Models of neuronal populations
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AACIMP 2011 Summer School. Neuroscience stream. Lecture by Anton Chizhov.

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

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  • -так много моделей используем для проверки -все базируются на LIF , картинка в центре -распределение потенциалов в популяции – задаётся средним значением пот. во множестве нейронов популяции - частота популяции – это -синхронное состояние – это когда спайки большей части нейронов поп. Происх. в одно и тоже время, всплески частоты -ассинхронное состояние – это когда спайки равномерно распределены вдоль временного интервала -Монте-Карло, численная проверка, RD модель, точное решение, но сложна для анализа, FR модель, приближённое решение, но проще анализировать
  • -так много моделей используем для проверки -все базируются на LIF , картинка в центре -распределение потенциалов в популяции – задаётся средним значением пот. во множестве нейронов популяции - частота популяции – это -синхронное состояние – это когда спайки большей части нейронов поп. Происх. в одно и тоже время, всплески частоты -ассинхронное состояние – это когда спайки равномерно распределены вдоль временного интервала -Монте-Карло, численная проверка, RD модель, точное решение, но сложна для анализа, FR модель, приближённое решение, но проще анализировать
  • -так много моделей используем для проверки -все базируются на LIF , картинка в центре -распределение потенциалов в популяции – задаётся средним значением пот. во множестве нейронов популяции - частота популяции – это -синхронное состояние – это когда спайки большей части нейронов поп. Происх. в одно и тоже время, всплески частоты -ассинхронное состояние – это когда спайки равномерно распределены вдоль временного интервала -Монте-Карло, численная проверка, RD модель, точное решение, но сложна для анализа, FR модель, приближённое решение, но проще анализировать
  • -применяем для связанной популяции нейронов -система демонстрирует период. Решение при опр. подборе параметров -адаптация и возбуждение компенсировано в модели -подобная активность характ. Для иктальной (что это) активности в гиппокампе -простая модель, но даже это воспроизводит
  • -так много моделей используем для проверки -все базируются на LIF , картинка в центре -распределение потенциалов в популяции – задаётся средним значением пот. во множестве нейронов популяции - частота популяции – это -синхронное состояние – это когда спайки большей части нейронов поп. Происх. в одно и тоже время, всплески частоты -ассинхронное состояние – это когда спайки равномерно распределены вдоль временного интервала -Монте-Карло, численная проверка, RD модель, точное решение, но сложна для анализа, FR модель, приближённое решение, но проще анализировать

Models of neuronal populations Models of neuronal populations Presentation Transcript

  • 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
  • Neurons Neuronal populations Large-scale simulations (NMM & FR-models for EEG & MRI)
  • 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
    • What can be modeled on population level?
    • Which details are important?
    • What kinds of population models do exist ?
    • Which one to choose ?
  • Experiment . Thalamic neuron responses on 3 trials of visual stimulation by movie.
  • [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
  • 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
  • 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
  • Voltage-sensitive Dye Optical Imaging [W.Tsau, L.Guan, J.-Y.Wu, 1999]
    • Evoked responses
    • Oscillations
    • Traveling waves
    Pure population events observed in experiments:
    • What can be modeled on population level?
    • Which details are important?
    • What kinds of population models do exist ?
    • Which one to choose ?
