Population dynamics: Variance and the sigmoid activation function

Marreiros, André C.; Daunizeau, Jean; Kiebel, Stefan J.; Friston, Karl J.

doi:10.1016/j.neuroimage.2008.04.239

Cited by 146 publications

(97 citation statements)

References 44 publications

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“…The present framework can easily accommodate more realistic neural mass model that go beyond such simplifications by considering the second moments of neural masses [Marreiros et al, 2008] or mean field approximations of neurons with intrinsic properties [Robinson et al, 2008].…”

Section: Discussionmentioning

confidence: 99%

Model driven EEG/fMRI fusion of brain oscillations

Valdés‐Sosa

Sánchez-Bornot

Sotero

et al. 2008

Human Brain Mapping

184

153

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This article reviews progress and challenges in model driven EEG/fMRI fusion with a focus on brain oscillations. Fusion is the combination of both imaging modalities based on a cascade of forward models from ensemble of post-synaptic potentials (ePSP) to net primary current densities (nPCD) to EEG; and from ePSP to vasomotor feed forward signal (VFFSS) to BOLD. In absence of a model, data driven fusion creates maps of correlations between EEG and BOLD or between estimates of nPCD and VFFS. A consistent finding has been that of positive correlations between EEG alpha power and BOLD in both frontal cortices and thalamus and of negative ones for the occipital region. For model driven fusion we formulate a neural mass EEG/fMRI model coupled to a metabolic hemodynamic model. For exploratory simulations we show that the Local Linearization (LL) method for integrating stochastic differential equations is appropriate for highly nonlinear dynamics. It has been successfully applied to small and medium sized networks, reproducing the described EEG/BOLD correlations. A new LL-algebraic method allows simulations with hundreds of thousands of neural populations, with connectivities and conduction delays estimated from diffusion weighted MRI. For parameter and state estimation, Kalman filtering combined with the LL method estimates the innovations or prediction errors. From these the likelihood of models given data are obtained. The LL-innovation estimation method has been already applied to small and medium scale models. With improved Bayesian computations the practical estimation of very large scale EEG/fMRI models shall soon be possible.

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Section: Discussionmentioning

confidence: 99%

Model driven EEG/fMRI fusion of brain oscillations

Valdés‐Sosa

Sánchez-Bornot

Sotero

et al. 2008

Human Brain Mapping

184

153

View full text Add to dashboard Cite

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“…This incorporates the step-like function of an individual neural response smeared over a Gaussian distribution of firing thresholds and neuronal states (Marreiros et al, 2008). The system of equations is closed by introducing the outward propagation of action potentials from the soma through axons, which then become presynaptic activity in distant regions.…”

Section: Corticothalamic Neural Field Modelmentioning

confidence: 99%

Biophysical Mechanisms of Multistability in Resting-State Cortical Rhythms

Freyer¹,

Roberts²,

Becker³

et al. 2011

J. Neurosci.

280

285

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The human alpha (8 -12 Hz) rhythm is one of the most prominent, robust, and widely studied attributes of ongoing cortical activity. Contrary to the prevalent notion that it simply "waxes and wanes," spontaneous alpha activity bursts erratically between two distinct modes of activity. We now establish a mechanism for this multistable phenomenon in resting-state cortical recordings by characterizing the complex dynamics of a biophysical model of macroscopic corticothalamic activity. This is achieved by studying the predicted activity of cortical and thalamic neuronal populations in this model as a function of its dynamic stability and the role of nonspecific synaptic noise. We hence find that fluctuating noisy inputs into thalamic neurons elicit spontaneous bursts between low-and high-amplitude alpha oscillations when the system is near a particular type of dynamical instability, namely a subcritical Hopf bifurcation. When the postsynaptic potentials associated with these noisy inputs are modulated by cortical feedback, the SD of power within each of these modes scale in proportion to their mean, showing remarkable concordance with empirical data. Our state-dependent corticothalamic model hence exhibits multistability and scale-invariant fluctuations-key features of resting-state cortical activity and indeed, of human perception, cognition, and behavior-thus providing a unified account of these apparently divergent phenomena.

