624 research outputs found
Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions
In this manuscript we analyze the collective behavior of mean-field limits of
large-scale, spatially extended stochastic neuronal networks with delays.
Rigorously, the asymptotic regime of such systems is characterized by a very
intricate stochastic delayed integro-differential McKean-Vlasov equation that
remain impenetrable, leaving the stochastic collective dynamics of such
networks poorly understood. In order to study these macroscopic dynamics, we
analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics
and sigmoidal interactions. In that case, we prove that the solution of the
mean-field equation is Gaussian, hence characterized by its two first moments,
and that these two quantities satisfy a set of coupled delayed
integro-differential equations. These equations are similar to usual neural
field equations, and incorporate noise levels as a parameter, allowing analysis
of noise-induced transitions. We identify through bifurcation analysis several
qualitative transitions due to noise in the mean-field limit. In particular,
stabilization of spatially homogeneous solutions, synchronized oscillations,
bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from
static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow
further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence
in space of Brownian motion
Refined Analytical Approximations to Limit Cycles for Non-Linear Multi-Degree-of-Freedom Systems
This paper presents analytical higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill's Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs. The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation. Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.Peer reviewedFinal Accepted Versio
Periodic solutions to a mean-field model for electrocortical activity
We consider a continuum model of electrical signals in the human cortex,
which takes the form of a system of semilinear, hyperbolic partial differential
equations for the inhibitory and excitatory membrane potentials and the
synaptic inputs. The coupling of these components is represented by sigmoidal
and quadratic nonlinearities. We consider these equations on a square domain
with periodic boundary conditions, in the vicinity of the primary transition
from a stable equilibrium to time-periodic motion through an equivariant Hopf
bifurcation. We compute part of a family of standing wave solutions, emanating
from this point.Comment: 9 pages, 5 figure
Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities
In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns.
With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem
On the spectra of certain integro-differential-delay problems with applications in neurodynamics
We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs
Order Parameter Equations for Front Transitions: Nonuniformly Curved Fronts
Kinematic equations for the motion of slowly propagating, weakly curved
fronts in bistable media are derived. The equations generalize earlier
derivations where algebraic relations between the normal front velocity and its
curvature are assumed. Such relations do not capture the dynamics near
nonequilibrium Ising-Bloch (NIB) bifurcations, where transitions between
counterpropagating Bloch fronts may spontaneously occur. The kinematic
equations consist of coupled integro-differential equations for the front
curvature and the front velocity, the order parameter associated with the NIB
bifurcation. They capture the NIB bifurcation, the instabilities of Ising and
Bloch fronts to transverse perturbations, the core structure of a spiral wave,
and the dynamic process of spiral wave nucleation.Comment: 20 pages. Aric Hagberg: http://cnls.lanl.gov/~aric; Ehud
Meron:http://www.bgu.ac.il/BIDR/research/staff/meron.htm
A normal form for excitable media
We present a normal form for travelling waves in one-dimensional excitable
media in form of a differential delay equation. The normal form is built around
the well-known saddle-node bifurcation generically present in excitable media.
Finite wavelength effects are captured by a delay. The normal form describes
the behaviour of single pulses in a periodic domain and also the richer
behaviour of wave trains. The normal form exhibits a symmetry preserving Hopf
bifurcation which may coalesce with the saddle-node in a Bogdanov-Takens point,
and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We
verify the existence of these bifurcations in numerical simulations. The
parameters of the normal form are determined and its predictions are tested
against numerical simulations of partial differential equation models of
excitable media with good agreement.Comment: 22 pages, accepted for publication in Chao
Dynamics of neural fields with exponential temporal kernel
Various experimental methods of recording the activity of brain tissue in
vitro and in vivo demonstrate the existence of traveling waves. Neural field
theory offers a theoretical framework within which such phenomena can be
studied. The question then is to identify the structural assumptions and the
parameter regimes for the emergence of traveling waves in neural fields. In
this paper, we consider the standard neural field equation with an exponential
temporal kernel. We analyze the time-independent (static) and time-dependent
(dynamic) bifurcations of the equilibrium solution and the emerging
Spatio-temporal wave patterns. We show that an exponential temporal kernel does
not allow static bifurcations such as saddle-node, pitchfork, and in
particular, static Turing bifurcations, in contrast to the Green's function
used by Atay and Hutt (SIAM J. Appl. Math. 65: 644-666, 2004). However, the
exponential temporal kernel possesses the important property that it takes into
account the finite memory of past activities of neurons, which the Green's
function does not. Through a dynamic bifurcation analysis, we give explicit
Hopf (temporally non-constant, but spatially constant solutions) and
Turing-Hopf (spatially and temporally non-constant solutions, in particular
traveling waves) bifurcation conditions on the parameter space which consists
of the coefficient of the exponential temporal kernel, the transmission speed
of neural signals, the time delay rate of synapses, and the ratio of excitatory
to inhibitory synaptic weights.Comment: 25 pages, 8 Figures, 44 Reference
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