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Origin of chaos in a twodimensional map modeling spikebursting neural activity
 Int. J. Bif. and Chaos
, 2003
"... Origin of chaos in a simple twodimensional map model replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: ..."
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Origin of chaos in a simple twodimensional map model replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. Individual dynamics of a fast subsystem of the map is characterized by two types of possible attractors: stable fixed point (replicating silence) and superstable limit cycle (replicating spikes). Coupling this subsystem with the slow subsystem leads to the generation of periodic or chaotic spikingbursting behavior. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos at alternating silence and spiking phases.
Full System Bifurcation Analysis of Endocrine Bursting Models
, 2009
"... Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltagegated calcium channels a ..."
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Cited by 5 (0 self)
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Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltagegated calcium channels and calciumsensitive potassium channels, they can be very different. For example, in insulinsecreting pancreatic βcells, plateau bursting is characterized by welldefined spikes during the depolarized phase whereas in pituitary cells, bursting features fast, irregular, small amplitude spikes. The latter has been termed “pseudoplateau bursting ” because the spikes are transients around a depolarized steady state rather than stable oscillations in the fast subsystem. In this study we systematically investigate the bursting patterns found in endocrine cell models. We show that this class of voltage and calcium gated conductance based models can be reduced to the polynomial model of Hindmarsh and Rose (25). This reduction preserves the main properties of the biophysical class of models that we consider and allows for detailed bifurcation analysis of the full fastslow system. Our analysis does not require decomposition of the full system into fast and slow subsystems and reveals properties of endocrine bursting that are not captured by the standard fastslow analysis.
Invariant template matching in systems with spatiotemporal coding: a matter of instability. Neural Netw
, 2009
"... We consider the design of a pattern recognition that matches templates to images, both of which are spatially sampled and encoded as temporal sequences. The image is subject to a combination of various perturbations. These include ones that can be modeled as parameterized uncertainties such as image ..."
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We consider the design of a pattern recognition that matches templates to images, both of which are spatially sampled and encoded as temporal sequences. The image is subject to a combination of various perturbations. These include ones that can be modeled as parameterized uncertainties such as image blur, luminance, translation, and rotation as well as unmodeled ones. Biological and neural systems require that these perturbations be processed through a minimal number of channels by simple adaptation mechanisms. We found that the most suitable mathematical framework to meet this requirement is that of weakly attracting sets. This framework provides us with a normative and unifying solution to the pattern recognition problem. We analyze the consequences of its explicit implementation in neural systems. Several properties inherent to the systems designed in accordance with our normative mathematical argument coincide with known empirical facts. This is illustrated in mental rotation, visual search and blur/intensity adaptation. We demonstrate how our results can be applied to a range of practical problems in template matching and pattern recognition.
Optimization for bursting neural models
, 2007
"... This thesis concerns parameter estimation for bursting neural models. Parameter estimation for dierential equations is a dicult task due to complicated objective function landscapes and numerical challenges. These diculties are particularly salient in bursting models and other multiple time scale ..."
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This thesis concerns parameter estimation for bursting neural models. Parameter estimation for dierential equations is a dicult task due to complicated objective function landscapes and numerical challenges. These diculties are particularly salient in bursting models and other multiple time scale systems. Here we make use of the geometry underlying bursting by introducing dening equations for burst initiation and termination. Fitting the timing of these burst events simpli es objective function landscapes considerably. We combine this with automatic dierentiation to accurately compute gradients for these burst events, and implement these features using standard unconstrained optimization algorithms. We use trajectories from a minimal spiking model and the HindmarshRose equations as test problems, and bursting respiratory neurons in the preBotzinger complex as an application. These geometrical ideas and numerical improvements signicantly enhance algorithm performance. Excellent ts are obtained to the preBotzinger data both in control conditions and when the neuromodulator norepinephrine is added. The results suggest dierent possible neuromodulatory mechanisms, and help analyze the roles of dierent currents in shaping burst duration and period.
