Mathematical foundations of neuroscience

  title={Mathematical foundations of neuroscience},
  author={G. Bard Ermentrout and David H. Terman},
The Hodgkin-Huxley Equations.- Dendrites.- Dynamics.- The Variety of Channels.- Bursting Oscillations.- Propagating Action Potentials.- Synaptic Channels.- Neural Oscillators: Weak Coupling.- Neuronal Networks: Fast/Slow Analysis.- Noise.- Firing Rate Models.- Spatially Distributed Networks. 

Statistics of spike trains in conductance-based neural networks: Rigorous results

  • B. Cessac
  • Computer Science
    Journal of mathematical neuroscience
  • 2011
We consider a conductance-based neural network inspired by the generalized Integrate and Fire model introduced by Rudolph and Destexhe in 1996. We show the existence and uniqueness of a unique Gibbs

Emergent Properties in a V1 Network of Hodgkin-Huxley Neurons

This article is devoted to the theoretical and numerical analysis of a network of excitatory and inhibitory neurons of Hodgkin and Huxley type inspired by the visual cortex V1, highlighting emergent properties such as partial synchronization and synchronization, waves of excitability, and oscillations in the gamma-band frequency.

Population Models and Neural Fields

The issue of how stochasticity at the single-cell level manifests itself at the population level is discussed, and topics such as balanced networks, Poisson statistics, and asynchronous states are discussed.

Extending Power Series Methods for the Hodgkin-Huxley Equations, Including Sensitive Dependence

A neural cell or neuron is the basic building block of the brain and transmits information to other neurons. This paper demonstrates the complicated dynamics of the neuron through a numerical study

Dynamics and synchronization of complex neural networks with boundary coupling

The asymptotic synchronization of the complex neuronal networks at a uniform exponential rate is proved under the condition that stimulation signal strength of the ensemble boundary coupling exceeds a quantitative threshold expressed by the biological parameters.

Slow fluctuations in recurrent networks of spiking neurons.

A transition is found in the long-term variability of a sparse recurrent network of perfect integrate-and-fire neurons at which the Fano factor switches from zero to infinity and the correlation time is minimized, corresponding to a bifurcation in a linear map arising from the self-consistency of temporal input and output statistics.

Analysis of biological integrate-and-fire oscillators

This work considers discontinuous dynamics of integrate-and-fire models, which consist of pulse-coupled biological oscillators, and concludes that significant advances for the solution of second Peskin's conjecture have been made.

Renewal theory of coupled neuronal pools: stable states and slow trajectories.

  • C. Leibold
  • Biology
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2011
A theory is provided to analyze the dynamics of delay-coupled pools of spiking neurons based on stability analysis of stationary firing. Transitions between stable and unstable regimes can be

Spatiotemporal dynamics of continuum neural fields

This work surveys recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields, an important example of spatially extended excitable systems with nonlocal interactions.



Structure of regular semigroups

This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come

Graph expansions of Right Cancellative Monoids

  • V. Gould
  • Mathematics
    Int. J. Algebra Comput.
  • 1996
The aim of this paper is to develop a construction of left adequate monoids from the Cayley graph of a presentation of a right cancellative monoid, inspired by the construction of inverse monoid from group presentations, given by Margolis and Meakin in [10].

On regular semigroups

Automata and Languages