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Journal of Mathematical Neuroscience

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https://www.readbyqxmd.com/read/28707194/regularization-of-ill-posed-point-neuron-models
#1
Bjørn Fredrik Nielsen
Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rate function is employed, then standard ODE theory implies that such models are well-posed and can thus, approximately, be solved with finite precision arithmetic. We investigate whether the solution of this well-posed model converges to a solution of the ill-posed limit problem as the steepness parameter of the firing rate function tends to infinity...
December 2017: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/28685484/finite-size-effects-on-traveling-wave-solutions-to-neural-field-equations
#2
Eva Lang, Wilhelm Stannat
Neural field equations are used to describe the spatio-temporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain...
December 2017: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/28647913/how-adaptation-makes-low-firing-rates-robust
#3
Arthur S Sherman, Joon Ha
Low frequency firing is modeled by Type 1 neurons with a SNIC, but, because of the vertical slope of the square-root-like f-I curve, low f only occurs over a narrow range of I. When an adaptive current is added, however, the f-I curve is linearized, and low f occurs robustly over a large I range. Ermentrout (Neural Comput. 10(7):1721-1729, 1998) showed that this feature of adaptation paradoxically arises from the SNIC that is responsible for the vertical slope. We show, using a simplified Hindmarsh-Rose neuron with negative feedback acting directly on the adaptation current, that whereas a SNIC contributes to linearization, in practice linearization over a large interval may require strong adaptation strength...
December 2017: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/28589465/timescales-and-mechanisms-of-sigh-like-bursting-and-spiking-in-models-of-rhythmic-respiratory-neurons
#4
Yangyang Wang, Jonathan E Rubin
Neural networks generate a variety of rhythmic activity patterns, often involving different timescales. One example arises in the respiratory network in the pre-Bötzinger complex of the mammalian brainstem, which can generate the eupneic rhythm associated with normal respiration as well as recurrent low-frequency, large-amplitude bursts associated with sighing. Two competing hypotheses have been proposed to explain sigh generation: the recruitment of a neuronal population distinct from the eupneic rhythm-generating subpopulation or the reconfiguration of activity within a single population...
December 2017: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/28220467/emergent-dynamical-properties-of-the-bcm-learning-rule
#5
Lawrence C Udeigwe, Paul W Munro, G Bard Ermentrout
The Bienenstock-Cooper-Munro (BCM) learning rule provides a simple setup for synaptic modification that combines a Hebbian product rule with a homeostatic mechanism that keeps the weights bounded. The homeostatic part of the learning rule depends on the time average of the post-synaptic activity and provides a sliding threshold that distinguishes between increasing or decreasing weights. There are, thus, two essential time scales in the BCM rule: a homeostatic time scale, and a synaptic modification time scale...
December 2017: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/28097513/stable-control-of-firing-rate-mean-and-variance-by-dual-homeostatic-mechanisms
#6
Jonathan Cannon, Paul Miller
Homeostatic processes that provide negative feedback to regulate neuronal firing rates are essential for normal brain function. Indeed, multiple parameters of individual neurons, including the scale of afferent synapse strengths and the densities of specific ion channels, have been observed to change on homeostatic time scales to oppose the effects of chronic changes in synaptic input. This raises the question of whether these processes are controlled by a single slow feedback variable or multiple slow variables...
