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Journal of Mathematical Neuroscience

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
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
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
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
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
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
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
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
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
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
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
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
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
Mireille Bossy, Olivier Faugeras, Denis Talay
In this note, we clarify the well-posedness of the limit equations to the mean-field N-neuron models proposed in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) and we prove the associated propagation of chaos property. We also complete the modeling issue in (Baladron et al. in J. Math. Neurosci. 2:10, 2012) by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters.
December 2015: Journal of Mathematical Neuroscience
David A Leen, Eric Shea-Brown
The collective dynamics of neural populations are often characterized in terms of correlations in the spike activity of different neurons. We have developed an understanding of the circuit mechanisms that lead to correlations among cell pairs, but little is known about what determines the population firing statistics among larger groups of cells. Here, we examine this question for a simple, but ubiquitous, circuit feature: common fluctuating input arriving to spiking neurons of integrate-and-fire type. We show that this leads to strong beyond-pairwise correlations-that is, correlations that cannot be captured by maximum entropy models that extrapolate from pairwise statistics-as for earlier work with discrete threshold crossing (dichotomous Gaussian) models...
December 2015: Journal of Mathematical Neuroscience
Mathew P Dafilis, Federico Frascoli, Peter J Cadusch, David T J Liley
BACKGROUND: In a previous work (Dafilis et al. in Chaos 23(2):023111, 2013), evidence was presented for four-dimensional chaos in Liley's mesoscopic model of the electroencephalogram. The study was limited to one parameter set of the model equations. FINDINGS: In this report we expand that result by presenting evidence for the extension of four-dimensional chaotic behavior to a large area of the biologically admissible parameter space. A two-parameter bifurcation analysis highlights the complexity of the dynamical landscape involved in the creation of such chaos...
December 2015: Journal of Mathematical Neuroscience
Peter De Maesschalck, Martin Wechselberger
We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures...
December 2015: Journal of Mathematical Neuroscience
Abigail C Snyder, Jonathan E Rubin
Rhythmic behaviors such as breathing, walking, and scratching are vital to many species. Such behaviors can emerge from groups of neurons, called central pattern generators, in the absence of rhythmic inputs. In vertebrates, the identification of the cells that constitute the central pattern generator for particular rhythmic behaviors is difficult, and often, its existence has only been inferred. For example, under experimental conditions, intact turtles generate several rhythmic scratch motor patterns corresponding to non-rhythmic stimulation of different body regions...
December 2015: Journal of Mathematical Neuroscience
Cynthia I Wood, Illya V Hicks
The concept of cell assembly was introduced by Hebb and formalized mathematically by Palm in the framework of graph theory. In the study of associative memory, a cell assembly is a group of neurons that are strongly connected and represent a "concept" of our knowledge. This group is wired in a specific manner such that only a fraction of its neurons will excite the entire assembly. We link the concept of cell assembly to the closure of a minimal k-core and study a particular type of cell assembly called k-assembly...
December 2015: Journal of Mathematical Neuroscience
Sergio Verduzco-Flores
Simple-spike synchrony between Purkinje cells projecting to a common neuron in the deep cerebellar nucleus is emerging as an important factor in the encoding of output information from cerebellar cortex. A phenomenon known as stochastic synchronization happens when uncoupled oscillators synchronize due to correlated inputs. Stochastic synchronization is a viable mechanism through which simple-spike synchrony could be generated, but it has received scarce attention, perhaps because the presence of feedforward inhibition in the input to Purkinje cells makes insights difficult...
December 2015: Journal of Mathematical Neuroscience
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