Effective Reduced Diffusion-Models: A Data Driven Approach to the Analysis of Neuronal Dynamics

We introduce in this paper a new method for reducing neurodynamical data to an effective diffusion equation, either experimentally or using simulations of biophysically detailed models. The dimensionality of the data is first reduced to the first principal component, and then fitted by the stationary solution of a mean-field-like one-dimensional Langevin equation, which describes the motion of a Brownian particle in a potential. The advantage of such description is that the stationary probability density of the dynamical variable can be easily derived. We applied this method to the analysis of cortical network dynamics during up and down states in an anesthetized animal. During deep anesthesia, intracellularly recorded up and down states transitions occurred with high regularity and could not be adequately described by a one-dimensional diffusion equation. Under lighter anesthesia, however, the distributions of the times spent in the up and down states were better fitted by such a model, suggesting a role for noise in determining the time spent in a particular state.Author Summary: We introduce a novel methodology that allows for an effective description of a neurodynamical system in a data-driven fashion. In particular, no knowledge of the dynamics operating at the neuronal or synaptic level is required. The idea is to fit the underlying dynamics of the data using a stochastic differential equation. We use a Langevin equation that describes the stochastic dynamics of the system with the assumption that there exists an underlying potential, or energy function. The advantage of this description is the fact that, for one-dimensional systems, the stationary distribution of the variable can be straightforwardly related to the underlying energy function. In cases where the dataset is high-dimensional we reduce the dimensionality with techniques like principal curves or principal components analysis. The methodology we propose is particularly relevant for cases where an ab initio approach cannot be applied like, for example, when an explicit description of the dynamics at the neuronal and synaptic levels is not available.