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230
Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons
, 1999
"... The dynamics of networks of sparsely connected excitatory and inhibitory integrateand re neurons is studied analytically. The analysis reveals a very rich repertoire of states, including: Synchronous states in which neurons re regularly; Asynchronous states with stationary global activity and very ..."
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Cited by 306 (16 self)
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The dynamics of networks of sparsely connected excitatory and inhibitory integrateand re neurons is studied analytically. The analysis reveals a very rich repertoire of states, including: Synchronous states in which neurons re regularly; Asynchronous states with stationary global activity and very irregular individual cell activity; States in which the global activity oscillates but individual cells re 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 eects in the asynchronous state are also discussed.
Fast global oscillations in networks of integrateandfire neurons with low firing rates
 Neural Computation
, 1999
"... We study analytically the dynamics of a network of sparsely connected inhibitory integrateandfire neurons in a regime where individual neurons emit spikes irregularly and at a low rate. In the limit when the number of neurons N → ∞, the network exhibits a sharp transition between a stationary and ..."
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Cited by 227 (17 self)
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We study analytically the dynamics of a network of sparsely connected inhibitory integrateandfire neurons in a regime where individual neurons emit spikes irregularly and at a low rate. In the limit when the number of neurons N → ∞, the network exhibits a sharp transition between a stationary and an oscillatory global activity regime where neurons are weakly synchronized. The activity becomes oscillatory when the inhibitory feedback is strong enough. The period of the global oscillation is found to be mainly controlled by synaptic times, but depends also on the characteristics of the external input. In large but finite networks, the analysis shows that global oscillations of finite coherence time generically exist both above and below the critical inhibition threshold. Their characteristics are determined as functions of systems parameters, in these two different regimes. The results are found to be in good agreement with numerical simulations.
Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking
 NEURAL COMPUTATION
, 2000
"... An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrateandfire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analy ..."
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Cited by 159 (25 self)
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An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrateandfire 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 integrateandfire 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.
Chaotic Balanced State in a Model of Cortical Circuits
 NEURAL COMPUT
, 1998
"... The nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are still not well understood. We consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells. We study a network model w ..."
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Cited by 122 (3 self)
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The nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are still not well understood. We consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells. We study a network model with excitatory and inhibitory populations of simple binary units. The internal feedback is mediated by relatively large synaptic strengths, so that the magnitude of the total excitatory as well as inhibitory feedback is much larger than the neuronal threshold. The connectivity is random and sparse. The mean number of connections per unit is large but small compared to the total number of cells in the network. The network also receives a large, temporally regular input from external sources. An analytical solution of the meanfield theory of this model which is exact in the limit of large network size is presented. This theory reveals a new cooperative stationary state of large networks, which we term a balanced state. In this state, a balance between the excitatory and inhibitory inputs emerges dynamically for a wide range of parameters, resulting in a net input whose temporal fluctuations are of the same order as its mean. The internal synaptic inputs act as a strong negative feedback, which linearizes the population responses to the external drive despite the strong nonlinearity of the individual cells. This feedback also greatly stabilizes 1 the system's state and enables it to track a timedependent input on time scales much shorter than the time constant of a single cell. The spatiotemporal statistics of the balanced state is calculated. It is shown that the autocorrelations decay on a short time scale yielding an approximate Poissonian temporal s...
Polychronization: computation with spikes
 Neural. Comput
, 2006
"... We present a minimal spiking network that can polychronize, i.e., exhibit persistent timelocked but not synchronous firing patterns with millisecond precision, as in synfire braids. The network consists of cortical spiking neurons with axonal conduction delays and spiketimingdependent plasticity ..."
