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203
NEURAL EXCITABILITY, SPIKING AND BURSTING
, 2000
"... Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neurocomputational properties of the cells. For example, when the rest state is near a saddlenode bifurcation, the cell can fire allorn ..."
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Cited by 145 (4 self)
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Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neurocomputational properties of the cells. For example, when the rest state is near a saddlenode bifurcation, the cell can fire allornone spikes with an arbitrary low frequency, it has a welldefined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not allornone, it does not have a welldefined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neurocomputational properties, and we show that different bursters can interact, synchronize and process information differently.
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.
The Effects of Spike Frequency Adaptation and Negative Feedback on the Synchronization of Neural Oscillators
"... this article, we de#ne synchrony to mean a zero phase lag when two identical neurons are identically coupled. This is a mathematical idealization but allows us to be precise when we describe the locking behavior of coupled oscillators ..."
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Cited by 68 (6 self)
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this article, we de#ne synchrony to mean a zero phase lag when two identical neurons are identically coupled. This is a mathematical idealization but allows us to be precise when we describe the locking behavior of coupled oscillators
Synchronization and Oscillatory Dynamics in Heterogeneous, Mutually Inhibited Neurons
 J. Comput. Neurosci
, 1998
"... . We study some mechanisms responsible for synchronous oscillations and loss of synchrony at physiologically relevant frequencies (10200 Hz) in a network of heterogeneous inhibitory neurons. We focus on the factors that determine the level of synchrony and frequency of the network response, as well ..."
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Cited by 58 (6 self)
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. We study some mechanisms responsible for synchronous oscillations and loss of synchrony at physiologically relevant frequencies (10200 Hz) in a network of heterogeneous inhibitory neurons. We focus on the factors that determine the level of synchrony and frequency of the network response, as well as the effects of mild heterogeneity on network dynamics. With mild heterogeneity, synchrony is never perfect and is relatively fragile. In addition, the effects of inhibition are more complex in mildly heterogeneous networks than in homogeneous ones. In the former, synchrony is broken in two distinct ways, depending on the ratio of the synaptic decay time to the period of repetitive action potentials (øs=T ), where T can be determined either from the network or from a single, selfinhibiting neuron. With øs=T ? 2, corresponding to large applied current, small synaptic strength or large synaptic decay time, the effects of inhibition are largely tonic and heterogeneous neurons spike relativ...
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 ..."
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Cited by 57 (13 self)
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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
A universal model for spikefrequency adaptation
 Neural Comput
, 2003
"... Spikefrequency adaptation is a prominent feature of neural dynamics. Among other mechanisms various ionic currents modulating spike generation cause this type of neural adaptation. Prominent examples are voltagegated potassium currents (Mtype currents), the interplay of calcium currents and int ..."
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Cited by 57 (9 self)
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Spikefrequency adaptation is a prominent feature of neural dynamics. Among other mechanisms various ionic currents modulating spike generation cause this type of neural adaptation. Prominent examples are voltagegated potassium currents (Mtype currents), the interplay of calcium currents and intracellular calcium dynamics with calciumgated potassium channels (AHPtype currents), and the slow recovery from inactivation of the fast sodium current. While recent modeling studies have focused on the effects of specic adaptation currents, we derive a universal model for the ringfrequency dynamics of an adapting neuron which is independent of the specic adaptation process and spike generator. The model is completely dened by the neuron's onset fIcurve, steadystate fIcurve, and the time constant of adaptation. For a specic neuron these parameters can be easily determined from electrophysiological measurements without any pharmacological manipulations. At the same time, the simplicity of the model allows one to analyze mathematically how adaptation inuences signal processing on the singleneuron level. In particular, we elucidate the specic nature of highpass lter properties caused by spikefrequency adaptation. The model is limited to ring frequencies higher than the reciprocal adaptation time constant and to moderate uctuations of the adaptation and the input current. As an extension of the model, we introduce a framework for combining an arbitrary spike generator with a generalized adaptation current. 1 J. Benda & A. V. M. Herz: A Universal Model for SpikeFrequency Adaptation 2
Dynamics of Membrane Excitability Determine Interspike Interval Variability: A Link Between Spike Generation Mechanisms and Cortical Spike Train Statistics
, 1998
"... We propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains. The key lies in the nonlinear dynamics of cortical spike generation, which are consistent with type I membranes where saddlenode dynamics underlie excitability (Rinzel & Ermentrout ..."
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Cited by 51 (4 self)
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We propose a biophysical mechanism for the high interspike interval variability observed in cortical spike trains. The key lies in the nonlinear dynamics of cortical spike generation, which are consistent with type I membranes where saddlenode dynamics underlie excitability (Rinzel & Ermentrout, 1989). We present a canonical model for type I membranes, the θneuron. The θneuron is a phase model whose dynamics reflect salient features of type I membranes. This model generates spike trains with coefficient of variation (CV) above 0.6 when brought to firing by noisy inputs. This happens because the timing of spikes for a type I excitable cell is exquisitely sensitive to the amplitude of the suprathreshold stimulus pulses. A noisy input current, giving random amplitude “kicks” to the cell, evokes highly irregular firing across a wide range of firing rates; an intrinsically oscillating cell gives regular spike trains. We corroborate the results with simulations of the MorrisLecar (ML) neural model with random synaptic inputs: type I ML yields high CVs. When this model is modified to have type II dynamics (periodicity arises via a Hopf bifurcation), however, it gives regular spike trains (CV below 0.3). Our results suggest that the high CV values such as those observed in cortical spike trains are an intrinsic characteristic of type I membranes driven to firing by “random” inputs. In contrast, neural oscillators or neurons exhibiting type II excitability should produce regular spike trains.
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 ..."
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Cited by 46 (4 self)
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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.
Synchrony, Stability, and Firing Patterns in PulseCoupled Oscillators
 PHYSICA D
, 2002
"... We study nontrivial firing patterns in small assemblies of pulsecoupled oscillatory maps. We find conditions for the existence of waves in rings of coupled maps that are coupled bidirectionally. We also find conditions for stable synchrony in general alltoall coupled oscillators. Surprisingly, ..."
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Cited by 39 (0 self)
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We study nontrivial firing patterns in small assemblies of pulsecoupled oscillatory maps. We find conditions for the existence of waves in rings of coupled maps that are coupled bidirectionally. We also find conditions for stable synchrony in general alltoall coupled oscillators. Surprisingly, we find that for maps that are derived from physiological data, the stability of synchrony depends on the number of oscillators. We describe rotating waves in twodimensional lattices of maps and reduce their existence to a reduced system of algebraic equations which are solved numerically.