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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 ..."
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Cited by 59 (15 self)
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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
Stabilization of planar collective motion: alltoall communication
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2007
"... This paper proposes a design methodology to stabilize isolated relative equilibria in a model of alltoall coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or circular motio ..."
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Cited by 48 (18 self)
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This paper proposes a design methodology to stabilize isolated relative equilibria in a model of alltoall coupled identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or circular motion of all particles with fixed relative phases. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The results of the paper provide a loworder parametric family of stabilizable collectives that offer a set of primitives for the design of higherlevel tasks at the group level.
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 46 (9 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.
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 41 (5 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.
Decentralized synchronization protocols with nearest neighbor communication
 In SenSys ’04: Proceedings of the 2nd international conference on Embedded networked sensor systems
, 2004
"... A class of synchronization protocols for dense, largescale sensor networks is presented. The protocols build on the recent work of Hong, Cheow, and Scaglione [5, 6] in which the synchronization update rules are modeled by a system of pulsecoupled oscillators. In the present work, we define a class ..."
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Cited by 40 (0 self)
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A class of synchronization protocols for dense, largescale sensor networks is presented. The protocols build on the recent work of Hong, Cheow, and Scaglione [5, 6] in which the synchronization update rules are modeled by a system of pulsecoupled oscillators. In the present work, we define a class of models that converge to a synchronized state based on the local communication topology of the sensor network only, thereby lifting the alltoall communication requirement implicit in [5, 6]. Under some rather mild assumptions of the connectivity of the network over time, these protocols still converge to a synchronized state when the communication topology is time varying. Categories and Subject Descriptors
Weakly pulsecoupled oscillators, FM interactions, synchronization, and oscillatory associative memory
 IEEE Trans. Neural Networks
, 1999
"... Abstract—We study pulsecoupled neural networks that satisfy only two assumptions: each isolated neuron fires periodically, and the neurons are weakly connected. Each such network can be transformed by a piecewise continuous change of variables into a phase model, whose synchronization behavior and ..."
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Cited by 31 (3 self)
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Abstract—We study pulsecoupled neural networks that satisfy only two assumptions: each isolated neuron fires periodically, and the neurons are weakly connected. Each such network can be transformed by a piecewise continuous change of variables into a phase model, whose synchronization behavior and oscillatory associative properties are easier to analyze and understand. Using the phase model, we can predict whether a given pulsecoupled network has oscillatory associative memory, or what minimal adjustments should be made so that it can acquire memory. In the search for such minimal adjustments we obtain a large class of simple pulsecoupled neural networks that can memorize and reproduce synchronized temporal patterns the same way a Hopfield network does with static patterns. The learning occurs via modification of synaptic weights and/or synaptic transmission delays. Index Terms — Canonical models, Class 1 neural excitability, integrateandfire neurons, multiplexing, synfire chain, transmission delay. I.
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 intra ..."
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Cited by 30 (7 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 specific adaptation currents, we derive a universal model for the firingfrequency dynamics of an adapting neuron which is independent of the specific adaptation process and spike generator. The model is completely defined by the neuron’s onset fIcurve, steadystate fIcurve, and the time constant of adaptation. For a specific 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 influences signal processing on the singleneuron level. In particular, we elucidate the specific nature of highpass filter properties caused by spikefrequency adaptation. The model is limited to firing frequencies higher than the reciprocal adaptation time constant and to moderate fluctuations 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
Collective motion and oscillator synchronization
 Proc. Block Island Workshop on Cooperative Control
, 2003
"... Summary. This paper studies connections between phase models of coupled oscillators and kinematic models of groups of selfpropelled particles. These connections are exploited in the analysis and design of feedback control laws for the individuals that stabilize collective motions for the group. 1 ..."
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Cited by 27 (8 self)
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Summary. This paper studies connections between phase models of coupled oscillators and kinematic models of groups of selfpropelled particles. These connections are exploited in the analysis and design of feedback control laws for the individuals that stabilize collective motions for the group. 1
Mechanisms of PhaseLocking and Frequency Control in Pairs of coupled Neural Oscillators
, 2000
"... INTRODUCTION Oscillations occur in many networks of neurons, and are associated with motor behavior, sensory processing, learning, arousal, attention and pathology (Parkinson's tremor, epilepsy). Such oscillations can be generated in many ways. This chapter discusses some mathematical issues a ..."
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Cited by 27 (6 self)
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INTRODUCTION Oscillations occur in many networks of neurons, and are associated with motor behavior, sensory processing, learning, arousal, attention and pathology (Parkinson's tremor, epilepsy). Such oscillations can be generated in many ways. This chapter discusses some mathematical issues associated with creation of coherent rhythmic activity in networks of neurons. We focus on pairs of cells, since many of the issues for larger networks are most clearly displayed in that context. As we will show, there are many mechanisms for interactions among the network components, and these can have different mathematical properties. A description of behavior of larger networks using some of the mechanisms described in this chapter can be found in the related chapter by Rubin and Terman. For reviews of papers about oscillatory behavior in specific networks in the nervous system, see Gray (1994), Marder and Calabrese (1996), Singer (1993), and Traub et al (1999). The chapter is organiz
Geometric Singular Perturbation Analysis of Neuronal Dynamics
 in Handbook of Dynamical Systems
, 2000
"... : In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which d ..."
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Cited by 24 (14 self)
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: In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several dierent classes of bursting solutions; these have been classied by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two dis...