AbstractThe importance of a mesoscopic description level of the brain has now been well established. Rate based models are widely used, but have limitations. Recently, several extremely efficient population-level methods have been proposed that go beyond the characterization of a population in terms of a single variable. Here, we present a method for simulating neural populations based on two dimensional (2D) point spiking neuron models that defines the state of the population in terms of a density function over the neural state space. Our method differs in that we do not make the diffusion approximation, nor do we reduce the state space to a single dimension (1D). We do not hard code the neural model, but read in a grid describing its state space in the relevant simulation region. Novel models can be studied without even recompiling the code. The method is highly modular: variations of the deterministic neural dynamics and the stochastic process can be investigated independently. Currently, there is a trend to reduce complex high dimensional neuron models to 2D ones as they offer a rich dynamical repertoire that is not available in 1D, such as limit cycles. We will demonstrate that our method is ideally suited to investigate noise in such systems, replicating results obtained in the diffusion limit and generalizing them to a regime of large jumps. The joint probability density function is much more informative than 1D marginals, and we will argue that the study of 2D systems subject to noise is important complementary to 1D systems.Author SummaryA group of slow, noisy and unreliable cells collectively implement our mental faculties, and how they do this is still one of the big scientific questions of our time. Mechanistic explanations of our cognitive skills, be it locomotion, object handling, language comprehension or thinking in general - whatever that may be - is still far off. A few years ago the following question was posed: Imagine that aliens would provide us with a brain-sized clump of matter, with complete freedom to sculpt realistic neuronal networks with arbitrary precision. Would we be able to build a brain? The answer appears to be no, because this technology is actually materializing, not in the form of an alien kick-start, but through steady progress in computing power, simulation methods and the emergence of databases on connectivity, neural cell types, complete with gene expression, etc. A number of groups have created brain-scale simulations, others like the Blue Brain project may not have simulated a full brain, but they included almost every single detail known about the neurons they modelled. And yet, we do not know how we reach for a glass of milk.Mechanistic, large-scale models require simulations that bridge multiple scales. Here we present a method that allows the study of two dimensional dynamical systems subject to noise, with very little restrictions on the dynamical system or the nature of the noise process. Given that high dimensional realistic models of neurons have been reduced successfully to two dimensional dynamical systems, while retaining all essential dynamical features, we expect that this method will contribute to our understanding of the dynamics of larger brain networks without requiring the level of detail that make brute force large-scale simulations so unwieldy.
On the Network Analysis of the State Space of Discrete Dynamical Systems
