Mean-Field Approximations for Coupled Populations of Generalized Linear Model Spiking Neurons with Markov Refractoriness

Toyoizumi, Taro; Rad, Kamiar Rahnama; Paninski, Liam

doi:10.1162/neco.2008.04-08-757

Cited by 44 publications

(43 citation statements)

References 50 publications

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“…by considering measures related to the Lyapunov exponent. Recently developed semi-analytical theories accounting for nonlinear neural features [71] may be helpful to answer this question. The second limiting factor of the current theory is the use of a perturbative approach to quantify the response of the integrate-and-fire model.…”

Section: Discussionmentioning

confidence: 99%

Decorrelation of Neural-Network Activity by Inhibitory Feedback

et al. 2012

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Correlations in spike-train ensembles can seriously impair the encoding of information by their spatio-temporal structure. An inevitable source of correlation in finite neural networks is common presynaptic input to pairs of neurons. Recent studies demonstrate that spike correlations in recurrent neural networks are considerably smaller than expected based on the amount of shared presynaptic input. Here, we explain this observation by means of a linear network model and simulations of networks of leaky integrate-and-fire neurons. We show that inhibitory feedback efficiently suppresses pairwise correlations and, hence, population-rate fluctuations, thereby assigning inhibitory neurons the new role of active decorrelation. We quantify this decorrelation by comparing the responses of the intact recurrent network (feedback system) and systems where the statistics of the feedback channel is perturbed (feedforward system). Manipulations of the feedback statistics can lead to a significant increase in the power and coherence of the population response. In particular, neglecting correlations within the ensemble of feedback channels or between the external stimulus and the feedback amplifies population-rate fluctuations by orders of magnitude. The fluctuation suppression in homogeneous inhibitory networks is explained by a negative feedback loop in the one-dimensional dynamics of the compound activity. Similarly, a change of coordinates exposes an effective negative feedback loop in the compound dynamics of stable excitatory-inhibitory networks. The suppression of input correlations in finite networks is explained by the population averaged correlations in the linear network model: In purely inhibitory networks, shared-input correlations are canceled by negative spike-train correlations. In excitatory-inhibitory networks, spike-train correlations are typically positive. Here, the suppression of input correlations is not a result of the mere existence of correlations between excitatory (E) and inhibitory (I) neurons, but a consequence of a particular structure of correlations among the three possible pairings (EE, EI, II).

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Section: Discussionmentioning

confidence: 99%

Decorrelation of Neural-Network Activity by Inhibitory Feedback

et al. 2012

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“…They applied a master equation for the adaptation variable, driven by a hazard function describing the instantaneous firing probability, which was obtained from fitting simulation data. Naud and Gerstner (2012) investigated the dynamics of a spike response model with adaptation (see also Toyoizumi et al, 2009). Richardson (2009) used a numerical approach (the threshold integration method introduced in Richardson, 2007) to investigate the static and dynamical transfer functions of a generalized EIF model (GEM), that include a calcium-based adaptation variable.…”

Section: Discussionmentioning

confidence: 99%

Analytical approximations of the firing rate of an adaptive exponential integrate-and-fire neuron in the presence of synaptic noise

Hertäg

Durstewitz

Brunel

2014

Front. Comput. Neurosci.

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Computational models offer a unique tool for understanding the network-dynamical mechanisms which mediate between physiological and biophysical properties, and behavioral function. A traditional challenge in computational neuroscience is, however, that simple neuronal models which can be studied analytically fail to reproduce the diversity of electrophysiological behaviors seen in real neurons, while detailed neuronal models which do reproduce such diversity are intractable analytically and computationally expensive. A number of intermediate models have been proposed whose aim is to capture the diversity of firing behaviors and spike times of real neurons while entailing the simplest possible mathematical description. One such model is the exponential integrate-and-fire neuron with spike rate adaptation (aEIF) which consists of two differential equations for the membrane potential (V) and an adaptation current (w). Despite its simplicity, it can reproduce a wide variety of physiologically observed spiking patterns, can be fit to physiological recordings quantitatively, and, once done so, is able to predict spike times on traces not used for model fitting. Here we compute the steady-state firing rate of aEIF in the presence of Gaussian synaptic noise, using two approaches. The first approach is based on the 2-dimensional Fokker-Planck equation that describes the (V,w)-probability distribution, which is solved using an expansion in the ratio between the time constants of the two variables. The second is based on the firing rate of the EIF model, which is averaged over the distribution of the w variable. These analytically derived closed-form expressions were tested on simulations from a large variety of model cells quantitatively fitted to in vitro electrophysiological recordings from pyramidal cells and interneurons. Theoretical predictions closely agreed with the firing rate of the simulated cells fed with in-vivo-like synaptic noise.

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“…The input-output function of a neuron population is sometimes described as a linear filter of the input [41], as a linear filter of the input reduced as a function of past activity [58], [59], as a non-linear function of the filtered input [60], or by any of the more recent population encoding frameworks [47], [48], [61]–[65]. These theories differ in their underlying assumptions.…”

Section: Discussionmentioning

confidence: 99%

Coding and Decoding with Adapting Neurons: A Population Approach to the Peri-Stimulus Time Histogram

Naud¹,

Gerstner

2012

PLoS Comput Biol

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The response of a neuron to a time-dependent stimulus, as measured in a Peri-Stimulus-Time-Histogram (PSTH), exhibits an intricate temporal structure that reflects potential temporal coding principles. Here we analyze the encoding and decoding of PSTHs for spiking neurons with arbitrary refractoriness and adaptation. As a modeling framework, we use the spike response model, also known as the generalized linear neuron model. Because of refractoriness, the effect of the most recent spike on the spiking probability a few milliseconds later is very strong. The influence of the last spike needs therefore to be described with high precision, while the rest of the neuronal spiking history merely introduces an average self-inhibition or adaptation that depends on the expected number of past spikes but not on the exact spike timings. Based on these insights, we derive a ‘quasi-renewal equation’ which is shown to yield an excellent description of the firing rate of adapting neurons. We explore the domain of validity of the quasi-renewal equation and compare it with other rate equations for populations of spiking neurons. The problem of decoding the stimulus from the population response (or PSTH) is addressed analogously. We find that for small levels of activity and weak adaptation, a simple accumulator of the past activity is sufficient to decode the original input, but when refractory effects become large decoding becomes a non-linear function of the past activity. The results presented here can be applied to the mean-field analysis of coupled neuron networks, but also to arbitrary point processes with negative self-interaction.

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Mean-Field Approximations for Coupled Populations of Generalized Linear Model Spiking Neurons with Markov Refractoriness

Cited by 44 publications

References 50 publications

Decorrelation of Neural-Network Activity by Inhibitory Feedback

Decorrelation of Neural-Network Activity by Inhibitory Feedback

Analytical approximations of the firing rate of an adaptive exponential integrate-and-fire neuron in the presence of synaptic noise

Coding and Decoding with Adapting Neurons: A Population Approach to the Peri-Stimulus Time Histogram

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