Skip to main content
Log in

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

  • Review
  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker–Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Google Scholar 

  • Abeles M (1982) Role of the cortical neuron: integrator or coincidence detector? Isr J Med Sci 18:83–92

    PubMed  CAS  Google Scholar 

  • Abeles M (1991) Corticonics: Neural circuits of the cerebral cortex. Cambridge University Press, New York

    Google Scholar 

  • Amit DJ, Brunel N (1997a) Dynamics of a recurrent network of spiking neurons before and following learning. Netw Comput Neur Sys 8:373–404

    Google Scholar 

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

    CAS  Google Scholar 

  • Amit DJ, Mongillo G (2003) Spike-driven synaptic dynamics generating working memory states. Neural Comput 15:565– 596

    PubMed  Google Scholar 

  • Amit DJ, Tsodyks MV (1991a) Quantitative study of attractor neural network retrieving at low spike rates: I. Substrate - spikes, rates and neuronal gain. Netw Comput Neur Sys 2:259– 273

    Google Scholar 

  • Amit DJ, Tsodyks MV (1991b) Quantitative study of attractor neural network retrieving at low spike rates: II Low-rate retrieval in symmetric networks. Netw Comput Neur Sys 2:275–294

    Google Scholar 

  • Anderson DJ (1973) Quantitative model for the effects of stimulus frequency upon synchronization of auditory nerve discharges. J Acoust Soc Am 54:361–364

    PubMed  CAS  Google Scholar 

  • Benzi R, Sutera A, Vulpiani A (1981) The mechanism of stochastic resonance. J Phys A 14:L453–L457

    Google Scholar 

  • Bi G-Q, Poo M-M (2001) Synaptic modification by correlated activity: Hebb’s postulate revisited. Ann Rev Neurosci 24:139–166

    PubMed  CAS  Google Scholar 

  • Braun HA, Wissing H, Schaefer K, Hirsch MC (1994) Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature 367:270–273

    PubMed  CAS  Google Scholar 

  • Bressloff PC (1999) Synaptically generated wave propagation in excitable neural media. Phys Rev Lett 82:2979–2982

    CAS  Google Scholar 

  • Bressloff PC (2000) Traveling waves and pulses in a one-dimensional networks of excitable integrate-and-fire neurons. J Math Biol 40:169–198

    PubMed  CAS  Google Scholar 

  • Bressloff PC, Coombes S (1998) Desynchronization, mode locking, and bursting in strongly coupled integrate-and-fire oscillators. Phys Rev Lett 81:2168–2171

    CAS  Google Scholar 

  • Bressloff PC, Coombes S (2000) A dynamical theory of spike train transitions in networks of integrate-and-fire oscillators. SIAM J Appl Math 60:820–841

    Google Scholar 

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

    PubMed  CAS  Google Scholar 

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

    PubMed  CAS  Google Scholar 

  • Brunel N, Wang X-J (2003) What determines the frequency of fast network oscillations with irregular neural discharges? I Synaptic dynamics and excitation-inhibition balance. J Neurophysiol 90:415–430

    PubMed  Google Scholar 

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

    PubMed  CAS  Google Scholar 

  • Brunel N, Hakim V, Richardson MJE (2003) Firing-rate resonance in a generalized integrate-and-fire neuron with subthreshold resonance. Phys Rev E 67:051916

    Google Scholar 

  • Bulsara AR, Zador A (1996) Threshold detection of wideband signals: a noise-induced maximum in the mutual information. Phys Rev E 54:R2185–2188

    CAS  Google Scholar 

  • Bulsara AR, Lowen SB, Rees CD (1994) Cooperative behavior in the periodically modulated Wiener process: noise-induced complexity in a model neuron. Phys Rev E 49:4989–5000

    Google Scholar 

  • Bulsara AR, Elston TC, Doering CR, Lowen SB, Lindenberg K (1996) Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys Rev E 53:3958–3969