December 2017: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/28004309/a-theoretical-study-on-the-role-of-astrocytic-activity-in-neuronal-hyperexcitability-by-a-novel-neuron-glia-mass-model
#7
Aurélie Garnier, Alexandre Vidal, Habib Benali
Recent experimental evidence on the clustering of glutamate and GABA transporters on astrocytic processes surrounding synaptic terminals pose the question of the functional relevance of the astrocytes in the regulation of neural activity. In this perspective, we introduce a new computational model that embeds recent findings on neuron-astrocyte coupling at the mesoscopic scale intra- and inter-layer local neural circuits. The model consists of a mass model for the neural compartment and an astrocyte compartment which controls dynamics of extracellular glutamate and GABA concentrations...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/27613652/analytic-modeling-of-neural-tissue-i-a-spherical-bidomain
#8
Benjamin L Schwartz, Munish Chauhan, Rosalind J Sadleir
Presented here is a model of neural tissue in a conductive medium stimulated by externally injected currents. The tissue is described as a conductively isotropic bidomain, i.e. comprised of intra and extracellular regions that occupy the same space, as well as the membrane that divides them, and the injection currents are described as a pair of source and sink points. The problem is solved in three spatial dimensions and defined in spherical coordinates [Formula: see text]. The system of coupled partial differential equations is solved by recasting the problem to be in terms of the membrane and a monodomain, interpreted as a weighted average of the intra and extracellular domains...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/27215548/responses-of-leaky-integrate-and-fire-neurons-to-a-plurality-of-stimuli-in-their-receptive-fields
#9
Kang Li, Claus Bundesen, Susanne Ditlevsen
A fundamental question concerning the way the visual world is represented in our brain is how a cortical cell responds when its classical receptive field contains a plurality of stimuli. Two opposing models have been proposed. In the response-averaging model, the neuron responds with a weighted average of all individual stimuli. By contrast, in the probability-mixing model, the cell responds to a plurality of stimuli as if only one of the stimuli were present. Here we apply the probability-mixing and the response-averaging model to leaky integrate-and-fire neurons, to describe neuronal behavior based on observed spike trains...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/27129667/ill-posed-point-neuron-models
#10
Bjørn Fredrik Nielsen, John Wyller
We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard ODE theory implies that point-neuron models are well posed. Nevertheless, in the steep firing rate regime, the problem may become close to ill posed, and the error amplification, in finite time, can be very large...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/27091694/entrainment-ranges-for-chains-of-forced-neural-and-phase-oscillators
#11
Nicole Massarelli, Geoffrey Clapp, Kathleen Hoffman, Tim Kiemel
Sensory input to the lamprey central pattern generator (CPG) for locomotion is known to have a significant role in modulating lamprey swimming. Lamprey CPGs are known to have the ability to entrain to a bending stimulus, that is, in the presence of a rhythmic signal, the CPG will change its frequency to match the stimulus frequency. Bending experiments in which the lamprey spinal cord has been removed and mechanically bent back and forth at a single point have been used to determine the range of frequencies that can entrain the CPG rhythm...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/27059027/wave-generation-in-unidirectional-chains-of-idealized-neural-oscillators
#12
Bastien Fernandez, Stanislav M Mintchev
We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the far-downstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/27043152/stochastic-network-models-in-neuroscience-a-festschrift-for-jack-cowan-introduction-to-the-special-issue
#13
EDITORIAL
Paul C Bressloff, Bard Ermentrout, Olivier Faugeras, Peter J Thomas
Jack Cowan's remarkable career has spanned, and molded, the development of neuroscience as a quantitative and mathematical discipline combining deep theoretical contributions, rigorous mathematical work and groundbreaking biological insights. The Banff International Research Station hosted a workshop in his honor, on Stochastic Network Models of Neocortex, July 17-24, 2014. This accompanying Festschrift celebrates Cowan's contributions by assembling current research in stochastic phenomena in neural networks...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26936267/neural-field-models-with-threshold-noise
#14
Rüdiger Thul, Stephen Coombes, Carlo R Laing
The original neural field model of Wilson and Cowan is often interpreted as the averaged behaviour of a network of switch like neural elements with a distribution of switch thresholds, giving rise to the classic sigmoidal population firing-rate function so prevalent in large scale neuronal modelling. In this paper we explore the effects of such threshold noise without recourse to averaging and show that spatial correlations can have a strong effect on the behaviour of waves and patterns in continuum models...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26739133/mathematical-frameworks-for-oscillatory-network-dynamics-in-neuroscience
#15
Peter Ashwin, Stephen Coombes, Rachel Nicks
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear-for example, heteroclinic network attractors...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26728012/wilson-cowan-equations-for-neocortical-dynamics
#16
Jack D Cowan, Jeremy Neuman, Wim van Drongelen
In 1972-1973 Wilson and Cowan introduced a mathematical model of the population dynamics of synaptically coupled excitatory and inhibitory neurons in the neocortex. The model dealt only with the mean numbers of activated and quiescent excitatory and inhibitory neurons, and said nothing about fluctuations and correlations of such activity. However, in 1997 Ohira and Cowan, and then in 2007-2009 Buice and Cowan introduced Markov models of such activity that included fluctuation and correlation effects. Here we show how both models can be used to provide a quantitative account of the population dynamics of neocortical activity...
