Spiking Neuron Models: An Introduction
Spiking Neuron Models: An Introduction
Finite Differences And Partial Differential Equations
Finite Differences And Partial Differential Equations
Effects of Fast Presynaptic Noise in Attractor Neural Networks
Neural Computation
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Oscillations are a ubiquitous feature of many neural systems, spanning many orders of magnitude in frequency. One of the most prominent oscillatory patterns, with possible functional implications, is that occurring in the mammalian thalamocortical system during sleep. This system is characterized by relatively long delays (reaching up to 40 msec) and gives rise to low-frequency oscillatory waves. Motivated by these phenomena, we study networks of excitatory and inhibitory integrate-and-fire neurons within a Fokker-Planck delay partial differential equation formalism and establish explicit conditions for the emergence of oscillatory solutions, and for the amplitude and period of the ensuing oscillations, for relatively large values of the delays. When a two-timescale analysis is employed, the full partial differential equation is replaced in this limit by a discrete time iterative map, leading to a relatively simple dynamic interpretation. This asymptotic result is shown numerically to hold, to a good approximation, over a wide range of parameter values, leading to an accurate characterization of the behavior in terms of the underlying physical parameters. Our results provide a simple mechanistic explanation for one type of slow oscillation based on delayed inhibition, which may play an important role in the slow spindle oscillations occurring during sleep. Moreover, they are consistent with experimental findings related to human motor behavior with visual feedback.