Abstract
Neural network dynamics are governed by the interaction of spiking neurons. Stochastic aspects of single-neuron dynamics propagate up to the network level and shape the dynamical and informational properties of the population. Mean-field models of population activity disregard the finite-size stochastic fluctuations of network dynamics and thus offer a deterministic description of the system. Here, we derive a stochastic partial differential equation (SPDE) describing the temporal evolution of the finite-size refractory density, which represents the proportion of neurons in a given refractory state at any given time. The population activity—the density of active neurons per unit time—is easily extracted from this refractory density. The SPDE includes finite-size effects through a two-dimensional Gaussian white noise that acts both in time and along the refractory dimension. For an infinite number of neurons the standard mean-field theory is recovered. A discretization of the SPDE along its characteristic curves allows direct simulations of the activity of large but finite spiking networks; this constitutes the main advantage of our approach. Linearizing the SPDE with respect to the deterministic asynchronous state allows the theoretical investigation of finite-size activity fluctuations. In particular, analytical expressions for the power spectrum and autocorrelation of activity fluctuations are obtained. Moreover, our approach can be adapted to incorporate multiple interacting populations and quasi-renewal single-neuron dynamics.
Highlights
Neurons communicate by sending and receiving pulses called spikes which occur in a rather stochastic fashion
Small fluctuations are neglected, and the randomness so present at the cellular level disappears from the description of the circuit dynamics
Finite-size spiking neural networks crucial to go beyond the mean-field approach and to propose a description that fully entails the stochastic aspects of network dynamics. We address this issue by showing that the dynamics of finite-size networks can be represented by stochastic partial differential equations
Summary
Neurons communicate by sending and receiving pulses called spikes which occur in a rather stochastic fashion. A stimulus is translated by neurons into spike trains with a certain randomness [1]. This variability is mostly attributed to the probabilistic nature of the opening and closing of ion channels underlying the emission of an action potential. Variability typically stems from the seemingly random barrage of synaptic inputs. This variability is fundamentally noise [2, 3]. Models are capable of reproducing both the statistics of the spiking activity and the subthreshold dynamics of different cell types
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