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Noise-induced precursors of state transitions in the stochastic Wilson-cowan model.

Negahbani E, Steyn-Ross DA, Steyn-Ross ML, Wilson MT, Sleigh JW - J Math Neurosci (2015)

Bottom Line: In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex.We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue.This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

View Article: PubMed Central - PubMed

Affiliation: School of Engineering, The University of Waikato, Hamilton, 3200 New Zealand.

ABSTRACT
The Wilson-Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here we use bifurcation theory and small-noise linear stochastics to study the range of a phase transitions-sudden qualitative changes in the state of a dynamical system emerging from a bifurcation-accessible to the Wilson-Cowan network. Specifically, we examine saddle-node, Hopf, Turing, and Turing-Hopf instabilities. We introduce stochasticity by adding small-amplitude spatio-temporal white noise, and analyze the resulting subthreshold fluctuations using an Ornstein-Uhlenbeck linearization. This analysis predicts divergent changes in correlation and spectral characteristics of neural activity during close approach to bifurcation from below. We validate these theoretical predictions using numerical simulations. The results demonstrate the role of noise in the emergence of critically slowed precursors in both space and time, and suggest that these early-warning signals are a universal feature of a neural system close to bifurcation. In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex. We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue. We show that in the period leading up to emergence of spontaneous seizure-like events, the mouse field potentials show a characteristic spectral focusing toward lower frequencies concomitant with a growth in fluctuation variance, consistent with critical slowing near a bifurcation point. This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

No MeSH data available.


Related in: MedlinePlus

Steady-state distribution and eigenvalues of homogeneous Wilson–Cowan cortex. (a) Steady-state excitatory firing rate () as a function of external excitatory input voltage P. (b) The real part of the eigenvalues corresponding to steady-state diagram of the model, determining stability and type of steady states. (c) The scaled imaginary part of eigenvalues, specifying the stationary oscillatory behavior of the system in Hz. Saddle-node (SN) and Hopf (HB) bifurcations switch the stability of the steady state near , respectively
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Fig1: Steady-state distribution and eigenvalues of homogeneous Wilson–Cowan cortex. (a) Steady-state excitatory firing rate () as a function of external excitatory input voltage P. (b) The real part of the eigenvalues corresponding to steady-state diagram of the model, determining stability and type of steady states. (c) The scaled imaginary part of eigenvalues, specifying the stationary oscillatory behavior of the system in Hz. Saddle-node (SN) and Hopf (HB) bifurcations switch the stability of the steady state near , respectively

Mentions: Following [15], we assume that the cortical rod normally operates close to a homogeneous equilibrium state with uniform firing rates (, ). For the deterministic model of Eq. (11), the equilibrium points are defined by setting all space- and time-derivatives to zero, and replacing the and firing rates with their fixed-point values, independent of time and space,12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} E^{\mathrm{o}}&=S_{E} \bigl(b_{EE} \phi_{EE}(x,t)-b_{IE} \phi_{IE}(x,t)+P\bigr), \\ I^{\mathrm{o}}&=S_{I}\bigl(b_{EI} \phi_{EI}(x,t)-b_{II} \phi_{II} (x,t)+Q\bigr). \end{aligned} $$\end{document}Eo=SE(bEEϕEE(x,t)−bIEϕIE(x,t)+P),Io=SI(bEIϕEI(x,t)−bIIϕII(x,t)+Q). Noting that at steady state, excitatory and inhibitory fluxes ( and ) are equal to steady-state excitatory and inhibitory firing rates and , we obtain the cline equations:13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} E^{\mathrm{o}}&=\mathcal{S}_{E} \bigl(b_{EE}E^{\mathrm{o}}-b_{IE}I^{\mathrm{o}}+P \bigr), \quad \text{$E$-cline}, \\ I^{\mathrm{o}}&=\mathcal{S}_{I}\bigl(b_{EI}E^{\mathrm{o}}-b_{II}I^{\mathrm{o}}+Q \bigr), \quad \text{$I$-cline} \end{aligned} $$\end{document}Eo=SE(bEEEo−bIEIo+P),E-cline,Io=SI(bEIEo−bIIIo+Q),I-cline whose intersections locate the (, ) steady state. Figure 1(a) shows the distribution of steady states as a function of excitatory drive P for the parameter values of Table 1. We observe both single- and multi-root regions, with bifurcation points predicted when the steady states lose stability. Fig. 1


Noise-induced precursors of state transitions in the stochastic Wilson-cowan model.