    CAS  Google Scholar 

  • Burkitt AN (2001) Balanced neurons: analysis of leaky integrate-and-fire neurons with reversal potentials. Biol Cybern 85: 247–255

    PubMed  CAS  Google Scholar 

  • Burkitt AN (2006) A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol Cybern DOI: 10.1007/s00422-006-0068-6

  • Burkitt AN, Clark GM (1999) Analysis of integrate-and-fire neurons: Synchronization of synaptic input and spike output in neural systems. Neural Comput 11:871–901

    PubMed  CAS  Google Scholar 

  • Burkitt AN, Clark GM (2000) Calculation of interspike intervals for integrate-and-fire neurons with Poisson distribution of synaptic inputs. Neural Comput 12:1789–1820

    PubMed  CAS  Google Scholar 

  • Burkitt AN, Clark GM (2001) Synchronization of theneural response to noisy periodic synaptic input. Neural Comput 13:2639–2672

    PubMed  CAS  Google Scholar 

  • Burkitt AN, van Hemmen JL (2003) How synapses in the auditory system wax and wane: Theoretical perspectives. Biol Cybern 89:318–332

    PubMed  CAS  Google Scholar 

  • Burkitt AN, Meffin H, Grayden DB (2004) Spike timing-dependent plasticity: The relationship to rate-based learning for models with weight dynamics determined by a stable fixed-point. Neural Comput 16:885–940

    PubMed  Google Scholar 

  • Cariani PA (1995) As if time really mattered: Temporal strategies for neural coding of sensory information. Commun Cog-Art Intell 12:157–219

    Google Scholar 

  • Cariani PA (2001) Neural timing nets. Neural Netw 14:737–753

    PubMed  CAS  Google Scholar 

  • Câteau H, Fukai T (2001) Fokker–Planck approach to the pulse packet propagation in synfire chain. Neural Netw 14:675–685

    PubMed  Google Scholar 

  • Chapeau-Blondeau F, Godivier X, Chambet N (1996) Stochastic resonance in a neuron model that transmits spike trains. Phys Rev E 53:1273–1275

    CAS  Google Scholar 

  • Chialvo DR, Longtin A, Müller-Gerking J (1997) Stochastic resonance in models of neuronal ensembles. Phys Rev E 55:1798–1808

    CAS  Google Scholar 

  • Compte A, Brunel N, Goldman-Rakic PS, Wang X-J (2000) Synaptic mechanisms and network dynamics underlying spatial memory in a cortical network model. Cereb Cortex 10:910–923

    PubMed  CAS  Google Scholar 

  • Cox DR, Smith WL (1954) On the superposition of renewal processes. Biometrika 41:91–99

    Google Scholar 

  • Cremers D, Herz AVM (2002) Traveling waves of excitation in neural field models: equivalence of rate descriptions and integrate-and-fire dynamics. Neural Comput 14:1651–1667

    PubMed  Google Scholar 

  • Destexhe A, Paré D (1999) Impact of network activity on the integrative properties of neocortical neurons in vivo. J Neurophysiol 81:1531–1547

    PubMed  CAS  Google Scholar 

  • Destexhe A, Rudolph M, Fellous J-M, Sejnowski TJ (2001) Fluctuating synaptic conductances recreate in-vivo-like activity in neocortical neurons. Neuroscience 107:13–24

    PubMed  CAS  Google Scholar 

  • Diesmann M, Gewaltig MO, Aertsen A (1996) Characterization of synfire activity by propagating ‘pulse packets’. In: Bower J (ed), Computational neuroscience: trends in research. Academic, San Diego, pp 59–64

    Google Scholar 

  • Diesmann M, Gewaltig MO, Aertsen A (1999) Stable propagation of synchronous spiking in cortical neural networks. Nature 402:529–533

    PubMed  CAS  Google Scholar 

  • Douglas JK, Wilkens L, Pantozelou E, Moss F (1993) Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365:337–340

