysteresis Network of Simple, Excitatory only N=16! N=16 FluctuationDriven: Relatively Strong Cortical Coupling: MeanDriven: N Bistability and Hysteresis Network of Simple, Excitatory only N=16! MeanDriven: Relatively Strong Cortical Coupling: Computational Efficiency • For statistical accuracy in these CG patch settings, Kinetic Theory is 103 -- 105 more efficient than I&F; 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 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) (3) Excitatory Cells Shown (1) 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 Sexc/pN Summary • Kinetic Theory is a numerically efficient (103 -- 105 more efficient than I&F), 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. • 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) Too good to be true? What’s missing? • First, the zeroth moment is more accurate than the first moment, as in many moment closures 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 Too good to be true? What’s missing? • But a far more serious problem • Kinetic Theory does not capture detailed “spike-timing” information Why does the kinetic theory (Boltzman-type approach in general) not work? Note E nsem ble A verage (N etw ork M echanism ) N etw ork M echanism (E nsem ble A verage) Too good to be true? What’s missing? • But a far more serious problem • Kinetic Theory does not capture detailed “spike-timing” statistics 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 Line-Motion-Illusion LMI Stimulus Model Voltage Direct ‘naïve’ coarse graining may not suffice: • Priming mechanism relies on Recruitment time 0 128 Model NMDA space • Recruitment relies on locally correlated cortical firing events • Naïve ensemble average destroys locally correlated events 0% ‘coarse’ 40% ‘coarse’ Trials Conclusion • Kinetic Theory is a numerically efficient (103 - 105 more efficient than I&F), and remarkably accurate. • Kinetic Theory accurately captures firing rates in fluctuation dominated systems • Kinetic Theory does not capture detailed spiketimed 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).