  • 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
  • Approximations for are from [L.Graham, 1999]; I AHP is from [N.Kopell et al., 2000] Model of a pyramidal neuron Color noise model ( Ornstein-Uhlenbeck process) : MODEL EXP Е RI МЕ N Т
    • ionic channel kinetics
    • input signal is 2-d
  • 2-comp. neuron with synaptic currents at somas Figure Transient activation of somatic and delayed activation of dendritic inhibitory conductances in experiment (solid lines) and in the model (small circles) . A, Experimental configuration. B , Responses to alveus stimulation without (left) and with ( right ) somatic V-clamp. C , In a different cell, responses to dynamic current injection in the dendrite; conductance time course (g) in green, 5-nS peak amplitude , V rev =-85 mV . [F.Pouille, M.Scanziani // Nature , 2004] Parameters of the model:  m = 33 ms ,  = 3.5 , G s = 6 nS in B and 2.4 nS in C
    • Two boundary problems:
    • current-clamp to register PSP :
    • voltage-clamp to register PSC :
    Solution:
    • neuron is spatially distributed
    [A.V.Chizhov // Biophysics 2004 ] C V d V d V d V d V s V s I s I s g=I d /(V d -V rev ) B A X=0 X=L V d V 0
      • Excitatory synaptic current :
      • Inhibitory synaptic current :
      • Non-dimensional synaptic conductances :
      • where
      • - rise and decay time constants
      • - presynaptic firing rate
    Pyramidal neurons Interneurons Synaptic conductance:
    • synaptic channel kinetics
  • Модель. Ответ зрительной коры на полосу горизонтальной, а затем вертикальной ориентации. Эксперимент. Зрительная кора. Карта ориентационной избирательности активности нейронов. Модель “Pinwheels” карты ориентационной избирательности входных сигналов.
    • spatial structure of connections
    1 mm
    • What can be modeled on population level?
    • Which details are important?
    • What kinds of population models do exist ?
    • Which one to choose ?
  • P opulation 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) Стимулирующий ток
  • 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
    • where the matrix represents the influence of noise
    • Problem! The equation is multi-dimensional.
    • Particular cases are
    • - membrane potential
      • - time passed since the last spike
      • - time till the next spike
    Idea of Probability Density Approach (PDA) For classical H-H : [A.Turbin 2003] Single neuron equation ( e.g. H-H model ) where is the common deterministic part, is the noisy term. Eq. for neural density [ B.Knight 1972] [ A.Omurtag et al. 2000] [ D.Nykamp, D.Tranchina 2000] [N.Brunel, V.Hakim 1999], … [J.Eggert, JL.Hemmen 2001] [ А.Чижов, А.Турбин 2003]
  • Leaky Integrate-and-Fire (LIF) neuron with noise ( stochastic LIF )
  • 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
  • Refractory Density model [Chizhov et al. // Neurocomputing 2006] Stimulation current
  • where according to Spike Response Model (SRM): [W.Gerstner, W.Kistler, 2002] Similar approach: Refractory Density model for SRM-neurons
  • 1-D Refractory Density Approach for c onductance-based neurons (CBRD)
    • Threshold single-neuron model
    • Refractory density approach (t* - parameterization)
    • Hazard-function
    t * is the time since the last spike; H(U) = ‘ frozen stationary ’ + ‘ self-similar ’ solutions of Kolmogorov-Fokker-Planck eq. for I&F neuron with white or color noise-current [Chizhov, Graham, Turbin // Neurocomputing, 2006] [Chizhov, Graham // Phys. Rev. E, 2007] [Chizhov, Graham // Phys. Rev. E, 2008]
  • 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
  •  
  • 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]
  • 3. Hazard function
  • 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 :
  • Self-similar solution (T=const) Assumption. U(t) = const (or T(t)=const) . Notation: Then the shape of , which is , is invariable. Equivalent formulation :
  • 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
  • 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 .
  • 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
  • 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
  • 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
  • Simulations with CBRD-model
  • Non-adaptive neurons (4000) Single population: comparison of CBRD with Monte-Carlo
  • Single population: current-step stimulation. Color noise. Adaptive neurons. LIF Adaptive conductance-based neuron
  • with I M Single population: oscillatory input
  • Single population: comparison of CBRD with analytical solution for stochastic LIF in steady-state
  • 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”
  • Single cell level Populations t * is the time since the last spike CBRD Large-scale simulations (NMM & FR-models for EEG & MRI)
  • From CBRD to Firing-Rate model
  • 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
    • What can be modeled on population level?
    • Which details are important?
    • What kinds of population models do exist ?
    • Which one to choose ?
  • 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 ):
  • Simulations with FR-model
  • 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
  • Approximation of synaptic current Presynaptic spike Postsynaptic current
      • Excitatory synaptic current :
      • Inhibitory synaptic current :
      • Non-dimensional synaptic conductances :
      • where
      • - rise and decay time constants
      • - presynaptic firing rate
    Pyramidal neurons Interneurons Synaptic conductance: Approximation of synaptic current
  • 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
  • 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 .
  • Firing-rate model of adaptive neuron population: « interictal » activity
  • Simulations with CBRD-model
  • 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
  • Simulations. Gamma rhythm. R ecurrent network of interneurons , including all-to-all connectivity by inhibitory synapses
  • Oscillations Model Experiments Control (“Kainate”) +“Bicuculline” Spikes in single neurons Conductances Power Spectrum of Extracellular Potentials Spike timing of pyramidal and inhibitory cells. [Khazipov, Holmes, 2003] Kainate-induced oscillations in CA3. [A.Fisahn et al., 1998] Cholinergically induced oscillations in CA3 [N. Hajos, J . Palhalmi, E . O.Mann, B . Nemeth, O . Paulsen, and T . F.Freund . J . Neuroscience , 24(41):9127–9137, 2004 ] con bic
    • 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, spike trains and monosynaptic responses .
    • The model reproduces the following characteristics of gamma-oscillations :
    • frequency of population spikes
    • a single pyramidal cell does not fire every cycle
    • every interneuron fires every cycle
    • amplitude of EPSC is less than that of IPSC
    • blockage of GABA-A receptors reduces the frequency
    • 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 homogeneous in space along the cortical surface (data not shown)
  • 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 :
  • PSPs and PSCs evoked by extracellular stimulation and registered at 3.5cm away, w/ and w/o kainate. [S.Karnup, A.Stelzer 1999] Effects of GABA-A receptor blockade on orthodromic potentials in CA1 pyramidal cells. Superimposed responses in a pyramidal cell soma before and after application of picrotoxin (PTX, 100 muM). Control and PTX recordings were obtained at V rest (-64 m V; 150 muA stimulation intensities; 1 mm distance between stratum radiatum stimulation site and perpendicular line through stratum pyramidale recording site). The recordings were carried out in ‘minislices’ in which the CA3 region was cut off by dissection. [V.Crepel, R.Khazipov, Y.Ben-Ari, 1997] In normal concentrations of Mg and in the absence of CNQX, block of GABA-A receptors induced a late synaptic response. B A C [B.Mlinar, A.M.Pugliese, R.Corradetti 2001] Components of complex synaptic responses evoked in CA1 pyramidal neurones in the presence of GABAA receptor block.
    • The model reproduces postsynaptic currents and postsynaptic potentials registered on somas of pyramidal cells, namely:
    • monosynaptic EPSCs and EPSPs
    • disynaptic IPSC/Ps followed be EPSC/Ps
    • polysynaptic EPSC/Ps
    • reduction of delays in polysynaptic EPSCs
    • decay of excitation after II component of poly-EPSCs in presence of GABA-A receptor block.
    • The model predicts that the evoked responses are essentially non-homogeneous in space:
    Spatial profiles of membrane potential and firing rate in pyramids. Evoked responses Model Experiments
  • 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.
  • Conclusion
    • « Gross » processes can be described by only population approach . When the dynamics of individual neurons is important, it should be modeled on a background of population activity .
    • Any population model must correctly reproduce unsteady regimes .
    • For a population of LIF-neurons one can choose the Fokker-Planck -based model .
    • For conductance-based neurons the CBRD -model is recommended.
    • As an approximate and simple model, a modified firing-rate ( FR ) model can be used ( with « non-stationary term »).
    Chizhov, Graham // Phys. Rev. E 2007 Chizhov, Graham // Phys. Rev. E 2008 Chizhov et al. // Physics Letters A 2007 Chizhov et al. // Neurocomputing 2006 Rodrigues et al. // Biol Cybern. 2010 Buchin, Chizhov // Opt . Memory 2010 Чижов // Мат. биол. и биоинф. 2010 Бучин, Чижов // Биофизика 2010 Чижов // Биофизика 2002 Чижов // Биофизика 2004 Чижов // Нейрокомпьютеры 2004 Чижов, Грэм //Известия РАЕН 2004 Чижов и др. // Биофизика 2009 Чижов // Вестник СПбГУ 2009 Смирнова, Чижов // Биофизика 2011
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