Abstract
The dynamics of networks of sparsely connected excitatory and inhibitory integrate-and-fire neurons are studied analytically. The analysis reveals a rich repertoire of states, including synchronous states in which neurons fire regularly; asynchronous states with stationary global activity and very irregular individual cell activity; and states in which the global activity oscillates but individual cells fire irregularly, typically at rates lower than the global oscillation frequency. The network can switch between these states, provided the external frequency, or the balance between excitation and inhibition, is varied. Two types of network oscillations are observed. In the fast oscillatory state, the network frequency is almost fully controlled by the synaptic time scale. In the slow oscillatory state, the network frequency depends mostly on the membrane time constant. Finite size effects in the asynchronous state are also discussed.
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References
Abbott LF, van Vreeswijk C (1993) Asynchronous states in a network of pulse-coupled oscillators. Phys. Rev. E 48:1483-1490.
Abramowitz M, Stegun IA (1970) Tables of Mathematical Functions. Dover Publications, New York.
Amit DJ, Brunel N (1997a) Dynamics of a recurrent network of spiking neurons before and following learning. Network 8:373-404.
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.
Amit DJ, Evans M, Abeles M (1990) Attractor neural networks with biological probe neurons. Network 1:381-405.
Amit DJ, Tsodyks MV (1991) Quantitative study of attractor neural network retrieving at low spike rates I: Substrate-spikes, rates and neuronal gain. Network 2:259-274.
Bernander O, Koch C, Usher M (1991) Synaptic background activity determines spatio-temporal integration in single pyramidal cells. Proc. Natl. Acad. Sci. USA 88:11569-11573.
Bragin A, Jando G, Nadasdy Z, Hetke J, Wise K, Buzsáki G (1995) Gamma (40-100 Hz) oscillation in the hippocampus of the behaving rat. J. Neurosci. 15:47-60.
Brunel N, Hakim V (1999) Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11:1621-1671.
Buzsaki G, Leung LW, Vanderwolf CH (1983) Cellular bases of hippocampal EEG in the behaving rat. Brain Res. 287:139-171.
Buzsaki G, Urioste R, Hetke J, Wise K (1992) High frequency network oscillation in the hippocampus. Science 256:1025-1027.
Chow C (1998) Phase-locking in weakly heterogeneous neuronal networks. Physica D 118:343-370.
Csicsvari J, Hirase H, Czurko A, Buzsáki G (1998) Reliability and state dependence of pyramidal cell-interneuron synapses in the hippocampus: An ensemble approach in the behaving rat. Neuron 21:179-189.
Draguhn A, Traub RD, Schmitz D, Jefferys JGR (1998) Electrical coupling underlies high-frequency oscillations in the hippocampus in vitro. Nature 394:189-193.
Fisahn A, Pike FG, Buhl EH, Paulsen O (1998) Cholinergic induction of network oscillations at 40 Hz in the hippocampus in vitro. Nature 394:186-189.
Fusi S, Mattia M (1999) Collective behavior of networks with linear (VLSI) integrate and fire neurons. Neural Comput. 11:633-652.
Gerstner W (1995) Time structure of the activity in neural network models. Phys. Rev. E 51:738-758.
Gerstner W, van Hemmen L, Cowan J (1996) What matters in neuronal locking? Neural Comput. 8:1653-1676.
Golomb D, Rinzel J (1994) Clustering in globally coupled inhibitory neurons. Physica D 72:259-282.
Gray CM (1994) Synchronous oscillations in neuronal systems: Mechanisms and functions. J. Comput. Neurosci. 1:11-38.
Hansel D, Mato G, Meunier C (1995) Synchrony in excitatory neural networks. Neural Comput. 7:307-337.
Hirsch MW, Smale S (1974) Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York.
Hopfield JJ, Herz AVM (1995) Rapid local synchronization of action potentials: Towards computation with coupled integrate-and-fire neurons. Proc. Natl. Acad. USA 92:6655-6662.
McLeod K, Laurent G (1996) Distinct mechanisms for synchronization and temporal patterning of odor-encoding neural assemblies. Science 274:976-979.
Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50:1645-1662.
Ricciardi LM (1977) Diffusion Processes and Related Topics on Biology. Springer-Verlag, Berlin.
Risken H (1984) The Fokker Planck Equation: Methods of Solution and Applications. Springer-Verlag, Berlin.
Terman D, Kopell N, Bose A (1998) Dynamics of two mutually coupled slow inhibitory neurons. Physica D 117:241-275.
Terman D, Wang DL (1995) Global competition and local cooperation in a network of neural oscillators. Physica D 81:243:1319-1325.
Traub RD, Whittington MA, Collins SB, Buzsáki G, Jefferys JGR (1996) Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo. J. Physiol. 493:471-484.
Treves A (1993) Mean-field analysis of neuronal spike dynamics. Network 4:259-284.
Tsodyks MV, Mit'kov I, Sompolinsky H (1993) Pattern of synchrony in inhomogeneous networks of oscillators with pulse interactions. Phys. Rev. Lett. 71:1280-1283.
Tsodyks MV, Sejnowski T (1995) Rapid state switching in balanced cortical network models. Network 6:111-124.
Tuckwell HC (1988) Introduction to Theoretical Neurobiology. Cambridge University Press, Cambridge.
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.
van Vreeswijk C (1996) Partial synchronization in populations of pulse-coupled oscillators. Phys. Rev. E 54:5522-5537.
van Vreeswijk C, Abbott L, Ermentrout GB (1994) When inhibition not excitation synchronizes neural firing. J. Comput. Neurosci. 1:313-321.
van Vreeswijk C, Sompolinsky H (1996) Chaos in neuronal networks with balanced excitatory and inhibitory activity. Science 274:1724-1726.
van Vreeswijk C, Sompolinsky H (1998) Chaotic balanced state in a model of cortical circuits. Neural Comput. 10:1321-1371.
Wang XJ, Buzsáki G (1996) Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J. Neurosci. 16:6402-6413.
White JA, Chow CC, Soto-Treviño C, Kopell N (1998) Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. J. Comput. Neurosci. 5:5-16.
Whittington MA, Traub RD, Jefferys JGR (1995) Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation. Nature 373:612-615.
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Brunel, N. Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons. J Comput Neurosci 8, 183–208 (2000). https://doi.org/10.1023/A:1008925309027
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DOI: https://doi.org/10.1023/A:1008925309027