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“…A notable exception is Buice et al (2010). The effect of variance of microscopic parameters and states within a population on the neural mass parameters may also be explicitly accounted for in such a derivation (Breakspear et al, 2003;Marreiros et al, 2008). Deco et al (2008) derived the evolving population density p(x, t) from a leaky integrate-and-fire single neuron model by assuming all neurons have identical input distributions.…”

Section: Construction Of Neural Mass Modelsmentioning

confidence: 99%

“…For example Marreiros et al (2008) discusses in depth the derivation of the neural mass sigmoid function from the distributions of neural state and membrane threshold in the neural population, with the static sigmoid shape embodying an assumption of static variance in neural state (see also Zandt et al (2014)). …”

Section: Biophysiological Features Modeled By Noisementioning

confidence: 99%

“…describes how the mean firing rate of a neural population depends on the mean soma membrane potential v, incorporating the dispersion of responses due to variability in the parameters and underlying neuronal states (Marreiros et al, 2008). Parameters e 0 , ρ 2 and ρ 1 determine the maximum firing rate, threshold potential and sensitivity, respectively.…”

Section: Jansen-rit Neural Mass Modelmentioning

confidence: 99%

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Critical fluctuations and coupling of stochastic neural mass models

Aburn¹

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Neuroimaging and implanted medical devices give a view onto large scale oscillations in the brain that rapidly change in both power and synchrony at different frequencies. These measured time series reveal a projection to just one dimension for each location, a moving shadow of the complex underlying activity. For this reason mathematical models of the activity are essential to make full use of the time series. The flexible onset and cessation of oscillations may have an important role in allowing the brain rapidly to rearrange its effective network structure. The onset of pathological oscillations is also central to neurological disorders such as epilepsy and Parkinson's disease. It is in the regime close to linear instability that transitions to and from oscillation can readily occur, by mechanisms of bifurcation and multistability. Empirical evidence of critical fluctuations and critical slowing down in neural dynamics also suggest proximity to linear instability. This thesis develops novel methods to analyze stochastic dynamical models near the point of transitionwhere oscillations start and stop, allowing to relate transitions in the models to changes in time series statistics that are accessible in clinical and experimental recordings.Taking as an example a widely used biophysical model of collective neural dynamics (the Jansen-Rit model) we show that near the point of oscillation onset at a supercritical Hopf bifurcation there can be a dramatic increase in autocorrelation length as well as power law scaling over four orders of magnitude in fluctuation size and duration, but that these time series indicators will depend sensitively on the direction in state space of input fluctuations and hence on which neuronal subpopulation is stochastically perturbed.In general near a supercritical Hopf bifurcation non-circular oscillations emerge on an arbitrarily curved two-dimensional surface embedded within the N -dimensional state space. This makes it difficult to determine exactly how noise will affect oscillation phase and amplitude. By applying Poincaré normal form transformations to find a parameter-dependent coordinate system in which the unperturbed oscillations are simple circles in a flat plane, then using Stratonovich stochastic calculus to transform noise perturbations to the same coordinate system, this makes explicit the effect of noise on phase and amplitude. We give a criterion for this reduced model to be a good approximation, balancing noise intensity and center manifold stability. Abstract iiOptionally averaging the diffusion process in Fokker-Planck form gives a still simpler weak approximation of the oscillations in a symmetrical standard form that retains the leading order stochastic effects on phase and amplitude dynamics. We demonstrate by large scale numerical simulation of the stochastic differential equations that the oscillation time series statistics of interest are preserved by transformation to this symmetrical weak approximation.Close to instability, dynamic synchrony of networ...

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Population dynamics: Variance and the sigmoid activation function

Cited by 146 publications

References 44 publications

Model driven EEG/fMRI fusion of brain oscillations

Model driven EEG/fMRI fusion of brain oscillations

Biophysical Mechanisms of Multistability in Resting-State Cortical Rhythms

Critical fluctuations and coupling of stochastic neural mass models

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