Convergence
, 2009
"... patterns. Matching is required to be tolerant to various combinations of image perturbations. These include ones that can be modeled as parameterized uncertainties such as image blur, luminance, and, as special cases, invariant transformation groups such as translation and rotations, as well as unmo ..."
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patterns. Matching is required to be tolerant to various combinations of image perturbations. These include ones that can be modeled as parameterized uncertainties such as image blur, luminance, and, as special cases, invariant transformation groups such as translation and rotations, as well as unmodeled uncertainties (noise). For a system to deal with such perturbations in an efficient way, they are to be handled through a minimal number of channels and by simple adaptation mechanisms. These normative requirements can be met within the mathematical framework of weakly attracting sets. We discuss explicit implementation of this principle in neural systems and show that it naturally explains a range of phenomena in biological vision, such as mental rotation, visual search, and the presence of multiple time scales in adaptation. We illustrate our results with an application to a realistic pattern recognition problem. © 2009 Elsevier Ltd. All rights reserved. 1. Notational preliminaries We define an image as a mapping S0(x, y) from a class of locally bounded mappings S ⊆ L∞(Ωx × Ωy), where Ωx ⊆ R, Ωy ⊆ R, and L∞(Ωx×Ωy) is the space of all functions f: Ωx×Ωy → R such
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DOI 10.1007/s1082700700608 Parameter estimation for bursting neural models
"... Abstract This paper presents work on parameter estimation methods for bursting neural models. In our approach we use both geometrical features specific to bursting, as well as general features such as periodic orbits and their bifurcations. We use the geometry underlying bursting to introduce defini ..."
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Abstract This paper presents work on parameter estimation methods for bursting neural models. In our approach we use both geometrical features specific to bursting, as well as general features such as periodic orbits and their bifurcations. We use the geometry underlying bursting to introduce defining equations for burst initiation and termination, and restrict the estimation algorithms to the space of bursting periodic orbits when trying to fit periodic burst data. These geometrical ideas are combined with automatic differentiation to accurately compute parameter sensitivities for the burst timing and period. In addition to being of inherent interest, these sensitivities are used in standard gradientbased optimization algorithms to fit model burst duration and period to data. As an application, we fit Butera et al.’s (Journal of Neurophysiology 81, 382–397, 1999) model of preBötzinger complex neurons to empirical data both in control conditions and when the neuromodulator norepinephrine is added (Viemari and Ramirez, Journal of Neurophysiology 95, Electronic Supplementary Material The online version of this article (doi:10.1007/s1082700700608) contains supplementary material, which is available to authorized users.
CHAOS IN 2D SLOWFAST MAPS FOR SPIKINGBURSTING NEURAL ACTIVITY
"... Abstract — Origin of chaos in a simple slowfast 2D map replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos through ca ..."
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Abstract — Origin of chaos in a simple slowfast 2D map replicating the spiking and spikingbursting activity of real biological neurons is studied. The map contains one fast and one slow variable. We study the bifurcation scenarios which reveal the dynamical mechanisms that lead to chaos through canards in alternation of silence and spiking phases. I.
Synchronized oscillation in a modular neural network composed of columns
, 2003
"... Abstract The columnar organization is a ubiquitous feature in the cerebral cortex. In this study, a neural network model simulating the cortical columns has been constructed. When fed with random pulse input with constant rate, a column generates synchronized oscillations, with a frequency varying f ..."
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Abstract The columnar organization is a ubiquitous feature in the cerebral cortex. In this study, a neural network model simulating the cortical columns has been constructed. When fed with random pulse input with constant rate, a column generates synchronized oscillations, with a frequency varying from 3 to 43 Hz depending on parameter values. The behavior of the model under periodic stimulation was studied and the inputoutput relationship was nonlinear. When identical columns were sparsely interconnected, the column oscillator could be locked in synchrony. In a network composed of heterogeneous columns, the columns were organized by intrinsic properties and formed partially synchronized assemblies.
2007 Special Issue Consciousness & the small network argument
"... This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproductio ..."
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This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: