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Register a new userLow-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation
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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.
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Voltage-sensitive dye imaging (VSDi) has revealed fundamental properties of neocortical processing at mesoscopic scales. Since VSDi signals report the average membrane potential, it seems natural to use a mean-field formalism to model such signals. Here, we investigate a mean-field model of networks of Adaptive Exponential (AdEx) integrate-and-fire neurons, with conductance-based synaptic interactions. The AdEx model can capture the spiking response of different cell types, such as regular-spiking (RS) excitatory neurons and fast-spiking (FS) inhibitory neurons. We use a Master Equation formalism, together with a semi-analytic approach to the transfer function of AdEx neurons. We compare the predictions of this mean-field model to simulated networks of RS-FS cells, first at the level of the spontaneous activity of the network, which is well predicted by the mean-field model. Second, we investigate the response of the network to time-varying external input, and show that the mean-field model accurately predicts the response time course of the population. One notable exception was that the tail of the response at long times was not well predicted, because the mean-field does not include adaptation mechanisms. We conclude that the Master Equation formalism can yield mean-field models that predict well the behavior of nonlinear networks with conductance-based interactions and various electrophysiolgical properties, and should be a good candidate to model VSDi signals where both excitatory and inhibitory neurons contribute.
We derive analytical formulae for the firing rate of integrate-and-fire neurons endowed with realistic synaptic dynamics. In particular we include the possibility of multiple synaptic inputs as well as the effect of an absolute refractory period into the description.
Collective oscillations and their suppression by external stimulation are analyzed in a large-scale neural network consisting of two interacting populations of excitatory and inhibitory quadratic integrate-and-fire neurons. In the limit of an infinite number of neurons, the microscopic model of this network can be reduced to an exact low-dimensional system of mean-field equations. Bifurcation analysis of these equations reveals three different dynamic modes in a free network: a stable resting state, a stable limit cycle, and bistability with a coexisting resting state and a limit cycle. We show that in the limit cycle mode, high-frequency stimulation of an inhibitory population can stabilize an unstable resting state and effectively suppress collective oscillations. We also show that in the bistable mode, the dynamics of the network can be switched from a stable limit cycle to a stable resting state by applying an inhibitory pulse to the excitatory population. The results obtained from the mean-field equations are confirmed by numerical simulation of the microscopic model.
We present two Bayesian procedures to infer the interactions and external currents in an assembly of stochastic integrate-and-fire neurons from the recording of their spiking activity. The first procedure is based on the exact calculation of the most likely time courses of the neuron membrane potentials conditioned by the recorded spikes, and is exact for a vanishing noise variance and for an instantaneous synaptic integration. The second procedure takes into account the presence of fluctuations around the most likely time courses of the potentials, and can deal with moderate noise levels. The running time of both procedures is proportional to the number S of spikes multiplied by the squared number N of neurons. The algorithms are validated on synthetic data generated by networks with known couplings and currents. We also reanalyze previously published recordings of the activity of the salamander retina (including from 32 to 40 neurons, and from 65,000 to 170,000 spikes). We study the dependence of the inferred interactions on the membrane leaking time; the differences and similarities with the classical cross-correlation analysis are discussed.
The dominant modeling framework for understanding cortical computations are heuristic firing rate models. Despite their success, these models fall short to capture spike synchronization effects, to link to biophysical parameters and to describe finite-size fluctuations. In this opinion article, we propose that the refractory density method (RDM), also known as age-structured population dynamics or quasi-renewal theory, yields a powerful theoretical framework to build rate-based models for mesoscopic neural populations from realistic neuron dynamics at the microscopic level. We review recent advances achieved by the RDM to obtain efficient population density equations for networks of generalized integrate-and-fire (GIF) neurons -- a class of neuron models that has been successfully fitted to various cell types. The theory not only predicts the nonstationary dynamics of large populations of neurons but also permits an extension to finite-size populations and a systematic reduction to low-dimensional rate dynamics. The new types of rate models will allow a re-examination of models of cortical computations under biological constraints.
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