Models of neuronal populations

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AACIMP 2010 Summer School lecture by Anton Chizhov. "Physics, Chemistry and Living Systems" stream. "Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex" course. Part 2.
More info at http://summerschool.ssa.org.ua

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Models of neuronal populations

  1. 1. Models of neuronal populations Anton V. Chizhov Ioffe Physico-Technical Institute of RAS, St.-Petersburg Definitions: Population is a great number of similar neurons receiving similar input Population activity (=population firing rate) is the number of spikes per unit time per total number of neurons
  2. 2. Neurons Neuronal populations Large-scale simulations (NMM & FR-models for EEG & MRI)
  3. 3. Overview • Experimental evidences of population firing rate coding • Conductance-based neuron model • Probability Density Approach (PDA) • Conductance-Based Refractory Density approach (CBRD) - threshold neuron - t*-parameterization - Hazard-function for white noise - Hazard-function for colored noise • Simulations of coupled populations • Firing-Rate model • Hierarchy of visual cortex models
  4. 4. • What can be modeled on population level? • Which details are important? • What kinds of population models do exist? • Which one to choose?
  5. 5. Commonly information is coded by firing rate [R.M.Bruno, B.Sakmann // Science 312:1622-1627, 2006] Population PSTH of thalamic neurons’ responses to a 2-Hz sinusoidal deflection of their [E.Aksay, R.Baker, H.S.Seung, D.W.Tank Activity of a J.Neurophysiol. 84:1035-1049, 2000] respective principal whiskers (n = 40 cells). position neuron during spontaneous saccades and fixations in the dark. A: horizontal eye position (top 2 traces), extracellular recording (middle), and firing rate (bottom) of an area I position neuron during a scanning pattern of horizontal eye movements.
  6. 6. Commonly populations are localized in cortical space Whole-cell (WC) recording of a layer 2/3 neuron of the C2 cortical barrel column was performed simultaneously with measurement of VSD fluorescence under conventional optics in a urethane anesthetized mouse.
  7. 7. Pure population events observed in experiments: • Evoked responses • Oscillations •Traveling waves Voltage-sensitive Dye Optical Imaging [W.Tsau, L.Guan, J.-Y.Wu, 1999]
  8. 8. • What can be modeled on population level? • Which details are important? • What kinds of population models do exist? • Which one to choose?
  9. 9. Synaptic conductance kinetics GABA-IPSC AMPA-EPSC AMPA-EPSC GABA-IPSC Membrane GABA-IPSP AMPA-EPSP equations AMPA-EPSP GABA-IPSP PSP PSP Threshold criterium Spike Spike Population model Firing rate Firing rate Eq. for spatial connections
  10. 10. • ionic channel kinetics • input signal is 2-d Model of a pyramidal neuron dV C = − I Na − I DR − I A − I M − I H − I L − I AHP − s(t ) (V − V0 ) + u(t ) + η (t ) dt u(t ) = ∑S g S (t ) (VS − V0 ) + I electrode (t ) I ... = g... x p (t ) y q (t ) (V (t ) − V... ) s(t ) = ∑S g S (t ) dx x∞ (U ) − x = , dt τ x (U ) dy y∞ (U ) − y = dt τ y (U ) Approximations for I Na , I DR , I A , I M , I H are from [L.Graham, 1999]; EXPЕRIМЕNТ IAHP is from [N.Kopell et al., 2000] MODEL Color noise model (Ornstein-Uhlenbeck process): dη τ = −η + 2τ σξ (t ) dt
  11. 11. • synaptic channel kinetics
  12. 12. • neuron is spatially distributed A 2-comp. neuron with synaptic currents at somas 15 0 B PSC, exp. PSP, exp. Vd -50 PSP, model 2 10 PSP, model 1 PSP, mV PSC, pA Vd -100 Vs Is 5 -150 -200 0 0 5 10 t, ms 15 C Two boundary problems: ∂V  ∂V  g=Id/(Vd-Vrev) Vd A) current-clamp to register PSP: ∂X = R Gs  V + ; X =0  ∂T  B) voltage-clamp to register PSC: V (T ,0) = 0; Vd ∂V Vd Is = R I S (T ) Vs ∂X X =L V0 [F.Pouille, ∂V ∂ V 2 M.Scanziani − +V = 0 //Nature, 2004] ∂T ∂X 2 X=0 X=L Figure Transient activation of somatic and delayed Solution: activation of dendritic inhibitory conductances in dV I [A.V.Chizhov // τm = −(V − V rest ) + ρ (Vd − V ) − S Biophysics 2004] experiment (solid lines) and in the model (small circles). dt Gs A, Experimental configuration. dV 1  ∂I d  B, Responses to alveus stimulation without (left) and with τ m d = −(Vd − V rest ) − (2 + ρ )(Vd − V ) − τ m  + 3I d   (right) somatic V-clamp. dt ρ Gs  ∂t  C, In a different cell, responses to dynamic current injection Parameters of the model: in the dendrite; conductance time course (g) in green, 5-nS τm= 33 ms, ρ = 3.5, Gs= 6 nS in B and 2.4 nS in C peak amplitude, Vrev=-85 mV.