Negahbani E, Steyn-Ross DA, Steyn-Ross ML, Wilson MT, Sleigh JW - J Math Neurosci (2015)

Steady-state distribution and eigenvalues of homogeneous Wilson–Cowan cortex. (a) Steady-state excitatory firing rate () as a function of external excitatory input voltage P. (b) The real part of the eigenvalues corresponding to steady-state diagram of the model, determining stability and type of steady states. (c) The scaled imaginary part of eigenvalues, specifying the stationary oscillatory behavior of the system in Hz. Saddle-node (SN) and Hopf (HB) bifurcations switch the stability of the steady state near , respectively
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4388113&req=5

Fig1: Steady-state distribution and eigenvalues of homogeneous Wilson–Cowan cortex. (a) Steady-state excitatory firing rate () as a function of external excitatory input voltage P. (b) The real part of the eigenvalues corresponding to steady-state diagram of the model, determining stability and type of steady states. (c) The scaled imaginary part of eigenvalues, specifying the stationary oscillatory behavior of the system in Hz. Saddle-node (SN) and Hopf (HB) bifurcations switch the stability of the steady state near , respectively
Mentions: Following [15], we assume that the cortical rod normally operates close to a homogeneous equilibrium state with uniform firing rates (, ). For the deterministic model of Eq. (11), the equilibrium points are defined by setting all space- and time-derivatives to zero, and replacing the and firing rates with their fixed-point values, independent of time and space,12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} E^{\mathrm{o}}&=S_{E} \bigl(b_{EE} \phi_{EE}(x,t)-b_{IE} \phi_{IE}(x,t)+P\bigr), \\ I^{\mathrm{o}}&=S_{I}\bigl(b_{EI} \phi_{EI}(x,t)-b_{II} \phi_{II} (x,t)+Q\bigr). \end{aligned} $$\end{document}Eo=SE(bEEϕEE(x,t)−bIEϕIE(x,t)+P),Io=SI(bEIϕEI(x,t)−bIIϕII(x,t)+Q). Noting that at steady state, excitatory and inhibitory fluxes ( and ) are equal to steady-state excitatory and inhibitory firing rates and , we obtain the cline equations:13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} E^{\mathrm{o}}&=\mathcal{S}_{E} \bigl(b_{EE}E^{\mathrm{o}}-b_{IE}I^{\mathrm{o}}+P \bigr), \quad \text{$E$-cline}, \\ I^{\mathrm{o}}&=\mathcal{S}_{I}\bigl(b_{EI}E^{\mathrm{o}}-b_{II}I^{\mathrm{o}}+Q \bigr), \quad \text{$I$-cline} \end{aligned} $$\end{document}Eo=SE(bEEEo−bIEIo+P),E-cline,Io=SI(bEIEo−bIIIo+Q),I-cline whose intersections locate the (, ) steady state. Figure 1(a) shows the distribution of steady states as a function of excitatory drive P for the parameter values of Table 1. We observe both single- and multi-root regions, with bifurcation points predicted when the steady states lose stability. Fig. 1

Bottom Line: In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex.We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue.This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

View Article: PubMed Central - PubMed

Affiliation: School of Engineering, The University of Waikato, Hamilton, 3200 New Zealand.

ABSTRACT
The Wilson-Cowan neural field equations describe the dynamical behavior of a 1-D continuum of excitatory and inhibitory cortical neural aggregates, using a pair of coupled integro-differential equations. Here we use bifurcation theory and small-noise linear stochastics to study the range of a phase transitions-sudden qualitative changes in the state of a dynamical system emerging from a bifurcation-accessible to the Wilson-Cowan network. Specifically, we examine saddle-node, Hopf, Turing, and Turing-Hopf instabilities. We introduce stochasticity by adding small-amplitude spatio-temporal white noise, and analyze the resulting subthreshold fluctuations using an Ornstein-Uhlenbeck linearization. This analysis predicts divergent changes in correlation and spectral characteristics of neural activity during close approach to bifurcation from below. We validate these theoretical predictions using numerical simulations. The results demonstrate the role of noise in the emergence of critically slowed precursors in both space and time, and suggest that these early-warning signals are a universal feature of a neural system close to bifurcation. In particular, these precursor signals are likely to have neurobiological significance as early warnings of impending state change in the cortex. We support this claim with an analysis of the in vitro local field potentials recorded from slices of mouse-brain tissue. We show that in the period leading up to emergence of spontaneous seizure-like events, the mouse field potentials show a characteristic spectral focusing toward lower frequencies concomitant with a growth in fluctuation variance, consistent with critical slowing near a bifurcation point. This observation of biological criticality has clear implications regarding the feasibility of seizure prediction.

No MeSH data available.


Related in: MedlinePlus