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Population Density Models

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Encyclopedia of Computational Neuroscience

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Definition

Population density models are reduced descriptions of neuronal ensembles, which consist in characterizing them by the instantaneous distribution of some neuronal variables over the population. The simplest and most useful form only uses the neuron membrane potentials. The distribution of membrane potentials and its dynamical evolution can be exactly computed in different cases, for noninteracting neurons submitted to time-varying stochastic inputs. This provides the basis for the most interesting application of population density models, namely, the quantitative description of the dynamics of networks of coupled spiking neurons. The approach has in particular been used to analyze neural rhythms.

Introduction

Cognitive phenomena depend on the concerted activity of large populations of neurons. Understanding the dynamics of such systems has been one of the major goals of theoretical neuroscience. Various methods have been developed to tackle this problem. These methods range...

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Notes

  1. 1.

    The important case of excitation-inhibition balance in a multi-population network, which allows for stronger synapses and fluctuations in the noise variance, is discussed in section “Strong Coupling Regime.”

References

  • Abbott LF, van Vreeswijk C (1993) Asynchronous states in a network of pulse-coupled oscillators. Phys Rev E 48:1483–1490

    Article  Google Scholar 

  • Amit DJ, Brunel N (1997) Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. Cereb Cortex 7:237–252

    Article  CAS  PubMed  Google Scholar 

  • Badel L, Lefort S, Berger TK, Petersen CC, Gerstner W, Richardson MJ (2008a) Extracting non-linear integrate-and-fire models from experimental data using dynamic I-V curves. Biol Cybern 99:361–370

    Article  PubMed Central  PubMed  Google Scholar 

  • Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ (2008b) Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. J Neurophysiol 99:656–666

    Article  PubMed  Google Scholar 

  • Brette R, Gerstner W (2005) Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J Neurophysiol 94:3637–3642

    Article  PubMed  Google Scholar 

  • Brunel N (2000a) Dynamics of networks of randomly connected excitatory and inhibitory spiking neurons. J Physiol Paris 94:445–463

    Article  CAS  PubMed  Google Scholar 

  • Brunel N (2000b) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J Comput Neurosci 8:183–208

    Article  CAS  PubMed  Google Scholar 

  • Brunel N, Hakim V (1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comp 11:1621–1671

    Article  CAS  Google Scholar 

  • Brunel N, Hansel D (2006) How noise affects the synchronization properties of networks of inhibitory neurons. Neural Comp 18:1066–1110

    Article  Google Scholar 

  • Brunel N, Latham P (2003) Firing rate of noisy quadratic integrate-and-fire neurons. Neural Comput 15:2281–2306

    Article  PubMed  Google Scholar 

  • Brunel N, Sergi S (1998) Firing frequency of integrate-and-fire neurons with finite synaptic time constants. J Theor Biol 195:87–95

    Article  CAS  PubMed  Google Scholar 

  • Brunel N, van Rossum MC (2007) Lapicque’s 1907 paper: from frogs to integrate-and-fire. Biol Cybern 97:337–339

    Article  PubMed  Google Scholar 

  • Brunel N, Wang XJ (2003) What determines the frequency of fast network oscillations with irregular neural discharges? J Neurophysiol 90:415–430

    Article  PubMed  Google Scholar 

  • Brunel N, Chance F, Fourcaud N, Abbott L (2001) Effects of synaptic noise and filtering on the frequency response of spiking neurons. Phys Rev Lett 86:2186–2189

    Article  CAS  PubMed  Google Scholar 

  • Brunel N, Hakim V, Richardson M (2003) Firing rate resonance in a generalized integrate-and-fire neuron with subthreshold resonance. Phys Rev E 67:051,916

    Article  Google Scholar 

  • Caceres MJ, Carillo JA, Tao L (2011) A numerical solver for a nonlinear Fokker-Planck equation representation of neural network dynamics. J Comp Phys 230:1084–1099

    Article  Google Scholar 

  • Casti ARR, Omurtag A, Sornborger A, Kaplan E, Knight B, Victor J, Sirovich L (2002) A population study of integrate-and-fire-or-burst neurons. Neural Comput 14:957–986

    Article  CAS  PubMed  Google Scholar 

  • de Solages C, Szapiro G, Brunel N, Hakim V, Isope P, Buisseret P, Rousseau C, Barbour B, Lena C (2008) High-frequency organization and synchrony of activity in the Purkinje cell layer of the cerebellum. Neuron 58:775–788

    Article  PubMed  Google Scholar 

  • Ermentrout GB (1994) Reduction of conductance based models with slow synapses to neural nets. Neural Comput 6:679–695

    Article  Google Scholar 

  • Ermentrout GB (1996) Type I membranes, phase resetting curves, and synchrony. Neural Comput 8:979–1001

    Article  CAS  PubMed  Google Scholar 

  • Ermentrout GB, Kopell N (1986) Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J Appl Math 46:233–253

    Article  Google Scholar 

  • Fourcaud N, Brunel N (2002) Dynamics of firing probability of noisy integrate-and-fire neurons. Neural Comput 14:2057–2110

    Article  PubMed  Google Scholar 

  • Fourcaud-TrocmĂ© N, Hansel D, van Vreeswijk C, Brunel N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. J Neurosci 23:11628–11640

    PubMed  Google Scholar 

  • Fusi S, Mattia M (1999) Collective behavior of networks with linear (VLSI) integrate and fire neurons. Neural Comput 11:633–652

    Article  CAS  PubMed  Google Scholar 

  • Galarreta M, Hestrin S (1999) A network of fast-spiking cells in the neocortex connected by electrical synapses. Nature 402:72–75

    Article  CAS  PubMed  Google Scholar 

  • Gardiner CW (1986) Handbook of stochastic methods for physics, chemistry and the natural sciences. Springer, New York

    Google Scholar 

  • Geisler C, Brunel N, Wang XJ (2005) Contributions of intrinsic membrane dynamics to fast network oscillations with irregular neuronal discharges. J Neurophysiol 94:4344–4361

    Article  PubMed  Google Scholar 

  • Gerstein GL, Mandelbrot B (1964) Random walk models for the spike activity of a single neuron. Biophys J 4:41–68

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Gibson JR, Beierlein M, Connors BW (1999) Two networks of electrically coupled inhibitory neurons in neocortex. Nature 402:75–79

    Article  CAS  PubMed  Google Scholar 

  • Gutkin B, Ermentrout GB (1998) Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Comp 10:1047–1065

    Article  CAS  Google Scholar 

  • Haider B, Duque A, Hasenstaub AR, McCormick DA (2006) Neocortical network activity in vivo is generated through a dynamic balance of excitation and inhibition. J Neurosci 26:4535–4545

    Article  CAS  PubMed  Google Scholar 

  • Haskell E, Nykamp DQ, Tranchina D (2001) Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size. Network Comput Neural Syst 12:141–174

    Article  CAS  Google Scholar 

  • Helias M, Deger M, Rotter S, Diesmann M (2011) Finite post synaptic potentials cause a fast neuronal response. Front Neurosci 5:19

    Article  PubMed Central  PubMed  Google Scholar 

  • Izhikevich EM (2001) Resonate-and-fire neurons. Neural Netw 14:883–894

    Article  CAS  PubMed  Google Scholar 

  • Izhikevich EM (2003) Simple model of spiking neurons. IEEE Trans Neural Netw 14:1569–1572

    Article  CAS  PubMed  Google Scholar 

  • Jahnke S, Memmesheimer R, Timme M (2008) Stable irregular dynamics in complex neural networks. Phys Rev Lett 100:048,102

    Article  Google Scholar 

  • Jahnke S, Memmesheimer RM, Timme M (2009) How chaotic is the balanced state? Front Comput Neurosci 3:13

    Article  PubMed Central  PubMed  Google Scholar 

  • Knight BW (1972) Dynamics of encoding in a population of neurons. J Gen Physiol 59:734–766

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Knight BW, Omurtag A, Sirovich L (2000) The approach of a neuron population firing rate to a new equilibrium: an exact theoretical result. Neural Comput 12:1045–1055

    Article  CAS  PubMed  Google Scholar 

  • Lapicque L (1907) Recherches quantitatives sur l’excitabilitĂ© Ă©lectrique des nerfs traitĂ©e comme une polarisation. J Physiol Pathol Gen 9:620–635

    Google Scholar 

  • Latham PE, Richmond BJ, Nelson PG, Nirenberg S (2000) Intrinsic dynamics in neuronal networks. I. Theory. J Neurophysiol 83:808

    CAS  PubMed  Google Scholar 

  • Ledoux E, Brunel N (2011) Dynamics of networks of excitatory and inhibitory neurons in response to time-dependent inputs. Front Comput Neurosci 5:25

    Article  PubMed Central  PubMed  Google Scholar 

  • Lewis T, Rinzel J (2003) Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J Comput Neurosci 14:283–309

    Article  PubMed  Google Scholar 

  • Lifshitz E, Pitaevskii L (1980) Statistical physics, part 2, (Landau and Lifshitz Course of Theoretical physics, vol 9) Pergamon, Oxford

    Google Scholar 

  • Lifshitz E, Pitaevskii L (1981) Physical Kinetics, (Landau and Lifshitz Course of Theoretical physics, vol 10) Pergamon, Oxford

    Google Scholar 

  • Lindner B, Schimansky-Geier L (2001) Transmission of noise coded versus additive signals through a neuronal ensemble. Phys Rev Lett 86:2934–2937

    Article  CAS  PubMed  Google Scholar 

  • Mattia M, Del Giudice P (2002) Population dynamics of interacting spiking neurons. Phys Rev E 66:051,917

    Article  Google Scholar 

  • Mattia M, Del Giudice P (2004) Finite-size dynamics of inhibitory and excitatory interacting spiking neurons. Phys Rev E Stat Nonlin Soft Matter Phys 70:052,903

    Article  Google Scholar 

  • Monteforte M, Wolf F (2010) Dynamical entropy production in spiking neuron networks in the balanced state. Phys Rev Lett 105:268,104

    Article  Google Scholar 

  • Monteforte M, Wolf F (2012) Dynamic flux tubes form reservoirs of stability in neuronal circuits. Phys Rev X 2:041007

    CAS  Google Scholar 

  • Moreno R, Parga N (2004) Role of synaptic filtering on the firing response of simple model neurons. Phys Rev Lett 92:028,102

    Article  Google Scholar 

  • Naud R, Marcille N, Clopath C, Gerstner W (2008) Firing patterns in the adaptive exponential integrate-and-fire model. Biol Cybern 99:335–347

    Article  PubMed Central  PubMed  Google Scholar 

  • Nykamp DQ, Tranchina D (2000) A population density approach that facilitates largescale modeling of neural networks: analysis and an application to orientation tuning. J Comp Neurosci 8:19–30

    Article  CAS  Google Scholar 

  • Omurtag A, Knight BW, Sirovich L (2000) On the simulation of large populations of neurons. J Comput Neurosci 8(1):51–63

    Article  CAS  PubMed  Google Scholar 

  • Ostojic S, Brunel N (2011) From spiking neuron models to linear-nonlinear models. PLOS Comp Biol 7:e1001,056

    Article  Google Scholar 

  • Ostojic S, Brunel N, Hakim V (2009) Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J Comput Neurosci 26:369–392

    Article  PubMed  Google Scholar 

  • Politi A, Livi R, Oppo GL, Kapral R (1993) Unpredictable behavior in stable systems. Europhys Lett 22:571–576

    Article  CAS  Google Scholar 

  • Rauch A, Camera GL, LĂĽscher HR, Senn W, Fusi S (2003) Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents. J Neurophysiol 90:1598–1612

    Article  PubMed  Google Scholar 

  • Ricciardi LM (1977) Diffusion processes and related topics on biology. Springer, Berlin

    Book  Google Scholar 

  • Richardson MJE (2004) Effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons. Phys Rev E Stat Nonlin Soft Matter Phys 69:051,918

    Article  Google Scholar 

  • Richardson MJE (2007) Firing-rate response of linear and nonlinear integrate-and-fire neurons to modulated current-based and conductance-based synaptic drive. Phys Rev E Stat Nonlin Soft Matter Phys 76:021,919

