Multidimensional Dynamical Systems with Noise : Population Density Techniques for Neuroscience.

The problem of how to create efficient multi-scale models of large networks of neurons is a pressing one. It requires a balance between computational efficiency and a reduction of the number of parameters involved against biological realism. Simulations of point-model neurons show very realistic features of neural dynamics but are very hard to configure and to analyse. Instead of using hundreds or thousands of point-model neurons, a population can often be modeled by a single density function in a way that accurately reproduces quantities aggregated over the population, such as population firing rate or average membrane potential. These techniques have been widely applied in neuroscience, mainly on populations comprised of one-dimensional point-model neurons, such as leaky-integrate-and-fire neurons. Here, we present very general density methods that can be applied to point-model neurons of higher dimensionality that can represent biological features not present in simpler ones, such as adaptation and bursting. The methods are geometrical in nature and lend themselves to immediate visualisation of the population state. By decoupling the neural dynamics and the stochastic processes that model inter-neuron communication, an efficient GPGPU implementation is possible that makes the study of such high-dimensional models feasible. This decoupling also allows the study of different noise models, such as Poisson, white noise, and gamma-distributed interspike intervals, which broadens the application domain considerably compared to the Fokker-Planck equations that have traditionally dominated this approach. We will present several examples based on high-dimensional neural models. We will use dynamical systems that represent point-model neurons, but inherently there is nothing to restrict the approach presented here to neuroscience. MIIND is an open-source simulator that contains an implementation of these techniques.