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Uncertainty and the Brain

Edited by: Olivier Faugeras, Michele Thieullen

Theoretical Neuroscientists have developed a wide range of mathematical, computational, and numerical tools for modeling and simulating sets of interacting neurons. While in most cases, with some notable exceptions, the framework of these efforts has been deterministic, drawing on the theory of dynamical systems, partial differential equations, integral or integro-differential equations, it has been felt from the early days of Hodgkin and Huxley that uncertainty has to be taken into account in the models.

Uncertainty has its source in the physics of the underlying phenomena, for example in the way the ion channels open and close, or the way in which neurotransmitters diffuse in the synaptic cleft. This is the physical uncertainty. Uncertainty also comes from the fact that many of the parameters in the models are out of reach of any of the current experimental techniques, most likely still for a long time. For example the exact values of the synaptic weights describing the way neurons influence each other at a given instant in a network will probably never be known. This second source is the intrinsic uncertainty.

Researchers are therefore in great need of stochastic models to account for these two sources in a mathematically rigorous framework allowing for quantitative descriptions and predictions.

The goal of this special issue is to bring together the key experimental and theoretical research linking state-of-the-art knowledge about uncertainty in the Brain with current forefront research in probability theory and statistics in the general area of stochastic nonlinear dynamical systems featuring several time scales. As a result we hope to shed some new light on the question of the role of noise in the activity of the Brain. We are certain that such an endeavour is highly timely and is presently missing, and that it will be highly attractive to a wide community of brain researchers and of mathematicians in probability and statistics.

  1. Research

    How Uncertainty Bounds the Shape Index of Simple Cells

    We propose a theoretical motivation to quantify actual physiological features, such as the shape index distributions measured by Jones and Palmer in cats and by Ringach in macaque monkeys. We will adopt the un...

    D Barbieri, G Citti and A Sarti

    The Journal of Mathematical Neuroscience 2014 4:5

    Published on: 17 April 2014

  2. Research

    Goodness-of-Fit Tests and Nonparametric Adaptive Estimation for Spike Train Analysis

    When dealing with classical spike train analysis, the practitioner often performs goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of p...

    Patricia Reynaud-Bouret, Vincent Rivoirard, Franck Grammont and Christine Tuleau-Malot

    The Journal of Mathematical Neuroscience 2014 4:3

    Published on: 17 April 2014

  3. Research

    Large Deviations for Nonlocal Stochastic Neural Fields

    We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape....

    Christian Kuehn and Martin G Riedler

    The Journal of Mathematical Neuroscience 2014 4:1

    Published on: 17 April 2014

  4. Research

    Fokker–Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing

    Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose ...

    Alexandre Iolov, Susanne Ditlevsen and André Longtin

    The Journal of Mathematical Neuroscience 2014 4:4

    Published on: 17 April 2014

  5. Research

    A Density Model for a Population of Theta Neurons

    Population density models that are used to describe the evolution of neural populations in a phase space are closely related to the single neuron model that describes the individual trajectories of the neurons...

    Grégory Dumont, Jacques Henry and Carmen Oana Tarniceriu

    The Journal of Mathematical Neuroscience 2014 4:2

    Published on: 17 April 2014

  6. Research

    Derived Patterns in Binocular Rivalry Networks

    Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Wilson proposed a class of neuronal network models that generalize rivalry to mul...

    Casey O Diekman, Martin Golubitsky and Yunjiao Wang

    The Journal of Mathematical Neuroscience 2013 3:6

    Published on: 8 May 2013