    Article  Google Scholar 

  • Richardson MJ (2009) Dynamics of populations and networks of neurons with voltage-activated and calcium-activated currents. Phys Rev E Stat Nonlin Soft Matter Phys 80:021,928

    Google Scholar 

  • Richardson MJE, Gerstner W (2005) Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance. Neural Comput 17:923–947

    Article  PubMed  Google Scholar 

  • Richardson MJ, Swarbrick R (2010) Firing-rate response of a neuron receiving excitatory and inhibitory synaptic shot noise. Phys Rev Lett 105:178,102

    Google Scholar 

  • Richardson M, Brunel N, Hakim V (2003) From subthreshold to firing-rate resonance. J Neurophysiol 89:2538–2554

    Article  PubMed  Google Scholar 

  • Risken H (1984) The Fokker-Planck equation: methods of solution and applications. Springer, Berlin

    Book  Google Scholar 

  • Romani S, Amit D, Mongillo G (2006) Mean-field analysis of selective persistent activity in presence of short-term synaptic depression. J Comput Neurosci 20:201–217

    Article  PubMed  Google Scholar 

  • Shadlen MN, Newsome WT (1994) Noise, neural codes and cortical organization. Curr Opin Neurobiol 4:569

    Article  CAS  PubMed  Google Scholar 

  • Shriki O, Hansel D, Sompolinsky H (2003) Rate models for conductance based cortical neuronal networks. Neural Comput 15:1806–1841

    Google Scholar 

  • Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrent balanced cortical activity. Nature 423:288–293

    Article  CAS  PubMed  Google Scholar 

  • Siegert AJF (1951) On the first passage time probability problem. Phys Rev 81:617–623

    Article  Google Scholar 

  • Silberberg G, Bethge M, Markram H, Pawelzik K, Tsodyks M (2004) Dynamics of population rate code in ensemble of neocortical neurons. J Neurophysiol 91:704–709

    Article  CAS  PubMed  Google Scholar 

  • Smith GD, Cox CL, Sherman SM, Rinzel J (2000) Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-fire-or-burst model. J Neurophysiol 83:588–610

    CAS  PubMed  Google Scholar 

  • Softky WR (1993) Sub-millisecond coincidence detection in active dendritic trees. Neuroscience 58:13

    Article  Google Scholar 

  • Stein R (1965) A theoretical analysis of neuronal variability. Biophys Journal 5:173–194

    Article  CAS  Google Scholar 

  • Touboul J (2008) Bifurcation analysis of a general class of nonlinear integrate-and-fire neurons. SIAM J Appl Math 68:1045–1079

    Article  Google Scholar 

  • Treves A (1993) Mean-field analysis of neuronal spike dynamics. Network 4:259–284

    Article  Google Scholar 

  • van Vreeswijk C, Sompolinsky H (1996) Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science 274:1724–1726

    Article  PubMed  Google Scholar 

  • Wang XJ, Buzsáki G (1996) Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J Neurosci 16:6402–6413

    CAS  PubMed  Google Scholar 

  • Zillmer R, Livi R, Politi A, Torcini A (2006) Desynchronization in diluted neural networks. Phys Rev E Stat Nonlin Soft Matter Phys 74:036,203

    Article  Google Scholar 

  • Zillmer R, Brunel N, Hansel D (2009) Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons. Phys Rev E Stat Nonlin Soft Matter Phys 79:031,909

    Article  Google Scholar 

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Authors and Affiliations

  1. Departments of Statistics and Neurobiology, University of Chicago, Chicago, USA

    Prof. Nicolas Brunel

  2. Laboratoire de Physique Statistique, Ecole Normale Supérieure, CNRS, University P and M Curie, Paris, Ile-de-France, France

    Prof. Vincent Hakim

Authors
  1. Prof. Nicolas Brunel
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  2. Prof. Vincent Hakim
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Correspondence to Nicolas Brunel .

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Editors and Affiliations

  1. Atlanta, Georgia, USA

    Dieter Jaeger

  2. Department of Biomedical Engineering, Florida International University, Miami, Florida, USA

    Ranu Jung

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Brunel, N., Hakim, V. (2013). Population Density Models. In: Jaeger, D., Jung, R. (eds) Encyclopedia of Computational Neuroscience. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7320-6_74-1

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