Macroscopic Description for Networks of Spiking Neurons

Abstract

A major goal of neuroscience, statistical physics and nonlinear dynamics is to understand how brain function arises from the collective dynamics of networks of spiking neurons. This challenge has been chiefly addressed through large-scale numerical simulations. Alternatively, researchers have formulated mean-field theories to gain insight into macroscopic states of large neuronal networks in terms of the collective firing activity of the neurons, or the firing rate. However, these theories have not succeeded in establishing an exact correspondence between the firing rate of the network and the underlying microscopic state of the spiking neurons. This has largely constrained the range of applicability of such macroscopic descriptions, particularly when trying to describe neuronal synchronization. Here we provide the derivation of a set of exact macroscopic equations for a network of spiking neurons. Our results reveal that the spike generation mechanism of individual neurons introduces an effective coupling between two biophysically relevant macroscopic quantities, the firing rate and the mean membrane potential, which together govern the evolution of the neuronal network. The resulting equations exactly describe all possible macroscopic dynamical states of the network, including states of synchronous spiking activity. Finally we show that the firing rate description is related, via a conformal map, with a lowdimensional description in terms of the Kuramoto order parameter, called Ott-Antonsen theory. We anticipate our results will be an important tool in investigating how large networks of spiking neurons self-organize in time to process and encode information in the brain.Processing and coding of information in the brain necessarily imply the coordinated activity of large ensembles of neurons. Within sensory regions of the cortex, many cells show similar responses to a given stimulus, indicating a high degree of neuronal redundancy at the local level. This suggests that information is encoded in the population response and hence can be captured via macroscopic measures of the network activity [1]. Moreover, the collective behavior of large neuronal networks is particularly relevant given that current brain measurement techniques, such as electroencephalography (EEG) or functional magnetic resonance imaging (fMRI), provide data which is necessarily averaged over the activity of a large number of neurons.The macroscopic dynamics of neuronal ensembles has been extensively studied through computational models of large networks of recurrently coupled spiking neurons, including Hodgkin-Huxley-type conductance-based neurons [2] as well as simplified neuron models, see e.g. [3][4][5]. In parallel, researchers have sought to develop statistical descriptions of neuronal networks, mainly in terms of a macroscopic observable that measures the mean rate at which neurons emit spikes, the firing rate [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. These descriptions, called firing-rate equations (FREs), have ...

Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

“…(15) in Ref. 7, we have also proven the attractiveness of the Lorentzian ansatz (21) for a network of QIF neurons.…”

“…They are, however, of minor interest for describing the transient behavior of the mean field. Several numerical results 7,22,23,31 suggest that the relaxation to the OA manifold is reasonably fast, in some cases even instantaneous.…”

Collective irregular dynamics in balanced networks of leaky integrate-and-fire neurons

“…This is, indeed the philosophy adopted by many studies based on pulse coupled units, such as leaky integrate-and-fire (LIF) neurons [5]. Accordingly, a general question arises as to whether the two approaches are consistent with one another and, in particular, to what extent spiking neurons reproduce the scenario observed in rate models [6,7].…”

A Mean-Field Description of Bursting Dynamics in Spiking Neural Networks with Short-Term Adaptation

“…We consider different short-term adaptation mechanisms that have been reported to affect neural excitability and incorporate them in the microscopic description of the QIF population dynamics. We show for which of these adaptation mechanisms a mean-field description can be derived according to the approach of Montbrió et al 15 . Employing those mean-field descriptions, we analyze the emergence of states of collective bursting and identify boundary conditions for such bursting to occur.…”

“…In the present study, we generalize these findings and show under which conditions bursting can emerge as a collective phenomenon from mere short-term adaptation within an excitatory population of globally coupled quadratic integrate-and-fire (QIF) neurons. To this end, we employ a recently derived mean-field description of the macroscopic dynamics of such a QIF population 15 . We consider different short-term adaptation mechanisms that have been reported to affect neural excitability and incorporate them in the microscopic description of the QIF population dynamics.…”