Dynamics of neuronal populations: eigenfunction theory; some solvable cases

A novel approach to cortical modelling was introduced by Knight and co-workers in 1996. In their presentation cortical dynamics is formulated in terms of interacting populations of neurons, a perspective that is in part motivated by modern cortical imaging. The approach may be regarded as the application of statistical mechanics to neuronal populations, and the simplest exemplar bears a kinship to the Boltzmann equation of kinetic theory. The disarming simplicity of this linear equation hides the complex behaviour it produces. A purpose of this paper is to investigate and reveal its intricacies by treating a series of solvable special cases. In particular we will focus on issues that relate to the spectral analysis of the underlying operators. A fairly thorough treatment is presented for a simple, but still useful example, that has important consequences for more general situations.

[1]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[2]  Lawrence Sirovich,et al.  On the Simulation of Large Populations of Neurons , 2004, Journal of Computational Neuroscience.

[3]  M. de Kamps,et al.  A simple and stable numerical solution for the population density equation. , 2003, Neural computation.

[4]  Bruce W. Knight,et al.  Dynamics of Encoding in Neuron Populations: Some General Mathematical Features , 2000, Neural Computation.

[5]  Lawrence Sirovich,et al.  Dynamics of Neuronal Populations: The Equilibrium Solution , 2000, SIAM J. Appl. Math..

[6]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[7]  P.I.M. Johannesma,et al.  Stochastic neural activity: a theoretical investigation , 1969 .

[8]  Abbott,et al.  Asynchronous states in networks of pulse-coupled oscillators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  G. Bollinger,et al.  Population Study , 2020, Definitions.

[10]  W. John Wilbur,et al.  An analysis of Stein's model for stochastic neuronal excitation , 1982, Biological Cybernetics.

[11]  Lawrence Sirovich,et al.  A Population Study of Integrate-and-Fire-or-Burst Neurons , 2002, Neural Computation.

[12]  Y. Kuramoto.  Collective synchronization of pulse-coupled oscillators and excitable units , 1991 .

[13]  Duane Q. Nykamp,et al.  A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Analysis and an Application to Orientation Tuning , 2004, Journal of Computational Neuroscience.

[14]  E. Haskell,et al.  Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size , 2001 .

[15]  D. Tranchina,et al.  Population density methods for large-scale modelling of neuronal networks with realistic synaptic kinetics: cutting the dimension down to size. , 2001, Network.

[16]  Wulfram Gerstner,et al.  Reduction of the Hodgkin-Huxley Equations to a Single-Variable Threshold Model , 1997, Neural Computation.

[17]  Bruce W. Knight,et al.  Dynamics of Encoding in a Population of Neurons , 1972, The Journal of general physiology.

[18]  E Kaplan,et al.  Contrast affects the transmission of visual information through the mammalian lateral geniculate nucleus. , 1987, The Journal of physiology.

[19]  W. Gerstner,et al.  Time structure of the activity in neural network models. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  W. Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. 1 , 1967 .

[21]  R. Stein.  A THEORETICAL ANALYSIS OF NEURONAL VARIABILITY. , 1965, Biophysical journal.

[22]  Philip M. Morse,et al.  Methods of Mathematical Physics , 1947, The Mathematical Gazette.

[23]  Mladen I. Glavinovic,et al.  Mechanisms Shaping Fast Excitatory Postsynaptic Currents in the Central Nervous System , 2002, Neural Computation.