    Google Scholar 

  • Ermentrout GB (1998) Linearization of F-I curves by adaptation. Neural Comput 10:1721–1729

    PubMed  CAS  Google Scholar 

  • Ernst U, Pawelzik K, Geisel T (1995) Synchronization induced by temporal delays in pulse-coupled oscillators. Phys Rev Lett 74:1570–1573

    PubMed  CAS  Google Scholar 

  • Fauve S, Heslot F (1983) Stochastic resonance in a bistable system. Phys Lett 97A:5–7

    Google Scholar 

  • Feng J, Sun Y, Buxton H, Wei G (2003) Training integrate-and-fire neurons with the informax principle II. IEEE Trans Neural Netw 14:326–336

    PubMed  Google Scholar 

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

    PubMed  Google Scholar 

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

    PubMed  CAS  Google Scholar 

  • Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223–287

    CAS  Google Scholar 

  • Gedeon T, Holzer M (2004) Phase locking in integrate-and-fire models with refractory periods and modulation. J Math Biol 49:577–603

    PubMed  Google Scholar 

  • Gerstner W (1995) Time structure of the activity in neural network models. Phys Rev E 51:738–758

    CAS  Google Scholar 

  • Gerstner W (2000) Population dynamics of spiking neurons: Fast transients, asynchronous states, and locking. Neural Comput 12:43–89

    PubMed  CAS  Google Scholar 

  • Gerstner W, van Hemmen JL (1992) Associative memory in a network of ‘spiking’ neurons. Netw Comput Neural Sys 3:139–164

    Google Scholar 

  • Gerstner W, van Hemmen JL (1993) Coherence and incoherence in a globally coupled ensemble of pulse-emitting units. Phys Rev Lett 71:312–315

    PubMed  Google Scholar 

  • Gerstner W, Kempter R, van Hemmen JL, Wagner H (1996a) A neuronal learning rule for sub-millisecond temporal coding. Nature 383:76–78

    CAS  Google Scholar 

  • Gerstner W, van Hemmen JL, Cowan JD (1996b) What matters in neuronal locking?. Neural Comput 8:1653–1676

    PubMed  CAS  Google Scholar 

  • Giraudo MT, Sacerdote L (2005) Effect of periodic stimuluson a neuronal diffusion model with signal-dependent noise. BioSys 79:73–81

    Google Scholar 

  • Giudice P Del, Fusi S, Mattia M (2003) Modelling the formation of working memory with networks of integrate-and-fire neurons connected by plastic synapses. J Physiol (Paris) 97:659–681

    Google Scholar 

  • Giugliano M, Darbon P, Arsiero M, Lüscher HR, Streit J (2004) Single-neuron discharge properties and network activity in dissociated cultures of neocortex. J Neurophysiol 92:977–996

    PubMed  CAS  Google Scholar 

  • Goldberg JM, Brown PB (1969) Response of binaural neurons of dog superior olivary complex to dichotic tonal stimuli: some physiological mechanisms of sound localization. J Neurophysiol 32:613–636

    PubMed  CAS  Google Scholar 

  • Golomb D, Ermentrout GB (1999) Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity. Proc Natl Acad Sci 96:13480–13485

    PubMed  CAS  Google Scholar 

  • Golomb D, Rinzel J (1993) Dynamics of globally coupled inhibitory neurons with heterogeneity. Phys Rev E 48:4810–4814

    Google Scholar 

  • Gummer AW (1991) Probability density function of successive intervals of a nonhomogeneous Poisson process under low-frequency conditions. Biol Cybern 65: 23–30

    PubMed  CAS  Google Scholar 

  • Hansel D, Mato G, Meunier C (1995) Synchrony in excitatory neural networks. Neural Comput 7:307–337

    PubMed  CAS  Google Scholar 

  • Hansel D, Mato G, Meunier C, Neltner L (1998) On numerical simulations of integrate-and-fire neural networks. Neural Comput 10:467–483

