AACIMP 2011 Summer School. Neuroscience stream. Lecture by Anton Chizhov.

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. Neurons Neuronal populations Large-scale simulations (NMM & FR-models for EEG & MRI)

5. Experiment . Thalamic neuron responses on 3 trials of visual stimulation by movie.

6. [E.Aksay, R.Baker, H.S.Seung, D.W.Tank J.Neurophysiol. 84:1035-1049, 2000] Activity of a 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. [R.M.Bruno, B.Sakmann // Science 312:1622-1627, 2006] Population PSTH of thalamic neurons’ r esponses to a 2-Hz sinusoidal deflection of their respective principal whiskers ( n = 40 cells). Commonly information is coded by firing rate

7. 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. spontaneous activity evoked activity Commonly populations are localized in cortical space

8. F. Chavane, D. Sharon, D. Jancke, O.Marre, Y. Frégnac and A. Grinvald // Frontiers in Systems Neuroscience , v.5, article 4 , 1-26, 2011. Local interactions in visual cortex

11. GABA-IPSC AMPA-EPSC AMPA-EPSC AMPA-EPSP AMPA-EPSP GABA-IPSP GABA-IPSC GABA-IPSP PSP PSP Firing rate Firing rate Spike Spike Threshold criterium Population model Synaptic conductance kinetics Membrane equations Eq. for spatial connections

18. Direct Monte-Carlo simulation of individual neurons: Firing-rate : Probability Density Approach (PDA ) : Types of population models (4000) Assumption. Neurons are de-synchronized. “ f-I-curve” Стимулирующий ток

20. Leaky Integrate-and-Fire (LIF) neuron with noise ( stochastic LIF )

21. Kolmogorov-Fokker-Planck eq. for ρ (t,V) of LIF-neurons V T V reset ρ Hz 0 V Problem! Voltage can not uniquely characterize neuron’s state. Stimulation current

22. Refractory Density model [Chizhov et al. // Neurocomputing 2006] Stimulation current

23. where according to Spike Response Model (SRM): [W.Gerstner, W.Kistler, 2002] Similar approach: Refractory Density model for SRM-neurons

25. 1. Threshold neuron model Approximations for are taken from [L.Graham, 1999]; I AHP is from [N.Kopell et al., 2000] Full single neuron model Threshold model

27. 2. Refractory density approach ( t* - parameterization) Boundary conditions: -- firing rate t * is the time since the last spike -- Hazard function [Chizhov, Graham // PRE 2007,2008] [Chizhov et al. // Neurocomputing 2006]

28. 3. Hazard function

29. A – solution in case of steady stimulation ( self-similar ); B – solution in case of abrupt excitation Single LIF neuron - Langevin equation Fokker-Planck equation Hazard-function in the case of white noise-current ( First-time passage problem ) Approximation :

30. Self-similar solution (T=const) Assumption. U(t) = const (or T(t)=const) . Notation: Then the shape of , which is , is invariable. Equivalent formulation :

31. Frozen Gaussian distribution (dT/dt = ∞) 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). Assumption. For the simplicity, we consider the case of arbitrary but monotonically increasing T(t) and the Gaussian distribution or [x] + for x>0 and zero otherwise U(t) U T

32. Approximation of hazard function in arbitrary case Weak stimulus Strong stimulus Approximation : A – solution in case of steady stimulation ( self-similar ); B – solution in case of abrupt excitation Approximation of H by A is green , by B is blue , by A+B is red , exact solution is black .

33. Langevin equation Fokker-Planck eq. Hazard-function in the case of c o l o r e d noise Without noise : With noise : or or

34. Self-similar solution (T=const) Assumption. U(t) (or T(t)) is constant or slow. Then the shape of , which is , is invariable. u q

35. Approximation of H by A is green , by B is blue , by A+B is red , exact solution is black . Hazard function in arbitrary case K=1: K=8: Weak stimulus Weak stimulus Strong stimulus Strong stimulus

36. Simulations with CBRD-model

37. Non-adaptive neurons (4000) Single population: comparison of CBRD with Monte-Carlo

38. Single population: current-step stimulation. Color noise. Adaptive neurons. LIF Adaptive conductance-based neuron

39. with I M Single population: oscillatory input

40. Single population: comparison of CBRD with analytical solution for stochastic LIF in steady-state

41. Firing rate depends on the noise time constant . dots – Monte-Carlo solid – eq .(*) dash – adiabatic approach [Moreno-Bote, Parga 2004] (*) Single population: color noise, comparison with “adiabatic approach”

42. Single cell level Populations t * is the time since the last spike CBRD Large-scale simulations (NMM & FR-models for EEG & MRI)

43. From CBRD to Firing-Rate model

44. Hazard-function: -- firing rate Oscillating input Firing-rate model [Chizhov, Rodrigues, Terry // Phys.Lett.A, 2007 ] Hazard-function: -- firing rate Oscillating input [ Чижов, Бучин // Нейроинформатика-2009 ] Not-adaptive neurons Adaptive neurons