  13. 13. • spatial structure of connections 1 mm Эксперимент. Зрительная кора. Карта ориентационной избирательности активности нейронов. Модель “Pinwheels” карты ориентационной избирательности входных сигналов. Модель. Ответ зрительной коры на полосу горизонтальной, а затем вертикальной ориентации.
  14. 14. • What can be modeled on population level? • Which details are important? • What kinds of population models do exist? • Which one to choose?
  15. 15. Population models • Definition A population is a set of similar neurons receiving a common input and dispersed due to noise and intrinsic parameter distribution. • Common assumptions: – Input – synaptic current (+conductance) – Infinite number of neurons – Output – population firing rate (4000) 1 nact (t; t + ∆t ) ν (t ) = lim lim ∆t →0 N →∞ ∆t N
  16. 16. Direct Monte-Carlo simulation of individual neurons: Types of population models ∂V C = I − g L (V − VL ) + σ I ξ (t ) ∂t если V > V T , т V = Vreset и спайк 1 nact ( t + ∆t ) ν (t ) = ∆t N Firing-rate: ν (t ) = f ( I (t )) f “f-I-curve” dν or τ = −ν + f ( I (t )) dt dU or C = I − g L (U − VL ), dt I ν (t ) = ~(U (t )) f Assumption. Neurons are de-synchronized. Probability Density Approach (PDA): (4000) RD модель : ∂ρ ∂ρ + = − ρH ∂t ∂t *  ∂U ∂U  C +  = I − g L (U − VL )  ∂t ∂t *  1 H (U ( t , t*)) = ( A(U ) + B(U , dU / dt )) τm ∞ v (t ) = ρ (t ,0) = ∫ ρ H dt * +0
  17. 17. Idea of Probability Density Approach (PDA) Single neuron equation (e.g. H-H model) r dX = F(X ) + S r r r dt r where F is the common deterministic part, r S is the noisy term. X = (V , m, h, n ) r For classical H-H: ρ ( X , t) r Eq. for neural density ∂ρ ∂t =− ∂ ∂X r r ( ∂ r ⋅ F(X ) ρ + r ∂X )  t ∂ρ  ⋅ W r   ∂X  [B.Knight 1972]   t r where the matrix W represents the influence of noise S Problem! The equation is multi-dimensional. Particular cases are [A.Omurtag et al. 2000] X ≡V - membrane potential [D.Nykamp, D.Tranchina 2000] [N.Brunel, V.Hakim 1999], … X ≡ t* - time passed since the last spike [J.Eggert, JL.Hemmen 2001] [А.Чижов, А.Турбин 2003] X ≡τ - time till the next spike [A.Turbin 2003]
  18. 18. ρ Simplest 1-d PDAs • Kolmogorov-Fokker-Planck eq. for ρ(t,V) Hz Leaky Integrate-and-Fire (LIF) neuron: dV τm = −V + RI (t ) + η (t ), dt if V > V T then V = Vreset 0 Vreset VT < η (t ) >= 0, < η (t ) η (t ' ) >= τ m σ 2 δ (t − t ' ) ν ∂ρ ∂ σ 2 ∂2ρ τm = [(V − RI ) ρ ] + + ν ⋅ δ (V − Vreset ) ∂t ∂V 2 ∂V 2 Hz σ 2 ∂ρ Problem! Voltage can not ν (t ) = uniquely characterize 2 ∂V V =V T neuron’s state. 0 t • Refractory density ρ(t,t*) for SRM - neurons ∂ρ ∂ρ ∞ + ∗ = −ρ H ρ (t ,0) ≡ ν (t ) = ∫ ρ (t , t * ) dt * ∂t ∂t 0 H = H (U (t , t*), V T ) Spike Response Model (SRM): U (t , t * ) = η (t * ) + ∫ k (t * , t ') I (t ' ) dt ' t* [W.Gerstner, W.Kistler, 2002] 0
