A statistical theory of interacting neural fields, motivated by Ventriglia's kinetic theory is formulated.
Ventriglia, F. Kinetic approach to neural systems I. Bull. Math. Biol. 36:534--544. (1974)
Ventriglia, F. Computational simulation of activity of cortical-like neural systems. Bull. Math. Biol. 50:143--185 (1988)
A statistical model is given to describe the electrical activity patterns of large neural populations of the hippocampal CA3 region. A continuous model has been formalized to describe the statistical processes governing the interactions within and between neural fields. The system of partial differential equations contains diffusion terms which determine the evolution of second moments of the probability distribution functions. The model is supplemented with a differential description of post-synaptic potentials as well. The discretization procedure has been designed so as to make the discrete equations scaling invariant.
A model with intermediate complexity is introduced to reproduce the basic firing modes of the CA3 pyramidal cell. Our model consists of a single compartment, has two variables (membrane potential and internal calcium concentration), and involves two separate stages for interspike mechanisms and firing. Interspike dynamics is governed by voltage- and calcium-dependent ionic channels but no channel kinetics is provided. This model is suitable to be included in our statistical population model. Bifurcation analysis reveals that interspike dynamics rather than sodium firing has dominant roe in the control of bursting/non-bursting behavior.
Population activities as well as underlying single cell voltages are simulated during normal and epileptiform activities in the hippocampal CA3 slice. It is demonstrated that our model can reproduce electrophysiological phenomena characteristic to both single cell and population activities. Specifically, fully synchronized population bursts, synchronized synaptic potentials, and low amplitude population oscillation were obtained.