46. Monte-Carlo simulations : conventional Firing-Rate model : CBRD : Mathematical complexity : 10 4 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 modified Firing-Rate model ( non-stationary and adaptive ):

47. Simulations with FR-model

48. GABA-IPSC AMPA-EPSC AMPA-EPSC AMPA-EPSP AMPA-EPSP GABA-IPSP GABA-IPSC GABA-IPSP PSP PSP Firing rate Firing rate Spike Spike Membrane equations Threshold criterium Population model Synaptic current kinetics Interconnected populations

49. Approximation of synaptic current Presynaptic spike Postsynaptic current

51. There is no plasticity in the model reproducing the experimental monosynaptic IPSCs evoked by extracellular pulse trains. Fig 1. IPSC-kinetics in the experiment and model. The maximum amplitudes of IPSC and IPSP in the model, shown at the right, are the same as registered in the experiment, 1.2nA and 14mV. Fig 2. Paired-pulse modulation of IPSCs in the experiment and model. Fig 3. Frequency-dependent IPSC modulation with repetitive stimulation in the experiment and model. M.Vreugdenhil, J.G.R.Jefferys, M.R.Celio, B.Schwaller. Parvalbumin-Deficiency Facilitates Repetitive IPSCs and Gamma Oscillations in the Hippocampus. J Neurophysiol 89: 1414-1422, 2003. Synaptic integration

52. Simple model of interacting cortical interneurons, evoked by thalamus Синаптические токи и проводимости: Мембранный потенциал: Популяционная частота спайков: Рис. 12. Схема активности популяции FS (fast spiking) нейронов, возбуждаемых внешним стимулом ν ext (t), приходящим из таламуса. Обозначения: ν (t) – популяционная частота спайков FS нейронов, g E (t), g I (t) – проводимости возбуждающих и тормозящих синапсов. FS ν ext ν g I g E Experiment Model Рис. 13. Постсинаптический (моносинаптический) ток в FS-нейроне при слабой таламической стимуляции током 30 μA и потенциале фиксации ‑88 mV в эксперименте (вверху) (adapted by permission from Macmillan Publishers Ltd: (Cruikshank et al., 2007), copyright 2007) и в модели (внизу). Рис. 14. Ответы FS-нейронов на таламическую стимуляцию током 120 μA в эксперименте (слева) (adapted by permission from Macmillan Publishers Ltd: (Cruikshank et al., 2007), © 2007) и в модели (справа). A, B - постсинаптические токи при потенциале фиксации -88, -62, и -35 mV; C, D - синаптические проводимости; E, F – постсинаптические потенциалы U и модельная популяционная частота ν .

53. Firing-rate model of adaptive neuron population: « interictal » activity

54. Simulations with CBRD-model

55. with I M and I AHP [S.Karnup, A.Stelzer 2001] Experiment Simulations. Interictal activity. R ecurrent network of pyramidal cells, including all-to-all connectivity by excitatory synapses . Model

56. Simulations. Gamma rhythm. R ecurrent network of interneurons , including all-to-all connectivity by inhibitory synapses

58. Spatial connections - firing rate on presynaptic terminals; - firing rate on somas . Assumption: distances from soma to synapses have exponentially decreasing distribution p(x) [ B.Hellwig 2000] . [V.Jirsa, G.Haken 1996] [P.Nunez 1995] [J.Wright, P.Robinson 1995] where γ = c/ λ ; c – the average velocity of spike propagation along the cortex surface by axons ; λ – characteristic axon length . [D.Contreras, R.Llinas 2001] Experiment :

60. Waves In the case of reduced GABA-reversal potential V GABA = -50mV and stimulation by extracellular electrode we obtain a traveling wave of stable amplitude and velocity 0.15 m/s. The velocity is much less than the axon propagation velocity (1m/s) and is determined mostly by synaptic interactions. B 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. A [Leinekugel et al. 1998]. Spontaneous GDPs propagate synchronously in both hippocampi from septal to temporal poles. Multiple extracellular field recordings from the CA3 region of the intact bilateral septohippocampal complex. Simultaneous extracellular field recordings at the four recording sites indicated in the scheme. Corresponding electrophysiological traces (1– 4) showing propagation of a GDP at a large time scale. [D.Golomb, Y.Amitai, 1997] Propagation of discharges in disinhibited neocortical slices. Model Experiments Waves with unchanging chape and velocity are observed in cortical tissue in disinhibiting or overexciting conditions. The waves are produced by complex interaction of pyramidal cells and interneurons. That is confirmed by much lower speed of the wave propagation comparing with the axon propagation velocity which is the coefficient in the wave-like equation. Analysis of wave solutions and more detailed comparison with experiments are expected in future.

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