  19. 19. 1-D Refractory Density Approach for conductance- based neurons (CBRD) [A.V.Chizhov, L.J.Graham // Phys. Rev. E 2007, 2008] 1. Threshold single-neuron model 2. Refractory density approach (t* - parameterization) 3. Hazard-function t* is the time since the last spike; H ≈ A+ B ∂ρ ∂ρ + = −ρ H ∂t ∂t ∗  ∂U ∂U  C + *  = − I DR − I A − I M − I H − I L − I AHP − I i H(U) = ‘frozen stationary’ + ‘self-similar’  ∂t ∂t  solutions of Kolmogorov-Fokker-Planck eq. ∂x ∂x x (U ) − x + * = ∞ , ∞ ∂t ∂t τ x (U ) for I&F neuron with white or color v (t ) = ρ H dt * ∫ noise-current ∂y ∂y y ∞ (U ) − y + = +0 ∂t ∂t * τ y (U )
  20. 20. 1. Threshold neuron model Full single neuron model dV C = − I Na − I DR − I A − I M − I H − I L − I AHP − I i dt Approximations for I Na , I DR , I A , I M , I H are taken from [L.Graham, 1999]; IAHP is from [N.Kopell et al., 2000] Threshold model dU C = − I Na − I DR − I A − I M − I H − I L − I AHP − I i dt dx x∞ (U ) − x dy y∞ (U ) − y = , = dt τ x (U ) dt τ y (U ) if U > V T then U = U reset = −40 mV for I DR : x = x reset = 0.262, y = y reset = 0.473; for IA : x = x reset = 0.743, y = y reset = 0.691; for IH : y = y reset = 0.002; for IM : x = x + ∆ x reset , ∆ x reset = 0.18 (1 − x ); for I AHP : w = w + ∆ w reset , ∆ w reset = 0.018 (1 − w).
  21. 21. 2. Refractory density approach (t* - parameterization) t* is the time since the last spike; ρ = ρ (t , t * ), U = U (t , t * ), x = x (t , t * ), y = y (t , t * ) ∂ρ ∂ρ d • ∂ • dt * ∂ • ∂ • ∂ • + ∗ = −ρ H = + = + ∂t dt ∂t * ∂t ∂t * ∂t ∂t dt  ∂U ∂U  C + *  = − I DR − I A − I M − I H − I L − I AHP − I i  ∂t ∂t  ∂x ∂x x∞ (U ) − x + *= , ∂t ∂t τ x (U ) I ... = g ... x y (U − V... ) for I DR , I A , I M , I H , I AHP ∂y ∂y y∞ (U ) − y + = ∂t ∂t * τ y (U ) H (U ) = 1 τ m ( A(U ) + B (U , dU dt ) ) -- Hazard function τ m = C /( g DR (t , t * ) + g A (t , t * ) + g M (t , t * ) + g H (t, t * ) + g L + g AHP (t , t * )) Boundary conditions: A(U ) = exp(6.1 ⋅ 10−3 − 1.12 T − 0.257 T 2 − 0.072 T 3 − 0.0117 T 4 ). ∞ B (U ) = -τ m 2 dT ~ F (T ), T= U −U T , ~ F (T ) = 2 exp( −T ) 2 ρ (t ,0) = ∫ ρ F dt ∗ ≡ ν (t ) -- firing rate dt σ π 1 + erf (T ) +0 U (t ,0) = U reset x (t ,0) = x reset , y (t ,0) = y reset for I DR , I A , I H ; Application x (t ,0) = x (t , t *T ) + ∆ x reset for I M ; w(t ,0) = w(t , t *T ) + ∆ w reset for I AHP ; t *T : U (t , t *T ) = U T ( dU (t , t *T ) dt ).
  22. 22. 3. Hazard-function in the case of white noise-current (First-time passage problem) A – solution in case of steady stimulation (self-similar); Approximation: H ≈ A+ B B – solution in case of abrupt excitation Single LIF neuron - Langevin equation dV < η (t ) > = 0 τm = −V + U (t ) + η (t ) < η (t )η (t ' ) >= σ 2 τ m δ (t − t ' ) dt if V < UT then spike Fokker-Planck equation ~ ∂ ∂ρ  ~ σ 2 ∂ρ  τm ~− + (U (t ) − V ) ρ 2 ∂V  = 0 ρ (t,U T ) = 0 ~ ∂t ∂V   ρ (t ,−∞) = 0 ~ σ 2 ∂ρ~ exp (− (V − U ) 2 σ 2 ) H (t ) ≡ − ~ 1 ρ (0,V ) = 2τ m ∂V V =U T πσ u (t ) = (V (t ) − U (t )) σ ˆ T (t ) = (U T − U (t )) σ ˆ ~ ~ ρ (t , T (t )) = 0 ~ ∂ρ ∂  ~ − 1 ∂ρ  = 0 τm + −uρ ρ (t ,−∞) = 0 ~ ∂t ∂u  2 ∂u   exp(− u 2 ) ~ 1 ρ (0, u ) = ~ 1 ∂ρ ~ π H (t ) ≡ H (t ) / τ m = − 2τ m ∂u u =T ( t )
  23. 23. Self-similar solution (T=const) Equivalent formulation: ~ ρ ( t , u ) = ρ (t ) p (t , u ) T (t ) where ρ (t ) = ∫ ~ ρ (t , u ) du −∞ ρ (t ) − amplitude , p(t , u ) − shape p (t , T ) = 0 ∂p ∂  1 ∂p  ~ ~ 1 ∂p τm + − u p − = H (t ) ⋅ p where H (t ) = − p(t ,−∞) = 0 ∂t ∂u  2 ∂u   2 ∂u u = T exp(− u 2 ) 1 p(0, u ) = dρ ~ π τm = − ρ H (t ), dt ~ Assumption. U(t)=const (or T(t)=const). Notation: A ≡ H ~ Then the shape of ρ , which is p(t , u) , is invariable. ∂  1 ∂p  1 ∂p p (t , T ) = 0  −u p−  = A(t ) ⋅ p where A(t ) = − ∂u  2 ∂u  2 ∂u u =T p(t ,−∞) = 0 dρ τm = − ρ A(T ) dt
  24. 24. Frozen Gaussian distribution (dT/dt = ∞) Assumption. T(t) decreases fast. The initial Gaussian distribution remains almost unchanged except cutting at u=T. The hazard function in this case is H=B(T,dT/dt). dρ τm = −ρ B T (t ) where ρ (t ) = ∫ ~ ρ (t , u ) du dt −∞ τ m dρ τ dρ  dT  or B=− =− m ρ dt ρ dT  dt  +   U(t) UT ~(t , u ) =  π exp(− u ), if  1 For the simplicity, we consider the case of 2 u ( t ) < T (t ) arbitrary but monotonically increasing T(t) and ρ the Gaussian distribution  0, otherwise  τ m dρ  dT   dT  ~ B=− = −τ m 2   F (T ) ρ dT  dt  +    dt  + ~ 2 exp( −T 2 ) where F (T ) = π 1 + erf(T ) [x]+ for x>0 and zero otherwise
  25. 25. Approximation of hazard function in arbitrary case ~ ∂ρ ∂  ~ ρ (t , T (t )) = 0 ~ где T (t ) = (U T − U (t )) σ τm + ~ − 1 ∂ρ  = 0 −uρ ˆ ∂t ∂u  2 ∂u  ρ (t ,−∞) = 0 ~   exp(− u 2 ) ~ 1 1 ∂ρ ~ ρ (0, u ) = H= ˆ π 2τ m ∂u u =T ( t ) A – solution in case of steady stimulation (self-similar); Approximation: H ≈ A+ B B – solution in case of abrupt excitation Weak stimulus Strong stimulus Approximation of H by A is green, by B is blue, by A+B is red, exact solution is black. ν (t ) = ∂ρ ∂t
  26. 26. 3. Hazard-function in the case of colored noise dU Langevin equation Without noise: C = − I tot (U , t ) U < UT dt dV du ~ With noise: C = − I tot (V , t ) + η (t ) V < UT τ m (U , t ) = −u + q(t ), u < T (t ) dt dt dη dq τ = −η + 2τ σ ξ (t ) or τ = − q + 2τ ξ (t ) dt dt < ξ (t ) > = 0 где u = gtot (U , t )(V − U ) / σ , q = η (t ) / σ , < ξ (t ) ξ (t ' ) >= τ δ (t − t ' ) ~ T (t ) = g (U , t )(U T − U ) / σ tot Fokker-Planck eq. ∂ρ ∂  − u + q ~  ∂  q ~  1 ∂ ρ ~ ~ 2 ρ (t, u = ∞, q) = ρ (t , u, q = −∞) = ~ ~ +  τ ρ  + − ρ  − =0 ∂t ∂u  m  ∂q  τ  τ ∂q 2 ~ ~ ~ ~ = ρ (t, u, q = +∞) = ρ (t , u = T , q ≤ T ) = 0 ~ 1 ∞ ~ ~ ~ ρ (t = 0, u, q) = ~ 1+ k 2π k 1+ k exp [  − (1 + k )u 2 − q 2 + 2 qu  ] H (U (t )) ≡ ρT ~ ∫ ( q − T ) ρ (t, T , q) dq, k (U , t ) ≡ τ m (U , t )/τ  2k  or ~ T ∞ ∞ ρ ( t , u, q ) = ρ ( t ) p ( t , u, q ) T (t ) ~ where ∫−∞ dq ∫− ∞ p (t , u, q ) du = 1 ρ (t ) = ∫ ∫ ρ (t, u, q) dq du. ~ ρ (t ) − amplitude , p(t , u ) − shape −∞−∞ dρ ~ ∞ τm = − ρ H (t ), where ~ ~ ~ H (U (t )) ≡ ∫ ( q − T ) p (t , T , q ) dq dt ~ T ∂p ∂ ∂ ∂2 p  ~ p(t , u = ∞, q) = p(t , u, q = −∞) = τm = (u − q) p + k  qp + 2  + H (t ) p ∂t ∂u  ∂q ∂q  ~ ~ = p(t , u, q = +∞) = p(t , u = T (t ), q ≤ T (t )) = 0