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Cited by 115 (0 self)
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We present a minimal spiking network that can polychronize, i.e., exhibit persistent timelocked but not synchronous firing patterns with millisecond precision, as in synfire braids. The network consists of cortical spiking neurons with axonal conduction delays and spiketimingdependent plasticity (STDP); a readytouse MATLAB code is included. It exhibits sleep oscillations, gamma (40 Hz) rhythms, and other interesting regimes. Due to the interplay between the delays and STDP, the spiking neurons spontaneously selforganize into groups and generate polychronous persistent activity. The number of coexisting polychronous groups far exceeds the number of neurons in the network, resulting in an unprecedented memory capacity of the system. 1
Populations of Spiking Neurons
 PULSED NEURAL NETWORKS, CHAPTER 10
, 1998
"... Introduction In standard neural network theory, neurons are described in terms of mean firing rates. The analog input variable I is mapped via a nonlinear gain function g to an analog output variable = g(I) which may be interpreted as the mean firing rate. If the input consists of output rates j ..."
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Cited by 93 (3 self)
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Introduction In standard neural network theory, neurons are described in terms of mean firing rates. The analog input variable I is mapped via a nonlinear gain function g to an analog output variable = g(I) which may be interpreted as the mean firing rate. If the input consists of output rates j of other neurons weighted by a factor w ij , we arrive at the standard formula i = g( X j w ij j ) (10.1) which is the starting point of most neural network theories. As we have seen in Chapter 1, the firing rate defined by a temporal average over many spikes of a single neuron is a concept which works well if the input is constant or changes on a time scale which is slow with respect to the size of the temporal averaging window. Sensory inpu
Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity
 Neural Computation
, 2003
"... In model networks of Ecells and Icells (excitatory and inhibitory neurons) , synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the Ecells synchronize the Icells and vice versa. Under ideal conditions  homogeneity in relevant network parameters, ..."
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Cited by 74 (8 self)
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In model networks of Ecells and Icells (excitatory and inhibitory neurons) , synchronous rhythmic spiking often comes about from the interplay between the two cell groups: the Ecells synchronize the Icells and vice versa. Under ideal conditions  homogeneity in relevant network parameters, and alltoall connectivity for instance  this mechanism can yield perfect synchronization.
HigherOrder Statistics of Input Ensembles and the Response of Simple Model Neurons
 Neural Computation
, 2003
"... Pairwise correlations among spike trains recorded in vivo have been frequently reported. It has been argued that correlated activity could play an important role in the brain, because it efficiently modulates the response of a postsynaptic neuron. We show here that a neuron’s output firing rate cr ..."
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Cited by 62 (18 self)
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Pairwise correlations among spike trains recorded in vivo have been frequently reported. It has been argued that correlated activity could play an important role in the brain, because it efficiently modulates the response of a postsynaptic neuron. We show here that a neuron’s output firing rate critically depends on the higherorder statistics of the input ensemble. We constructed two statistical models of populations of spiking neurons that fired with the same rates and had identical pairwise correlations, but differed with regard to the higherorder interactions within the population. The first ensemble was characterized by clusters of spikes synchronized over the whole population. In the second ensemble, the size of spike clusters was, on average, proportional to the pairwise correlation. For both input models, we assessed the role of the size of the population, the firing rate, and the pairwise correlation on the output rate of two simple model neurons: a continuous firingrate model and a conductancebased leaky integrateandfire neuron. An approximation
Is there chaos in the brain? II. Experimental evidence and related models
 C. R. Biol
, 2003
"... The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The meth ..."
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Cited by 53 (0 self)
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The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773–793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system.
Noise in IntegrateandFire Neurons: From Stochastic Input to Escape Rates
 TO APPEAR IN NEURAL COMPUTATION.
, 1999
"... We analyze the effect of noise in integrateandfire neurons driven by timedependent input, and compare the diffusion approximation for the membrane potential to escape noise. It is shown that for timedependent subthreshold input, diffusive noise can be replaced by escape noise with a hazard funct ..."
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Cited by 51 (6 self)
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We analyze the effect of noise in integrateandfire neurons driven by timedependent input, and compare the diffusion approximation for the membrane potential to escape noise. It is shown that for timedependent subthreshold input, diffusive noise can be replaced by escape noise with a hazard function that has a Gaussian dependence upon the distance between the (noisefree) membrane voltage and threshold. The approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage. Stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics.