Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size.

Tilo Schwalger,Moritz Deger,Wulfram Gerstner,Lyle J. Graham

doi:10.1371/journal.pcbi.1005507

Abstract

Neural population equations such as neural mass or field models are widely used to study brain activity on a large scale. However, the relation of these models to the properties of single neurons is unclear. Here we derive an equation for several interacting populations at the mesoscopic scale starting from a microscopic model of randomly connected generalized integrate-and-fire neuron models. Each population consists of 50–2000 neurons of the same type but different populations account for different neuron types. The stochastic population equations that we find reveal how spike-history effects in single-neuron dynamics such as refractoriness and adaptation interact with finite-size fluctuations on the population level. Efficient integration of the stochastic mesoscopic equations reproduces the statistical behavior of the population activities obtained from microscopic simulations of a full spiking neural network model. The theory describes nonlinear emergent dynamics such as finite-size-induced stochastic transitions in multistable networks and synchronization in balanced networks of excitatory and inhibitory neurons. The mesoscopic equations are employed to rapidly integrate a model of a cortical microcircuit consisting of eight neuron types, which allows us to predict spontaneous population activities as well as evoked responses to thalamic input. Our theory establishes a general framework for modeling finite-size neural population dynamics based on single cell and synapse parameters and offers an efficient approach to analyzing cortical circuits and computations.

Highlights

When neuroscientists report electrophysiological, genetic, or anatomical data from a cortical neuron, they typically refer to the cell type, say, a layer 2/3 fast-spiking interneuron, a parvalbumin-positive neuron in layer 5, or a Martinotti cell in layer 4, together with the area, say primary visual cortex or somatosensory cortex [1,2,3,4]
On the “microscopic” level of nerve cells, neural spike trains can be well predicted by phenomenological spiking neuron models
Neural activity can be modeled by phenomenological equations that summarize the total activity of many thousands of neurons

Summary

Introduction

Genetic, or anatomical data from a cortical neuron, they typically refer to the cell type, say, a layer 2/3 fast-spiking interneuron, a parvalbumin-positive neuron in layer 5, or a Martinotti cell in layer 4, together with the area, say primary visual cortex or somatosensory cortex [1,2,3,4]. We will refer to a model where each neuron in each population is simulated explicitly by a spiking neuron model as a microscopic model. On a much coarser level, neural mass models [8,9,10], called field models [11,12,13], population activity equations [14], rate models [15], or Wilson-Cowan models [16] represent the activity of a cortical column at location x by one or at most a few variables, such as the mean activity of excitatory and inhibitory neurons located in the region around x. Since neural mass models give a compact summary of coarse neural activity, they can be efficiently simulated and fit to experimental data [17, 18]

Methods

Results

Discussion

Conclusion