  27. 27. Self-similar solution (T=const) Assumption. U(t) (or T(t)) is constant or slow. ~ Then the shape of ρ , which is p(t , u, q), is invariable. ∂ ∂ ∂2 p  p(t, u = ∞, q) = p(t, u, q = −∞) = (u − q) p + k  qp + 2  + A p = 0 ~ ~ ∂u  ∂q ∂q  = p(t , u, q = +∞) = p(t, u = T , q ≤ T ) = 0 ∞ ~ ~ A = ∫ ( q − T ) p (t , T , q ) dq where ~ 1+ k T =T ~ T 2 q u A ∞ (T) = exp(0.0061 - 1.12 T - 0.257 T 2 - 0.072 T 3 - 0.0117 T 4 )
  28. 28. Hazard function in arbitrary case H ≈ A+ B K=1: Weak stimulus Strong stimulus K=8: Weak stimulus Strong stimulus Approximation of H by A is green, by B is blue, by A+B is red, exact solution is black. ν (t ) = ∂ρ ∂t
  29. 29. CBRD Single cell level t* is the time since the last spike ∂ρ ∂ρ + ∗ = −ρ H ∂t ∂t  ∂U ∂U  C +  = − I DR − I A − I M − I H − I L − I AHP − I i  ∂t ∂t *  ∂x ∂x x∞ (U ) − x Populations + *= , I ... = g ... x y (U − V... ) ∂t ∂t τ x (U ) for I DR , I A , I M , I H , I AHP ∂y ∂y y∞ (U ) − y + *= ∂t ∂t τ y (U ) ∞ ρ (t ,0) = ∫ ρ F dt ∗ ≡ ν (t ) +0 Large-scale simulations (NMM & FR-models for EEG & MRI)
  30. 30. Simulations by CBRD-model
  31. 31. Simulations. Current-step stimulation. Comparison with Monte-Carlo. Non-adaptive neurons (4000)
  32. 32. Simulations. Current-step stimulation. Color noise. LIF Adaptive neurons. Adaptive conductance-based neuron
  33. 33. Simulations. Oscillatory input. with IM
  34. 34. Simulations. Constant current stimulation. Comparison with analytical solution. [Johannesma 1968]
  35. 35. Simulations. Constant current stimulation. Color noise. Comparison with analytical solution. −1 (*)  a  u ' τ m H (u )   ν = τ m ∫0 exp − ∫0 du  /(a − u ′) du′   a−u   a = I a /g L (U T − VL ) dots – Monte-Carlo solid – eq.(*) dash – adiabatic approach [Moreno-Bote, Parga 2004] Firing rate depends on the noise time constant.
  36. 36. Interconnected populations Synaptic current kinetics GABA-IPSC AMPA-EPSC AMPA-EPSC GABA-IPSC Membrane GABA-IPSP AMPA-EPSP equations AMPA-EPSP GABA-IPSP PSP PSP Threshold criterium Spike Spike Population model Firing rate Firing rate
  37. 37. Pyramidal neurons Approximations of synaptic currents 200 AMPA-PSC 40 NMDA-PSC (with PTX, APV) (with PTX, CNQX) 150 20 Vh=-40 mV Vh=-80 mV experiment model 0 PSC, pA 100 PSC, pA experiment model Excitatory synaptic current: -20 50 iE = i AMPA + i NMDA Vh=-40 mV -40 i AMPA = g AMPA m AMPA (t ) (V − V AMPA ) 0 Vh=+20 mV -60 Vh=+20 mV iNMDA = g NMDAm NMDA (t ) f NMDA (V ) (V − VNMDA ) -50 -80 0 10 20 30 40 50 0 25 50 75 100 t, ms t, ms gj - maximum specific conductance, 0 0 mj - non-dimensional conductance GABA-PSC (with CNQX, D-AP5) Vj - reversal potential -100 f NMDA (V ) = 1 /(1 + Mg / 3.57 exp( −0.062V )) -50 Vh=-64 mV fast GABA-A -IPSC PSC, pA (with CNQX, D,L-APV) PSC, pA -200 Vh=-60 mV Inhibitory synaptic current: -300 i I = g GABAmGABA (t ) (V − VGABA ) -100 experiment model experiment Non-dimensional synaptic conductances: -400 model d 2m j dm j ττ + (τ rj + τ d ) + m j = S (ν j ), r d 0 10 20 30 40 50 -500 0 10 20 30 40 50 j j j t, ms t, ms dt 2 dt Interneurons j = AMPA , GABA , NMDA 500 AMPA-PSC (with PTX, APV) 150 NMDA-PSC where S ( ν j ) = 2 ( 1 + exp( −2τ ν j ) ) − 1 τ = 1 µs 400 (with PTX, CNQX) τ r , τ d - rise and decay time constants j j experiment 100 Vh=-40 mV 300 Vh=-80 mV ν j (t ) - firing rate on j-type axonal terminals model PSC, pA 50 PSC, pA 200 experiment 0 model 100 -50 Vh=-40 mV 0 -100 Vh=+20 mV Vh=+20 mV -100 -150 0 10 20 30 40 50 0 25 50 75 100 t, ms t, ms
