Results 1  10
of
122
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 ..."
Abstract

Cited by 226 (18 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 158 (25 self)
 Add to MetaCart
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.
What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitationinhibition balance
 J NEUROPHYSIOL 90: 415–430, 2003
, 2003
"... When the local field potential of a cortical network displays coherent fast oscillations (~40Hz gamma or ~200Hz sharpwave ripples), the spike trains of constituent neurons are typically irregular and sparse. The dichotomy between rhythmic local field and stochastic spike trains presents a challe ..."
Abstract

Cited by 130 (7 self)
 Add to MetaCart
When the local field potential of a cortical network displays coherent fast oscillations (~40Hz gamma or ~200Hz sharpwave ripples), the spike trains of constituent neurons are typically irregular and sparse. The dichotomy between rhythmic local field and stochastic spike trains presents a challenge to the theory of brain rhythms in the framework of coupled oscillators. Previous studies have shown that when noise is large and recurrent inhibition is strong, a coherent network rhythm can be generated while single neurons fire intermittently at low rates compared to the frequency of the oscillation. However, these studies used too simplified synaptic kinetics to allow quantitative predictions of the population rhythmic frequency. Here we show how to derive quantitatively the coherent
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 ..."
Abstract

Cited by 120 (2 self)
 Add to MetaCart
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...
Reduction of the HodgkinHuxley Equations to a SingleVariable Threshold Model
 NEURAL COMPUTATION
, 1997
"... It is generally believed that a neuron is a threshold element which fires when some variable u reaches a threshold. Here we pursue the question of whether this picture can be justified and study the fourdimensional neuron model of Hodgkin and Huxley as a concrete example. The model is approximat ..."
Abstract

Cited by 85 (25 self)
 Add to MetaCart
It is generally believed that a neuron is a threshold element which fires when some variable u reaches a threshold. Here we pursue the question of whether this picture can be justified and study the fourdimensional neuron model of Hodgkin and Huxley as a concrete example. The model is approximated by a response kernel expansion in terms of a single variable, the membrane voltage. The firstorder term is linear in the input and has the typical form of an elementary postsynaptic potential. Higherorder kernels take care of nonlinear interactions between input spikes. In contrast to the standard Volterra expansion the kernels depend on the firing time of the most recent output spike. In particular, a zeroorder kernel which describes the shape of the spike and the typical afterpotential is included. Our model neuron fires, if the membrane voltage, given by the truncated response kernel expansion crosses a threshold. The threshold model is tested on a spike train generated by t...
Generalized IntegrateandFire Models of Neuronal Activity Approximate Spike Trains of a . . .
"... We demonstrate that singlevariable integrateandfire models can quantitatively capture the dynamics of a physiologicallydetailed model for fastspiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrateandfire mode ..."
Abstract

Cited by 84 (16 self)
 Add to MetaCart
We demonstrate that singlevariable integrateandfire models can quantitatively capture the dynamics of a physiologicallydetailed model for fastspiking cortical neurons. Through a systematic set of approximations, we reduce the conductance based model to two variants of integrateandfire models. In the first variant (nonlinear integrateandfire model), parameters depend on the instantaneous membrane potential whereas in the second variant, they depend on the time elapsed since the last spike (Spike Response Model). The direct reduction links features of the simple models to biophysical features of the full conductance based model. To quantitatively
Stationary Bumps in Networks of Spiking Neurons
"... Introduction Neuronal activity due to recurrent excitations in the form of a spatially localized pulse or bump has been proposed as a mechanism for feature selectivity in models of the visual system (Somers, Nelson, & Sur, 1995; Hansel & Sompolinsky, 1998), the head direction system (Skaggs ..."
Abstract

Cited by 58 (14 self)
 Add to MetaCart
Introduction Neuronal activity due to recurrent excitations in the form of a spatially localized pulse or bump has been proposed as a mechanism for feature selectivity in models of the visual system (Somers, Nelson, & Sur, 1995; Hansel & Sompolinsky, 1998), the head direction system (Skaggs, Knieram, Kudrimoti, & McNaughton, 1995; Zhang, 1996; Redish, Elga, & Touretzky, 1996), and working memory (Wilson & Cowan, 1973; Amit & Brunel, 1997; Camperi & Wang, 1998). Many of the previous mathematical formulations of such structures have employedpopulation rate models (Wilson &Cowan, 1972, 1973; Amari, 1977; Kishimoto & Amari, 1979; Hansel & Sompolinsky, 1998). (See Ermentrout, 1998, for a recent review.) Here, we consider a network of spiking neurons that shows such structures and investigate their properties. In our network we #nd localized timestationary states
What Matters in Neuronal Locking?
"... Present and permanent address: PhysikDepartment der TU Munchen Exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessa ..."
Abstract

Cited by 49 (10 self)
 Add to MetaCart
Present and permanent address: PhysikDepartment der TU Munchen Exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire. If the postsynaptic potential is decreasing, oscillations are bound to be unstable. This is a kind of locking theorem and boils down to a subtle interplay of axonal delays, postsynaptic potentials, and refractory behavior. The theorem also allows for mixtures of excitatory and inhibitory interactions. On the basis of the locking theorem we present a simple geometric method to verify existence and local stability of a coherent oscillation. 2 1
Firing Rate of the Noisy Quadratic IntegrateandFire Neuron
, 2003
"... We calculate the firing rate of the quadratic integrateandfire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the syna ..."
Abstract

Cited by 48 (4 self)
 Add to MetaCart
We calculate the firing rate of the quadratic integrateandfire neuron in response to a colored noise input current. Such an input current is a good approximation to the noise due to the random bombardment of spikes, with the correlation time of the noise corresponding to the decay time of the synapses. The key parameter that determines the firing rate is the ratio of the correlation time of the colored noise, ¿s, to the neuronal time constant, ¿m. We calculate the firing rate exactly in two limits: when the ratio, ¿s=¿m, goes to zero (white noise) and when it goes to infinity. The correction to the short correlation time limit is O.¿s=¿m/, which is qualitatively different from that of the leaky integrateandfire neuron, where the correction is O. p ¿s=¿m/. The difference is due to the different boundary conditions of the probability density function of the membrane potential of the neuron at firing threshold. The correction to the long correlation time limit is O.¿m=¿s/. By combining the short and long correlation time limits, we derive an expression that provides a good approximation to the firing rate over the whole range of ¿s=¿m in the suprathreshold regime— that is, in a regime in which the average current is sufficient to make the cell fire. In the subthreshold regime, the expression breaks down somewhat when ¿s becomes large compared to ¿m.