    PubMed  CAS  Google Scholar 

  • Hanson FB, Tuckwell HC (1983) Diffusion approximations for neuronal activity including synaptic reversal potentials. J Theor Neurobiol 2:127–153

    Google Scholar 

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

    CAS  Google Scholar 

  • Heneghan C, Chow CC, Collins JJ, Imhoff TT, Lowen SB, Teich MC (1996) Information measures quantifying aperiodic stochastic resonance. Phys Rev E 54:R2228–2231

    CAS  Google Scholar 

  • Hohn N, Burkitt AN (2001) Shot noise in the leaky integrate-and-fire neuron. Phys Rev E 63:031902

    CAS  Google Scholar 

  • Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci USA 79:2554–2558

    PubMed  CAS  Google Scholar 

  • Johnson DH (1980) The relationship between spike rate and synchrony in responses of auditory-nerve fibers to single tones. J Acoust Soc Am 68:1115–1122

    PubMed  CAS  Google Scholar 

  • Joris PX, Smith PH, Yin TCT (1998) Coincidence detection in the auditory system: 50 years after Jeffress. Neuron 21:1235–1238

    PubMed  CAS  Google Scholar 

  • Keener JP, Hoppensteadt FC, Rinzel J (1981) Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J Appl Math 41:503–517

    Google Scholar 

  • Kempter R, Gerstner W, van Hemmen JL, Wagner H (1998) Extracting oscillations: neuronal coincidence detection with noisy periodic spike input. neural Comput 10:1987–2017

    PubMed  CAS  Google Scholar 

  • Kempter R, Gerstner W, van Hemmen JL (1999a) Hebbian learning and spiking neurons. Phys Rev E 59:4498–4514

    CAS  Google Scholar 

  • Kempter R, Gerstner W, van Hemmen JL, Wagner H (1999b) Quality of coincidence detection and ITD tuning: a theoretical framework. In: Dau T, Hohmann V, Kollmeier B (eds) Psychophysics, physiology and models of hearing. World Scientific, Singapore, pp 185–194

    Google Scholar 

  • Kistler WM, Gerstner W (2002) Stable propagation of activity pulses in populations of spiking neurons. Neural Comput 14:987–997

    PubMed  Google Scholar 

  • Kistler WM, van Hemmen JL (1998) Modeling collective excitations in cortical tissue. Physica D 114:273–295

    Google Scholar 

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

    CAS  Google Scholar 

  • Knight BW (1972b) The relationship between the firing rate of a single neuron and the level of activity in a population of neurons. J Gen Physiol 59:767–778

    CAS  Google Scholar 

  • Knight BW (2000) Dynamics of encoding in neuron populations: Some general mathematical features. Neural Comput 12:473–518

    PubMed  CAS  Google Scholar 

  • Knight BW, Manin D, Sirovich L (1996) Dynamical models of interacting neuron populations. In: Symposium on robotics and cybernetics: computational engineering in systems applications. Lille, Cite Scientifique, France

  • 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

    PubMed  CAS  Google Scholar 

  • König P, Engel AE, Singer W (1996) Integrator or coincidence detector? The role of the cortical neuron revisited. TINS 19:130–137

    PubMed  Google Scholar 

  • Kuhlmann L, Burkitt AN, Paolini A, Clark GM (2002) Summation of spatiotemporal input patterns in leaky integrate-and-fire neurons: Application to neurons in the cochlear nucleus receiving converging auditory nerve fiber input. J Comput Neurosci 12:55–73

    PubMed  Google Scholar 

  • Kuramoto Y (1984) Chemical oscillations, waves and turbulance. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Kuramoto Y (1991) Collective synchronization of pulse-coupled oscillators and excitable units. Physica D 50:15–30

    Google Scholar 

  • Lánský P (1997) Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics. Phys Rev E 55:2040–2043