  38. 38. Simulations. Interictal activity. Recurrent network of pyramidal cells, including all-to-all connectivity by excitatory synapses. I i (t ) = I ext (t ) + I S (t ), Model with IM and IAHP I S (t ) = g S (t ) (U (t ) − VS ), 2 2 d g S (t ) dg (t ) τS + 2τ S S + g S (t ) = g S τ ρ (t ,0) dt 2 dt Experiment I = 150 pA ext τ S = 5.4 ms, τ = 1 ms, VS = 5 mV, g S = 1 mS/cm 2 σ V = 2 mV ( at rest ) [S.Karnup, A.Stelzer 2001]
  39. 39. Simulations. Gamma rhythm. Recurrent network of interneurons, including all-to-all connectivity by inhibitory synapses τ S = 3ms, d 2 g S (t ) dg (t ) I i (t ) = I ext (t ) + I S (t ), τ2 S + 2τ S S + g S (t ) = g S τ ν (t − τ d ) τ d = 1ms, dt 2 dt τ = 1ms, I S (t ) = g S (t ) (U (t ) − VS ), for density approach ν (t ) = ρ (t , t * = 0) VS = -80mV, g S = 7mS/cm 2
  40. 40. Model Experiments Oscillations Control (“Kainate”) +“Bicuculline” All the simulations were done with a single set of parameters. All the parameters except synaptic maximum conductances have been obtained by fitting to experimental registration of elementary events such as patch- electrode current-induced traces, Spikes in single neurons spike trains and monosynaptic responses . Conductances The model reproduces the following characteristics of gamma-oscillations : frequency of population spikes a single pyramidal cell does not fire Power Spectrum of Extracellular Potentials every cycle every interneuron fires every cycle bic con amplitude of EPSC is less than that of IPSC blockage of GABA-A receptors [Khazipov, Holmes, 2003] reduces the frequency Kainate-induced oscillations [A.Fisahn et al., 1998] in CA3. Cholinergically induced oscillations in CA3 peak of pyramidal cell’s firing frequency corresponds to the descending phase of EPSC and the ascending phase of IPSC firing of interneurons follows the firing of pyramidal cells gamma-oscillations are [N.Hajos, J.Palhalmi, E.O.Mann, B.Nemeth, homogeneous in space along the Spike timing of pyramidal and inhibitory cells. O.Paulsen, and T.F.Freund. J.Neuroscience, cortical surface (data not shown) 24(41):9127–9137, 2004]
  41. 41. Spatial connections ϕij (t , x, y ) = ∫ ∫ ν i (t − d ( x, y , X , Y ) / c, X , Y ) W ( x, y , X , Y ) dX dY , d ( x, y , X , Y ) = ( x − X ) 2 + ( y − Y ) 2 Experiment: φ ( t , x , y ) - firing rate on presynaptic terminals; ν ( t , x , y ) - firing rate on somas. Assumption: distances from soma to synapses have exponentially decreasing distribution p(x) [B.Hellwig 2000]. d ( x , y , X ,Y ) − W ( x, y , X , Y ) = e λ ∂ 2φ ∂φ 2 ∂ φ 2 ∂ 2φ   2 ∂ + 2γ + γ φ − c  2 + 2  =  γ + γ  ν (t , x, y ) 2  ∂x ∂t 2 ∂t  ∂y    ∂t  [V.Jirsa, G.Haken 1996] where γ = c/λ; c – the average velocity of spike [P.Nunez 1995] propagation along the cortex surface by axons; [J.Wright, P.Robinson 1995] λ – characteristic axon length. [D.Contreras, R.Llinas 2001]
  42. 42. Model Experiments Evoked responses A B The model reproduces postsynaptic currents and postsynaptic potentials registered on somas of pyramidal cells, namely: monosynaptic EPSCs and EPSPs [S.Karnup, A.Stelzer 1999] Effects of GABA-A disynaptic IPSC/Ps followed be EPSC/Ps receptor blockade on orthodromic potentials in CA1 polysynaptic EPSC/Ps pyramidal cells. Superimposed C responses in a pyramidal cell reduction of delays in polysynaptic EPSCs soma before and after decay of excitation after II component of application of picrotoxin (PTX, 100 muM). Control and PTX