Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation

Abstract

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. [...] Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. [...] The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that is generated by recurrent synaptic excitation and neuronal adaptation. [...] We have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models.

concatenation of main text (including 8 figures), supplementary methods text and supporting figure

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• Bumps and Oscillons in Networks of Spiking Neurons
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• Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons
• Self-consistent stochastic dynamics for finite-size networks of spiking neurons
• Equivalence between synaptic current dynamics and heterogeneous propagation delays in spiking neuron networks
• Comparison between an exact and a heuristic neural mass model with second order synapses
• Low-dimensional model for adaptive networks of spiking neurons
• On the robustness of the emergent spatiotemporal dynamics in biophysically realistic and phenomenological whole-brain models at multiple network resolutions
• How random connectivity shapes the fluctuating dynamics of finite-size neural populations
• Mean-field limit of interacting 2D nonlinear stochastic spiking neurons