Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the spectral properties of the linearized equation. We transform the characteristic integral equation for the delay differential equation (DDE) into a linear partial differential equation (PDE) with boundary conditions. We demonstrate that finding eigenvalues and eigenvectors of the DDE is equivalent with obtaining nontrivial solutions of this boundary value problem (BVP). When the connectivity kernel consists of a single exponential, we construct a basis of the solutions of this BVP that forms a complete set in $L^2$. This gives a complete characterization of the spectrum and is used to construct a solution to the resolvent problem. As an application we give an example of a Hopf bifurcation and compute the first Lyapunov coefficient.
In this paper, we investigate a reaction-diffusion-advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our result implies that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction-diffusion-advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation.
In this paper, by using pinning impulse controller and hybrid controller respectively, the research difficulties of asymptotic synchronization and finite time cluster synchronization of time-varying delayed neural networks are studied. On the ground of Lyapunov stability theorem and Lyapunov-Razumikhin method, a novel sufficient criterion on asymptotic cluster synchronization of time-varying delayed neural networks is obtained. Utilizing Finite time stability theorem and hybrid control technology, a sufficient criterion on finite-time cluster synchronization is also obtained. In order to deal …
In the auditory streaming paradigm alternating sequences of pure tones can be perceived as a single galloping rhythm (integration) or as two sequences with separated low and high tones (segregation). Although studied for decades, the neural mechanisms underlining this perceptual grouping of sound remains a mystery. With the aim of identifying a plausible minimal neural circuit that captures this phenomenon, we propose a firing rate model with two periodically forced neural populations coupled by fast direct excitation and slow delayed inhibition. By analyzing the model in a non-smooth, slow-fa…
Neural field equations are integro-differential systems describing the macroscopic activity of spatially extended pieces of cortex. In such cortical assemblies, the propagation of information and the transmission machinery induce communication delays, due to the transport of information (propagation delays) and to the synaptic machinery (constant delays). We investigate the role of these delays on the formation of structured spatiotemporal patterns for these systems in arbitrary dimensions. We focus on localized activity, either induced by the presence of a localized stimulus (pulses) or by tr…
Learning how complex dynamical systems evolve over time is a key challenge in system identification. For safety critical systems, it is often crucial that the learned model is guaranteed to converge to some equilibrium point. To this end, neural ODEs regularized with neural Lyapunov functions are a promising approach when states are fully observed. For practical applications however, partial observations are the norm. As we will demonstrate, initialization of unobserved augmented states can become a key problem for neural ODEs. To alleviate this issue, we propose to augment the system's state …
The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for $t>τ$ (where $τ$ is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained…
In the context of a spatially extended model for the electrical activity in a pituitary lactotroph cell line, we establish that two delayed bifurcation phenomena from ODEs ---folded node canards and slow passage through Hopf bifurcations--- persist in the presence of diffusion. For canards, the single cell (ODE) model exhibits canard-induced bursting. Numerical simulations of the PDE reveal rich spatio-temporal canard dynamics, and the transitions between different bursts are mediated by spatio-temporal maximal canards. The ODE model also exhibits delayed loss of stability due to slow passage …
In this article, the phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation with a slowly-varying parameter. We use the classical asymptotic methods of stationary phase and steepest descents to show that solutions which approach the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linear PDE has a saddle point, and the Stokes and anti-Stokes lines are cen…
The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic nonlocal delayed reaction-diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition on the finite end. This equation models the spatial-temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first estab…
This paper deals with the existence, monotonicity, uniqueness and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations.
We obtain a result on the behavior of the solutions of a general nonautonomous Hopfield neural network model with delay, assuming some general bound for the product of consecutive terms in the sequence of neuron charging times and some conditions to control the nonlinear part of the equations. We then apply this result to improve some existent results on the stability of Hopfield models. Our results are based on a new abstract result on the behavior of nonautonomous delayed equations.
Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction-diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. We find that the dynamical behavior near bifurcation points are proved to be governed by twelve disti…
Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction-diffusion neural networks with time delays under Dirichlet zero boundary value
Chengdu Normal University（成都师范学院）；Sichuan Sanhe Vocational College（四川三河职业学院）；Shandong Normal University（山东师范大学）
We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of centre-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are …
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 solutio…
Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with ti…
Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been successfully developed for conquering some limitations emergent in application of the original framework. Here we propose a new class of continuous-depth neural networks with delay, named as Neural Delay Differential Equations (NDDEs), and, for computing the corresponding gradients, we use the adjoint sensitivity method to obtain the delayed dynamics of the a…