poly-EPSCs in presence of GABA-A receptor recordings were obtained at V block. rest (-64 mV; 150 muA stimulation intensities; 1 mm The model predicts that the evoked responses distance between stratum [B.Mlinar, are essentially non-homogeneous in space: radiatum stimulation site and A.M.Pugliese, perpendicular line through stratum pyramidale recording R.Corradetti site). The recordings were 2001] Components of carried out in ‘minislices’ in complex synaptic which the CA3 region was cut responses evoked in CA1 off by dissection. pyramidal neurones in the presence of GABAA receptor block. PSPs and PSCs evoked by extracellular stimulation and registered at 3.5cm away, w/ and w/o kainate. [V.Crepel, R.Khazipov, Y.Ben-Ari, 1997] In normal concentrations of Mg and in the absence of CNQX, block of GABA-A Spatial profiles of membrane potential and receptors induced a late synaptic response. firing rate in pyramids.
  43. 43. Model Experiments Waves In the case of reduced GABA-reversal potential VGABA= -50mV and stimulation Waves with unchanging chape and by extracellular electrode we obtain a velocity are observed in cortical tissue traveling wave of stable amplitude and in disinhibiting or overexciting velocity 0.15 m/s. The velocity is much conditions. The waves are produced less than the axon propagation velocity by complex interaction of pyramidal (1m/s) and is A determined mostly by cells and interneurons. That is synaptic interactions. confirmed by much lower speed of the wave propagation comparing with the 140 voltage, pyramids axon propagation velocity which is the voltage, interneurons 120 firing rate, pyramids firing rate, interneurons coefficient in the wave-like equation. 100 80 -40 Analysis of wave solutions and more mV Hz 60 B detailed comparison with experiments 40 20 -60 are expected in future. 0 0 25 50 75 100 ms [Leinekugel et al. 1998]. Spontaneous GDPs propagate synchronously in both hippocampi from septal to 120 voltage, pyramids temporal poles. Multiple extracellular field recordings from the CA3 100 voltage, interneurons firing rate, pyramids region of the intact bilateral septohippocampal complex. firing rate, interneurons 0.15m/s -40 Simultaneous extracellular field recordings at the four recording 80 sites indicated in the scheme. Corresponding electrophysiological mV Hz 60 traces (1–4) showing propagation of a GDP at a large time scale. 40 -60 20 0 10 20 30 40 mm Fig.5. Wave propagating from the site of extracellular stimulation at right border of the “slice”. A, Evoked responses of pyramidal cells and interneurons at the site of stimulation. B, Profiles of mean voltage and firing rate in pyramidal cells and interneurons at the time 200 ms after the stimulus. [D.Golomb, Y.Amitai, 1997] Propagation of discharges in disinhibited neocortical slices.
  44. 44. From CBRD to Firing-Rate model
  45. 45. Macro- and meso-scale macro-scale meso-scale micro-scale external granular layer external pyramidal layer internal granular layer internal pyramidal layer AP generation zone synapses [S.Kiebel] [C.Friston]
  46. 46. Not-adaptive neurons Firing-rate model Adaptive neurons dU C = −( g L + g S )(U − VL ) − I C dU = −( g L + g S )(U − VL ) − g M n 2 (t )(U − VM ) − g AHP w(t )(U − V AHP ) + I dt dt Hazard-function: Hazard-function: ν (t ) = A (U ) + B(U , dU dt ) -- firing rate ν (t ) = A (U ) + B(U , dU dt ) -- firing rate τ m = C / gL τ m = C / gL −1 −1     T T (V −U ) / σ (V −U ) / σ A(U ) = τ m π ∫ e (1 + erf (u))du  ; A(U ) = τ m π ∫ e (1 + erf (u))du  ; u2 u2 (steady) (steady)      (V reset −U ) / σ   (V reset −U ) / σ  1  dU   (V T − U )2  1  dU   (V T − U )2  B(U ) = -τ m exp − ; ( sudden) B(U ) = -τ m exp − ; ( sudden) π σ  dt  +     σ2   π σ  dt  +     σ2   d 2w dw χ (1 − w) τ 1 τ AHP 0 + (τ 1 + τ AHP ) 0 − w∞ + w = v(t ) K (1 / τ 1 ,1 / τ AHP ) AHP 2 AHP 0 dt dt AHP d 2n dn ξ (1 − n) τ1 τM 0 + (τ 1 + τ M ) − n∞ + n = 0 v(t ) K (1 / τ 1 ,1 / τ M ) M 2 M 0 dt dt M Oscillating input Oscillating input [Chizhov, Rodrigues, Terry // Phys.Lett.A, 2007 ] [Чижов, Бучин // Нейроинформатика-2009 ]
  47. 47. Синаптические токи и проводимости: Simple model of interacting iE (t ) = g E (t ) (V (t ) − VE ) i I (t ) = g I (t ) (V (t ) − VI ) d 2 gE dg d 2gI dg cortical interneurons, τ 1Eτ 2E + (τ 1E + τ 2E ) E + g E = τ g E ν ext (t ) ττ I I + (τ 1I + τ 2I ) I + g I = τ g I ν (t ) dt 2 dt 1 2 dt 2 dt evoked by thalamus Мембранный потенциал: dU C = − g L (U − VL ) + i E (t ) + i I (t ), dt Experiment Популяционная частота спайков: ν (t ) = A (U ) + B (U ), -1  (VT −U ) / σ V 2  A(U ) = τ m π ∫ e (1 + erf (u ) ) du  ; u2    (Vreset −U ) / σ V 2  1  dU   (V T − U ) 2  B(U ) = × exp −    2π σ V  dt  +    2σ V2  Model gE νext FS ν gI Рис. 12. Схема активности популяции FS (fast spiking) нейронов, возбуждаемых внешним стимулом νext(t), Рис. 13. Постсинаптический приходящим из таламуса. (моносинаптический) ток в FS- Обозначения: ν(t) – нейроне при слабой популяционная частота спайков таламической стимуляции FS нейронов, gE(t), gI(t) – током 30 µA и потенциале Рис. 14. Ответы FS-нейронов на таламическую стимуляцию проводимости возбуждающих и фиксации -88 mV в током 120 µA в эксперименте (слева) (adapted by permission from тормозящих синапсов. эксперименте (вверху) (adapted Macmillan Publishers Ltd: (Cruikshank et al., 2007), © 2007) и в by permission from Macmillan модели (справа). A, B - постсинаптические токи при Publishers Ltd: (Cruikshank et al., потенциале фиксации -88, -62, и -35 mV; C, D - синаптические 2007), copyright 2007) и в модели проводимости; E, F – постсинаптические потенциалы U и (внизу). модельная популяционная частота ν.
  48. 48. Частотная модель популяции адаптивных нейронов: «интериктальная» активность I AHP (ν ), I M (ν ) I E I S (ν ) FR модель : ∂V C = I − I AHP (ν ) − I M (ν ) − g L (V − VL ) − I S (ν ) ∂t I S = g S (t )(V − VS ) d 2 g S (t ) d g S (t ) τS 2 2 + 2τ S + g S (t ) = g Sτv(t ) dt dt ν (t ) = A(U ) + B(U , dU / dt )
  49. 49. • What can be modeled on population level? • Which details are important? • What kinds of population models do exist? • Which one to choose?
  50. 50. Monte-Carlo conventional simulations: Firing-Rate modified Firing-Rate CBRD: model: model (non- Метод Монте − Карло : ∂V FR модель : stationary and RD модель : C = I − ( g L + g S )(V − VL ) + σ I ξ ( t ) ∂t adaptive): ∂ρ ∂ρ dU + = − ρH если V > V T , т V = Vreset и спайк C = I − ( g L + g S )(U − VL ) ∂t ∂t * dt 1 nact ( t + ∆ t ) ν (t ) = A(U ) + B (U , dU / dt ) FR модель :  ∂U ∂U  ν (t ) = C +  = I − ( g L + g S )(U − VL ) ∆t ∂V  ∂t ∂t *  N C = I − ( g L + g S )(V − VL ) − I M (ν ) − I AHP (ν ) ∂t 1 2 H (U (t , t*)) = ( A(U ) + B(U , dU / dt )) τS 2 d g S (t ) + 2τ S d g S (t ) + g S (t ) = g Sτv (t ) τm dt 2 dt ∞ ν (t ) = A(U ) + B (U , dU / dt ) v (t ) = ρ (t ,0) = ∫ ρ H dt * +0 Mathematical complexity: 104 ODEs 1 ODE a few ODEs 1-d PDEs Precision: 4 2 3 5 Precision for non-stationary problems: 5 2 4 5 Precision for adaptive neurons : 5 1 3 4 Computational efficiency: 2 5 5 4 Mathematical analyzability: 1 5 4 4

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