Research
The Journal of Mathematical Neuroscience 2017 7:11
Published on: 11 October 2017
Articles
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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...
The Journal of Mathematical Neuroscience 2017 7:10
Published on: 10 October 2017
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An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex
Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an ...
The Journal of Mathematical Neuroscience 2017 7:9
Published on: 25 August 2017
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A Stochastic Version of the Jansen and Rit Neural Mass Model: Analysis and Numerics
Neural mass models provide a useful framework for modelling mesoscopic neural dynamics and in this article we consider the Jansen and Rit neural mass model (JR-NMM). We formulate a stochastic version of it whi...
The Journal of Mathematical Neuroscience 2017 7:8
Published on: 8 August 2017
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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...
The Journal of Mathematical Neuroscience 2017 7:7
Published on: 25 July 2017
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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...
The Journal of Mathematical Neuroscience 2017 7:6
Published on: 14 July 2017
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Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations
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...
The Journal of Mathematical Neuroscience 2017 7:5
Published on: 6 July 2017
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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 f–I curve, low f only occurs over a narrow range of I. When an adaptive current is added, ...
The Journal of Mathematical Neuroscience 2017 7:4
Published on: 24 June 2017
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Timescales and Mechanisms of Sigh-Like Bursting and Spiking in Models of Rhythmic Respiratory Neurons
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, wh...
The Journal of Mathematical Neuroscience 2017 7:3
Published on: 6 June 2017
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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...
The Journal of Mathematical Neuroscience 2017 7:2
Published on: 20 February 2017
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Stable Control of Firing Rate Mean and Variance by Dual Homeostatic Mechanisms
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 afferen...
The Journal of Mathematical Neuroscience 2017 7:1
Published on: 17 January 2017
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A Theoretical Study on the Role of Astrocytic Activity in Neuronal Hyperexcitability by a Novel Neuron-Glia Mass Model
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 r...
The Journal of Mathematical Neuroscience 2016 6:10
Published on: 21 December 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:9
Published on: 9 September 2016
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Responses of Leaky Integrate-and-Fire Neurons to a Plurality of Stimuli in Their Receptive Fields
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 ha...
The Journal of Mathematical Neuroscience 2016 6:8
Published on: 23 May 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:7
Published on: 30 April 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:6
Published on: 18 April 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:5
Published on: 8 April 2016
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Editorial
Stochastic Network Models in Neuroscience: A Festschrift for Jack Cowan. Introduction to the Special Issue
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 gr...
The Journal of Mathematical Neuroscience 2016 6:4
Published on: 4 April 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:3
Published on: 2 March 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:2
Published on: 6 January 2016
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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...
The Journal of Mathematical Neuroscience 2016 6:1
Published on: 4 January 2016
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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 ...
The Journal of Mathematical Neuroscience (JMN) 2015 5:20
Published on: 5 October 2015
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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...
The Journal of Mathematical Neuroscience (JMN) 2015 5:19
Published on: 1 September 2015
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A Simple Mechanism for Beyond-Pairwise Correlations in Integrate-and-Fire Neurons
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...
The Journal of Mathematical Neuroscience (JMN) 2015 5:17
Published on: 1 September 2015
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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...
The Journal of Mathematical Neuroscience (JMN) 2015 5:18
Published on: 12 August 2015