    Google Scholar 

  • Lánský P, Sato S (1999) The stochastic diffusion models of nerve membrane depolarization and interspike interval generation. J Periph Nerv Syst 4:27–42

    Google Scholar 

  • Laurent G (2002) Olfactory network dynamics and the coding of multidimensional signals. Nat Rev 3:884–895

    CAS  Google Scholar 

  • Lindner B (2004) Moments of the first passage time under external driving. J Stat Phys 117:703–737

    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

    PubMed  CAS  Google Scholar 

  • Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L (2004) Effects of noise in excitable systems. Phys Rep 392:321–424

    Google Scholar 

  • Litvak V, Sompolinsky H, Segev I, Abeles M (2003) On the transmission of rate code in long feedforward networks with excitatory-inhibitory balance. J Neurosci 23:3006–3015

    PubMed  CAS  Google Scholar 

  • Longtin A (1993) Stochastic resonance in neuron models. J Stat Phys 70:309–327

    Google Scholar 

  • Longtin A, Bulsara A, Pierson D, Moss F (1994) Bistability and the dynamics of periodically forced sensory neurons. Biol Cybern 70:569–578

    PubMed  CAS  Google Scholar 

  • Maršálek P, Koch C, Maunsell J (1997) On the relationship between synaptic input and spike output jitter in individual neurons. Proc Natl Acad Sci USA 94:735–740

    PubMed  Google Scholar 

  • Mattia M, Del Giudice P (2000) Efficient event-driven simulation of large networks of spiking neurons and dynamical synapses. Neural Comput 12:2305–2329

    PubMed  CAS  Google Scholar 

  • Mattia M, Del Giudice P (2002) Population dynamics of interacting spiking neurons. Phys Rev E 66:051917

    Google Scholar 

  • Mattia M, Del Giudice P (2004) Finite-size dynamics of inhibitory and excitatory Interacting spiking neurons. Phys Rev E 70:052903

    Google Scholar 

  • Meffin H, Burkitt AN, Grayden DB (2004) An analytical model for the ‘large, fluctuating conductance state’ typical of neocortical neurons in vivo. J Comput Neurosci 16:159–175

    PubMed  Google Scholar 

  • Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50:1645–1662

    Google Scholar 

  • Mongillo G, Amit DJ (2003) Retrospective and prospective persistent activity induced by Hebbian learning in a recurrent cortical network. Eur J Neurosci 18:2011–2024

    PubMed  Google Scholar 

  • Mongillo G, Amit DJ (2005) Learning in realistic networks of spiking neurons and spike-driven plastic synapses. Eur J Neurosci 21:3143–3160

    PubMed  Google Scholar 

  • Nykamp DQ, Trachina D (2000) A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning. J Comput Neurosci 8:19–50

    PubMed  CAS  Google Scholar 

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

    PubMed  CAS  Google Scholar 

  • Plesser HE, Geisel T (1999) Markov analysis of stochastic resonance in a periodically driven integrate-fire neuron. Phys Rev E 59:7008–7017

    CAS  Google Scholar 

  • Plesser HE, Geisel T (2001) Stochastic resonance in model neurons: Endogenous stimulation revisited. Phys Rev E 63:031916

    CAS  Google Scholar 

  • Plesser HE, Tanaka S (1997) Stochastic resonance in a model neuron with reset. Phys Lett A 225:228–234

    CAS  Google Scholar 

  • Prete VD, Coolen ACC (2004) Non-equilibrium statistical mechanics of recurrent networks with realistic neurons. Neurocomputing 58–60:239–244

    Google Scholar 

  • Rauch A, La Camera G, Lüscher 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

    PubMed  Google Scholar 

  • Rennie CJ, Robinson PA, Wright JJ (2002) Unified neurophysical model of EEG spectra and evoked potentials. Biol Cybern 86:457–471

    PubMed  CAS  Google Scholar 

  • Reyes AD (2003) Synchrony-dependent propagation of firing rate in iteratively constructed networks in vitro. Nat Neurosci 6:593– 599

