Abstract

An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized `locked' solutions is analyzed.

Citations

584	Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex - Hubel, Wiesel - 1962
228	Speed of processing in the human visual system - Thorpe, Fize, et al. - 1996
211	Absolute stability of global pattern formation and parallel memory storage by competitive neural networks - Cohen, Grossberg
196	Excitatory and inhibitory interactions in localized populations of model neurons - Wilson, Cowan - 1972
181	Renewal Theory - Cox - 1962
158	AT (2001) The Geometry of Biological Time - Winfree
156	The Statistical Analysis of Series of Events - Cox, Lewis - 1966
116	Mathematical theory of functional dynamics of cortical and thalamic nervous-tissue - Wilson, Cowan - 1973
99	Introduction to Theoretical Neurobiology - Tuckwell - 1988
96	Synchronization in excitatory neural networks - Hansel, Mato, et al. - 1995
90	Fast global oscillations in networks of integrate-and-fire neurons with low firing rates - Brunel, Hakim - 1999
89	Time structure of the activity in neural network models - Gerstner - 1995
88	Global competition and local cooperation in a network of neural oscillators - Terman, Wang - 1995
82	Chaos in neuronal networks with balanced excitatory and inhibitory activity - Vreeswijk, Sompolinsky - 1996
78	Neuronal spike trains and stochastic point processes. I. The single spike train - Perkel, Gerstein, et al. - 1967
73	When inhibition not excitation synchronizes neural firing - Vreeswijk, Abbott, et al. - 1994
69	Mean-field analysis of neuronal spike dynamics - Treves - 1993
68	Asynchronous states in networks of pulsecoupled oscillators - Abbott, Vreeswijk - 1993
58	of the HodgkinHuxley equations to a single-variable threshold model - Kistler, Gerstner, et al. - 1997
56	Rapid synchronization through fast threshold modulation - Somers, Kopell - 1993
53	Modality and topographic properties of single neurons of cat’s somatic sensory cortex - Mountcastle - 1957
52	Associative memory in a network of “spiking” neurons - Gerstner, Hemmen - 1992
46	A biologically motivated and analytically soluble model of collective oscillations in the cortex: I. Theory of weak locking - Gerstner, Ritz, et al. - 1993
36	What matters in neuronal locking - Gerstner, Hemmen, et al. - 1996
36	Rapid state switching in balanced cortical network models - Tsodyks, Sejnowski - 1995
34	Some models of neuronal variability - Stein - 1967
33	A.V.M.: Rapid local synchronization of action potentials: toward computation with coupled integrate-and-firing neurons - Hopfield, Herz - 1995
31	W (2000) Noise in integrate-and-fire neurons: from stochastic input to escape rates - HE, Gerstner
30	Collective Synchronization of Pulse-Coupled Oscillators and Excitable Units - Kuramoto - 1991
28	Dynamics of membrane excitability determine interspike interval variability: A link between spike generation mechanisms and cortical spike train statistics - Gutkin, Ermentrout - 1998
27	Clustering in globally coupled inhibitory neurons - Golomb, Rinzel - 1994
24	Phase-locking in weakly heterogeneous neuronal networks - Chow - 1998
23	Self-entrainment of a population of coupled nonlinear oscillators - Kuramoto - 1975
23	Pattern of synchrony in inhomogeneous networks of oscillators with pulse interaction - Tsodyks, Mit’kov, et al. - 1993
21	A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations - Pinto, Brumberg, et al. - 1996
18	Coherence and incoherence in a globally coupled ensemble of pulse-emitting Units - Gerstner, Hemmen - 1993
15	The relationship between the firing rate of a single neuron and the level of activity in a population of neurons - Knight - 1972
14	Aperiodic stochastic resonance - Collins, Chow, et al. - 1996
14	Novel integrate-and-fire-like model of repetitive firing in cortical neurons - Stevens, Zador - 1998
13	Relation between shapes of postsynaptic potentials and changes in the firing probability of cat motoneurons - Fetz - 1983
12	Clustering in globally coupled phase oscillators - Golomb, Hansel, et al. - 1992
11	n : m phase locking of weakly coupled oscillators - Ermentrout - 1981
11	Synchronization of pulse coupled biological oscillators - Mirollo, Strogatz - 1990
10	Coding and information processing in neural networks - Gerstner, Hemmen - 1994
10	Dynamics of a random neural network with synaptic depression - Senn, Wyler, et al. - 1996
8	Functional identification of the input-output transforms of motoneurones in the rat and cat - Poliakov, Powers, et al. - 1997
8	Stability of incoherence in a population of coupled oscillators - Strogatz, Mirollo - 1991
7	Alternating oscillatory and stochastic dynamics in a model for a neuronal assembly - Bauer, Pawelzik - 1993
7	Cross-correlation functions for a neuronal model - Knox - 1974
7	Local neocortical processing: a time for recognition, Int - Treves - 1992