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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.
On dynamics of integrateandfire neural networks with adaptive conductances
 Frontiers in Neuroscience
, 2008
"... We present a mathematical analysis of a networks with IntegrateandFire neurons with conductance based synapses. Taking into account the realistic fact that the spike time is only known within some finite precision, we propose a model where spikes are effective at times multiple of a characteristic ..."
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Cited by 34 (17 self)
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We present a mathematical analysis of a networks with IntegrateandFire neurons with conductance based synapses. Taking into account the realistic fact that the spike time is only known within some finite precision, we propose a model where spikes are effective at times multiple of a characteristic time scale δ, where δ can be arbitrary small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the modeldynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the “edge of chaos”, a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a onetoone correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely “in the spikes ” in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and IntegrateandFire models and conductance based models. The present study considers networks with constant input, and without timedependent plasticity, but the framework has been designed for both extensions.
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 30 (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
Existence and Wandering of Bumps in a Spiking Neural Network Model
"... We study spatially localized states of a spiking neuronal network populated by a pulsecoupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to those of the standard Amari model. For nonslow synap ..."
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We study spatially localized states of a spiking neuronal network populated by a pulsecoupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to those of the standard Amari model. For nonslow synaptic connections we are able to go beyond the standard firing rate analysis of localized solutions allowing us to explicitly construct a family of coexisting onebump solutions, and then track bump width and firing pattern as a function of system parameters. We also present an analysis of the model on a discrete lattice. We show that multiple width bump states can coexist and uncover a mechanism for bump wandering linked to the speed of synaptic processing. Moreover, beyond a wandering transition point we show that the bump undergoes an effective random walk with a diffusion coefficient that scales exponentially with the rate of synaptic processing and linearly with the lattice spacing.
A view of Neural Networks as dynamical systems
 in "International Journal of Bifurcations and Chaos", 2009, http://lanl.arxiv.org/abs/0901.2203
"... We present some recent investigations resulting from the modelling of neural networks as dynamical systems, and dealing with the following questions, adressed in the context of specific models. (i). Characterizing the collective dynamics; (ii). Statistical analysis of spikes trains; (iii). Interplay ..."
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Cited by 4 (2 self)
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We present some recent investigations resulting from the modelling of neural networks as dynamical systems, and dealing with the following questions, adressed in the context of specific models. (i). Characterizing the collective dynamics; (ii). Statistical analysis of spikes trains; (iii). Interplay between dynamics and network structure; (iv). Effects of synaptic plasticity.
Metastability in a Stochastic Neural Network Modeled as a Velocity Jump Markov Process
, 2014
"... Brain dynamics is noisy at the single cell level...but often observe coherent states at the macroscopic level noisy spike trains coherent waves and oscillations at network level SINGLE CELL RECORDINGS Single cell recordings in vivo suggest that individual cortical neurons are noisy with interspike ..."
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Cited by 3 (1 self)
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Brain dynamics is noisy at the single cell level...but often observe coherent states at the macroscopic level noisy spike trains coherent waves and oscillations at network level SINGLE CELL RECORDINGS Single cell recordings in vivo suggest that individual cortical neurons are noisy with interspike intervals (ISIs) close to Poisson (Softy and Koch 1993) The Journal of Neuroscience, January 1993, 13(l) 335 neocortical units have a very high degree of irregularity, with C, ranging between 0.5 and 1.O. We attempt to understand the origin ofthese values by two different theoretical methods: modified integrateandfire models, and simulations ofdetailed compartmental models of cortical pyramidal cells. Our analysis reveals a strong contradiction between the large observed interspike variability at high firing rates and the much smaller values pre
Neural oscillators and integrators in the dynamics of decision tasks
, 2004
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A Synchronization Metric for Meshed Networks of PulseCoupled Oscillators
"... Natural phenomena such as the synchronization of fireflies, interactions between neurons, and the formation of earthquakes are commonly described by the mathematical model of pulsecoupled oscillators. This article investigates the behavior of this model when oscillators form a meshed network, i.e. ..."
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Natural phenomena such as the synchronization of fireflies, interactions between neurons, and the formation of earthquakes are commonly described by the mathematical model of pulsecoupled oscillators. This article investigates the behavior of this model when oscillators form a meshed network, i.e. nodes are not directly coupled to all others. In order to characterize the synchronization process of populations of coupled oscillators we propose a metric that allows to characterize the level of local synchronization. We demonstrate the merits of the proposed local metric by means of two case studies that examine the effect of imperfections on the synchronization process, namely the presence of frequency drifts and propagation delays.