Dynamics of Encoding in Neuron Populations: Some General Mathematical Features

The use of a population dynamics approach promises efficient simulation of large assemblages of neurons. Depending on the issues addressed and the degree of realism incorporated in the simulated neurons, a wide range of different population dynamics formulations can be appropriate. Here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common. This underlying structure serves as a guide toward efficient means of simulation. As an example, we derive the general population firing-rate frequency-response and show how it may be used effectively to address a broad range of interacting-population response and stability problems. A few specific cases will be worked out. A summary of this work appears at the end, before the appendix.

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

[2]  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.

[3]  Lawrence Sirovich,et al.  Management and Analysis of Large Scientific Datasets , 1992 .

[4]  D. E. Barton,et al.  The Elements of Stochastic Processes with Applications to the Natural Sciences , 1964 .

[5]  B. Knight.  The Relationship between the Firing Rate of a Single Neuron and the Level of Activity in a Population of Neurons , 1972, The Journal of general physiology.

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

[7]  L. Sirovich,et al.  Modeling the functional organization of the visual cortex , 1996 .

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

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

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

[11]  Duane Q. Nykamp,et al.  A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Extension to Slow Inhibitory Synapses , 2001, Neural Computation.

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

[13]  Duane Q. Nykamp,et al.  A population density method for large-scale modeling of neuronal networks with realistic synaptic kinetics , 2001, Neurocomputing.

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

[15]  Lawrence Sirovich,et al.  The Approach of a Neuron Population Firing Rate to a New Equilibrium: An Exact Theoretical Result , 2000, Neural Computation.

[16]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[17]  Wulfram Gerstner,et al.  Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking , 2000, Neural Computation.

[18]  J. Elgin.  The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[19]  Julian Eggert,et al.  Modeling Neuronal Assemblies: Theory and Implementation , 2001, Neural Computation.

[20]  B W Knight,et al.  Representation of spatial frequency and orientation in the visual cortex. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[21]  D. McCormick,et al.  A model of the electrophysiological properties of thalamocortical relay neurons. , 1992, Journal of neurophysiology.