December 2016: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26458902/uncertainty-propagation-in-nerve-impulses-through-the-action-potential-mechanism
#17
Aldemar Torres Valderrama, Jeroen Witteveen, Maria Navarro, Joke Blom
We investigate the propagation of probabilistic uncertainty through the action potential mechanism in nerve cells. Using the Hodgkin-Huxley (H-H) model and Stochastic Collocation on Sparse Grids, we obtain an accurate probabilistic interpretation of the deterministic dynamics of the transmembrane potential and gating variables. Using Sobol indices, out of the 11 uncertain parameters in the H-H model, we unravel two main uncertainty sources, which account for more than 90 % of the fluctuations in neuronal responses, and have a direct biophysical interpretation...
December 2015: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26458901/coarse-grained-clustering-dynamics-of-heterogeneously-coupled-neurons
#18
Sung Joon Moon, Katherine A Cook, Karthikeyan Rajendran, Ioannis G Kevrekidis, Jaime Cisternas, Carlo R Laing
The formation of oscillating phase clusters in a network of identical Hodgkin-Huxley neurons is studied, along with their dynamic behavior. The neurons are synaptically coupled in an all-to-all manner, yet the synaptic coupling characteristic time is heterogeneous across the connections. In a network of N neurons where this heterogeneity is characterized by a prescribed random variable, the oscillatory single-cluster state can transition-through [Formula: see text] (possibly perturbed) period-doubling and subsequent bifurcations-to a variety of multiple-cluster states...
December 2015: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26458900/shifting-spike-times-or-adding-and-deleting-spikes-how-different-types-of-noise-shape-signal-transmission-in-neural-populations
#19
Sergej O Voronenko, Wilhelm Stannat, Benjamin Lindner
We study a population of spiking neurons which are subject to independent noise processes and a strong common time-dependent input. We show that the response of output spikes to independent noise shapes information transmission of such populations even when information transmission properties of single neurons are left unchanged. In particular, we consider two Poisson models in which independent noise either (i) adds and deletes spikes (AD model) or (ii) shifts spike times (STS model). We show that in both models suprathreshold stochastic resonance (SSR) can be observed, where the information transmitted by a neural population is increased with addition of independent noise...
December 2015: Journal of Mathematical Neuroscience
https://www.readbyqxmd.com/read/26438186/a-mechanistic-neural-field-theory-of-how-anesthesia-suppresses-consciousness-synaptic-drive-dynamics-bifurcations-attractors-and-partial-state-equipartitioning
#20
Saing Paul Hou, Wassim M Haddad, Nader Meskin, James M Bailey
With the advances in biochemistry, molecular biology, and neurochemistry there has been impressive progress in understanding the molecular properties of anesthetic agents. However, there has been little focus on how the molecular properties of anesthetic agents lead to the observed macroscopic property that defines the anesthetic state, that is, lack of responsiveness to noxious stimuli. In this paper, we use dynamical system theory to develop a mechanistic mean field model for neural activity to study the abrupt transition from consciousness to unconsciousness as the concentration of the anesthetic agent increases...
December 2015: Journal of Mathematical Neuroscience
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