    PubMed  CAS  Google Scholar 

  • Risken H (1996) The Fokker–Planck equation 3rd edn. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Ritz R, Gerstner W, Fuentes U, van Hemmen JL (1994) A biologically motivated and analytically soluble model of collective oscillations in the cortex II Application to binding and pattern segmentation. Biol Cybern 71:349–358

    PubMed  CAS  Google Scholar 

  • Rodriguez R, Lánský P (2000) Effect of spatial extension on noise-enhanced phase locking in a leaky integrate-and-fire model of a neuron. Phys Rev E 62:1–11

    Google Scholar 

  • Sakaguchi H (2004) Oscillatory phase transition and pulse propagation in noisy integrate-and-fire neurons. Phys Rev E 70:022901

    Google Scholar 

  • Scharstein H (1979) Input–output relationship of the leaky-integrator neuron model. J Math Biol 8:403–420

    PubMed  CAS  Google Scholar 

  • Schindler M, Talkner P, Hänggi P (2004) Firing time statistics for driven neuron models: analytic expressions versus numerics. Phys Rev Lett 93:048102

    PubMed  Google Scholar 

  • Schindler M, Talkner P, Hänggi P (2005) Escape rates in periodically driven Markov processes. Physica A 351:40–50

    Google Scholar 

  • Schrödinger E (1915) Zur Theorie der Fall- und Steigversuche an Teilchen mit Brownscher Bewegung. Phys Zeitschr 16:289–295

    Google Scholar 

  • Segundo JP, Perkel DH, Wyman J, Hegsted H, Moore GP (1968) Input-output relations in computer simulated nerve cells. Kybernetik 4:157–171

    PubMed  CAS  Google Scholar 

  • Senn W, Urbanczik R (2000) Similar non-leaky integrate-and-fire neurons with instantaneous couplings always synchronize. SIAM J Appl Math 61:1143–1155

    Google Scholar 

  • Shelley MJ, Tao L (2001) Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks. J Comput Neurosci 11:111–119

    PubMed  CAS  Google Scholar 

  • Shimokawa T, Rogel A, Pakdaman K, Sato S (1999a) Stochastic resonance and spike-timing precision in an ensemble of leaky integrate and fire neuron models. Phys Rev E 59:3461–3470

    CAS  Google Scholar 

  • Shimokawa T, Pakdaman K, Sato S (1999b) Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise. Phys Rev E 59:3427–3443

    CAS  Google Scholar 

  • Siebert WM (1970) Frequency discrimination in the auditory system: place or periodicity mechanisms? Proc IEEE 58:723–30

    Article  Google Scholar 

  • Singer W (1993) Synchronization of cortical activity and its putative role in information processing and learning. Annu Rev Physiol 55:349–374

    PubMed  CAS  Google Scholar 

  • Smith LS (1996) Onset-based sound segmentation. In: Touretzky DS, Mozer MC, Haselmo ME (eds), Advances in neural information processing systems, vol 8. MIT Press, Cambridge

  • Spiridon M, Gerstner W (1999) Noise spectrum and signal transmission through a population of spiking neurons. Netw Comput Neural Syst 10:257–272

    CAS  Google Scholar 

  • Stemmler M (1996) A single spike suffices: the simplest form of stochastic resonance in model neurons. Netw Comput Neural Syst 7:687–716

    Google Scholar 

  • Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20

    Google Scholar 

  • Theunissen F, Miller JP (1995) Temporal encoding in nervous systems: A rigorous definition. J Comput Neurosci 2:149–162

    PubMed  CAS  Google Scholar 

  • Tiesinga PHE (2002) Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons. Phys Rev E 65:041913

    CAS  Google Scholar 

  • Tiesinga PHE, Sejnowski TJ (2001) Precision of pulse-coupled networks of integrate-and-fire neurons. Netw Comput Neural Syst 12:215–233

