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  1. Research

    Symmetries Constrain Dynamics in a Family of Balanced Neural Networks

    We examine a family of random firing-rate neural networks in which we enforce the neurobiological constraint of Dale’s Law—each neuron makes either excitatory or inhibitory connections onto its post-synaptic t...

    Andrea K. Barreiro, J. Nathan Kutz and Eli Shlizerman

    The Journal of Mathematical Neuroscience 2017 7:10

    Published on: 10 October 2017

  2. Research

    Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

    Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system...

    Maria Luisa Saggio, Andreas Spiegler, Christophe Bernard and Viktor K. Jirsa

    The Journal of Mathematical Neuroscience 2017 7:7

    Published on: 25 July 2017

  3. Research

    Regularization of Ill-Posed Point Neuron Models

    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 rat...

    Bjørn Fredrik Nielsen

    The Journal of Mathematical Neuroscience 2017 7:6

    Published on: 14 July 2017

  4. Research

    How Adaptation Makes Low Firing Rates Robust

    Low frequency firing is modeled by Type 1 neurons with a SNIC, but, because of the vertical slope of the square-root-like fI curve, low f only occurs over a narrow range of I. When an adaptive current is added, ...

    Arthur S. Sherman and Joon Ha

    The Journal of Mathematical Neuroscience 2017 7:4

    Published on: 24 June 2017

  5. Research

    Emergent Dynamical Properties of the BCM Learning Rule

    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 homeosta...

    Lawrence C. Udeigwe, Paul W. Munro and G. Bard Ermentrout

    The Journal of Mathematical Neuroscience 2017 7:2

    Published on: 20 February 2017

  6. Research

    Analytic Modeling of Neural Tissue: I. A Spherical Bidomain

    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 extracell...

    Benjamin L. Schwartz, Munish Chauhan and Rosalind J. Sadleir

    The Journal of Mathematical Neuroscience 2016 6:9

    Published on: 9 September 2016

  7. Research

    Ill-Posed Point Neuron Models

    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...

    Bjørn Fredrik Nielsen and John Wyller

    The Journal of Mathematical Neuroscience 2016 6:7

    Published on: 30 April 2016

  8. Research

    Entrainment Ranges for Chains of Forced Neural and Phase Oscillators

    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 bendin...

    Nicole Massarelli, Geoffrey Clapp, Kathleen Hoffman and Tim Kiemel

    The Journal of Mathematical Neuroscience 2016 6:6

    Published on: 18 April 2016

  9. Research

    Wave Generation in Unidirectional Chains of Idealized Neural Oscillators

    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 stud...

    Bastien Fernandez and Stanislav M. Mintchev

    The Journal of Mathematical Neuroscience 2016 6:5

    Published on: 8 April 2016

  10. Research

    Neural Field Models with Threshold Noise

    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 class...

    Rüdiger Thul, Stephen Coombes and Carlo R. Laing

    The Journal of Mathematical Neuroscience 2016 6:3

    Published on: 2 March 2016

  11. Review

    Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

    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 s...

    Peter Ashwin, Stephen Coombes and Rachel Nicks

    The Journal of Mathematical Neuroscience 2016 6:2

    Published on: 6 January 2016

  12. Research

    Wilson–Cowan Equations for Neocortical Dynamics

    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 o...

    Jack D. Cowan, Jeremy Neuman and Wim van Drongelen

    The Journal of Mathematical Neuroscience 2016 6:1

    Published on: 4 January 2016

  13. Research

    A Mechanistic Neural Field Theory of How Anesthesia Suppresses Consciousness: Synaptic Drive Dynamics, Bifurcations, Attractors, and Partial State Equipartitioning

    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 ...

    Saing Paul Hou, Wassim M. Haddad, Nader Meskin and James M. Bailey

    The Journal of Mathematical Neuroscience (JMN) 2015 5:20

    Published on: 5 October 2015

  14. Short report

    Clarification and Complement to “Mean-Field Description and Propagation of Chaos in Networks of Hodgkin–Huxley and FitzHugh–Nagumo Neurons”

    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...

    Mireille Bossy, Olivier Faugeras and Denis Talay

    The Journal of Mathematical Neuroscience (JMN) 2015 5:19

    Published on: 1 September 2015

  15. Short report

    Extensive Four-Dimensional Chaos in a Mesoscopic Model of the Electroencephalogram

    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 paramete...

    Mathew P. Dafilis, Federico Frascoli, Peter J. Cadusch and David T. J. Liley

    The Journal of Mathematical Neuroscience (JMN) 2015 5:18

    Published on: 12 August 2015

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