    CAS  Google Scholar 

  • Tsodyks MV, Skaggs WE, Sejnowski TJ, McNaughton BL (1996) Population dynamics and theta rhythm phase precession of hippocampal place cell firing: a spiking neuron model. Hippocampus 6:271–280

    PubMed  CAS  Google Scholar 

  • Tuckwell HC (1979) Synaptic transmission in a model for stochastic neural activity. J Theor Biol 77:65–81

    PubMed  CAS  Google Scholar 

  • Tuckwell HC (1988a) Introduction to theoretical neurobiology: linear cable theory and dendritic structure vol 1. Cambridge University Press, Cambridge

    Google Scholar 

  • Tuckwell HC (1988b) Introduction to theoretical neurobiology: nonlinear and stochastic theories vol 2. Cambridge University Press, Cambridge

    Google Scholar 

  • Tuckwell HC (1989) Stochastic processes in the neurosciences. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  • Tuckwell HC, Lánský P (1997) On the simulation of biological diffusion processes. Comput Biol Med 27:1–7

    PubMed  CAS  Google Scholar 

  • Uhlenbeck GE, Ornstein LS (1930) On the theory of Brownian motion. Phys Rev 36:823–841

    CAS  Google Scholar 

  • Usher M, Schuster HG, Niebur E (1993) Dynamics of populations of integrate-and-fire neurons, partial synchronizaton and memory. Neural Comput 5:570–586

    Google Scholar 

  • Usher M, Stemmler M, Koch C, Olami Z (1994) Network amplification of local fluctuations causes high spike rate variability, fractal firing patterns and oscillatory local field potentials. Neural Comput 6:795–836

    Google Scholar 

  • van Hemmen JL (2001) Theory of synaptic plasticity. In: Moss F, Gielen S (eds), Handbook of biological physics: neuro-informatics and neural modelling, vol 4. Elsevier, Amsterdam, pp 771–823

    Google Scholar 

  • van Kampen NG (1992) Stochastic processes in physics and chemistry. North-Holland, Amsterdam

    Google Scholar 

  • van Rossum MCW (2001) The transient precision of integrate-and-fire neurons: Effect of background activity and noise. J Comput Neurosci 10:303–311

    PubMed  Google Scholar 

  • van Rossum MCW, Renart A (2004) Computation with populations codes in layered networks of integrate-and-fire neurons. Neurocomputing 58–60:265–270

    Google Scholar 

  • van Rossum MCW, Bi GQ, Turrigiano GG (2000) Stable Hebbian learning from spike timing-dependent plasticity. J Neurosci 20:8812–8821

    PubMed  Google Scholar 

  • van Rossum MCW, Turrigiano GG, Nelson SB (2002) Fast propagation of firing rates through layered networks of noisy neurons. J Neurosci 22:1956–1966

    PubMed  Google Scholar 

  • van Vreeswijk C, Abbott LF (1993) Self-sustained firing in populations of integrate-and-fire neurons. SIAM J Appl Math 53:253–264

    Google Scholar 

  • Weiss TF (1966) A model of the peripheral auditory system. Kybernetik 3:153–175

    PubMed  CAS  Google Scholar 

  • Wenning G, Obermayer K (2003) Activity driven adaptive stochastic resonance. Phys Rev Lett 90:120602

    PubMed  CAS  Google Scholar 

  • Wenning G, Hoch T, Obermayer K (2005) Detection of pulses in a colored noise setting. Phys Rev E 71:021902

    Google Scholar 

  • Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24

    Article  PubMed  CAS  Google Scholar 

  • Winfree AT (1967) Biological rhythms and the behavior of populations of coupled oscillators. J Theor Biol 16:14–42

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. N. Burkitt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burkitt, A.N. A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties. Biol Cybern 95, 97–112 (2006). https://doi.org/10.1007/s00422-006-0082-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00422-006-0082-8

Keywords

Navigation