Stability and Bifurcations in Neural Fields with Finite Propagation Speed and General Connectivity

Atay, Fatihcan M.; Hutt, Axel

doi:10.1137/s0036139903430884

Cited by 101 publications

(115 citation statements)

References 44 publications

Supporting

Mentioning

112

Contrasting

Order By: Relevance

“…This condition defines the critical wave number k c and does not depend on the synaptic time scales and the conduction speed in accordance with previous studies on populations Cogn Neurodyn (2010) 4:37-59 47 of a single neuron type (Hutt et al 2003;Atay and Hutt 2005;Venkov et al 2007). Considering the loss of stability to the spatially homogeneous state (k = 0) the threshold condition reads 1 = d E (p)a e -f(p)d I (p)a i which coincides with the condition (20) for the points A and B, cf.…”

Section: The Resting Statesupporting

confidence: 88%

Effects of the anesthetic agent propofol on neural populations

Hutt

Longtin

2009

Cogn Neurodyn

Self Cite

View full text Add to dashboard Cite

The neuronal mechanisms of general anesthesia are still poorly understood. Besides several characteristic features of anesthesia observed in experiments, a prominent effect is the bi-phasic change of power in the observed electroencephalogram (EEG), i.e. the initial increase and subsequent decrease of the EEG-power in several frequency bands while increasing the concentration of the anaesthetic agent. The present work aims to derive analytical conditions for this bi-phasic spectral behavior by the study of a neural population model. This model describes mathematically the effective membrane potential and involves excitatory and inhibitory synapses, excitatory and inhibitory cells, nonlocal spatial interactions and a finite axonal conduction speed. The work derives conditions for synaptic time constants based on experimental results and gives conditions on the resting state stability. Further the power spectrum of Local Field Potentials and EEG generated by the neural activity is derived analytically and allow for the detailed study of bi-spectral power changes. We find bi-phasic power changes both in monostable and bistable system regime, affirming the omnipresence of bi-spectral power changes in anesthesia. Further the work gives conditions for the strong increase of power in the d-frequency band for large propofol concentrations as observed in experiments.

show abstract

Section: The Resting Statesupporting

confidence: 88%

Effects of the anesthetic agent propofol on neural populations

Hutt

Longtin

2009

Cogn Neurodyn

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed, the solutions of (3) exhibit a much richer range of dynamics when τ is nonzero. Figure 2 shows the bifurcation diagram for several values of τ when f is given by (12). Of course, for τ = 0 the familiar bifurcation diagram of the logistic map is obtained, displaying the period-doubling route to chaos.…”

mentioning

confidence: 99%

“…To demonstrate, we take the logistic map (12) with ρ = 4, for which it has a chaotic attractor. Thus, (3) is chaotic for all ε when τ = 0.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

2004

Self Cite

View full text Add to dashboard Cite

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scalefree and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation. (http://link.aps.org/abstract/PRL/v92/e144101) Recent years have witnessed a growing interest in the dynamics of interacting units. Particularly, a large number of studies have been devoted to synchronization in a variety of systems (see [1] and the references therein), including the coupled map lattices introduced by Kaneko [2]. Usually, such systems have been investigated under the assumption of a certain regularity in the connection topology, where units are coupled to their nearest neighbors or to all other units. Lately, more general networks with random, small-world, scale-free, and hierarchical architectures have been emphasized as appropriate models of interaction [3,4,5,6,7]. On the other hand, realistic modeling of many large networks with non-local interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the units. Some numerical studies have regarded synchronization under delays for special cases such as globally coupled logistic maps [8] or carefully chosen delays [9]. In this Letter we consider synchronization of coupled chaotic maps for general network architectures and connection delays. Because of the presence of the delays, the constituent units are unaware of the present state of the others; so it is not evident a priori that such a collection of chaotic units can operate in unison, i. e. synchronize. Based on analytical calculations, we show that this is indeed possible, and in fact may be facilitated by the presence of delays. Moreover, while the connection topology is important for synchronization, the delays have a crucial role in determining the resulting collective dynamics. As a result, the synchronized system can exhibit a plethora of new behavior in the presence of delays. We illustrate the results by numerical simulation of large networks of logistic maps.

show abstract

“…2. The form of w(r, r ′ ) typically describes a Laplacian (exponential) or Gaussian decay in connectivity strength from any point in the field Atay and Hutt, 2005). By assuming the form of h(t) is the same across layers (scales) the Amari style neural field model is formed, incorporating a multi-layer structure into the connectivity kernel.…”

Section: Ide Neural Field Modelmentioning

confidence: 99%

Spatiotemporal multi-resolution approximation of the Amari type neural field model

Aram

Freestone

Dewar³

et al. 2013

NeuroImage

View full text Add to dashboard Cite

Neural fields are spatially continuous state variables described by integro-differential equations, which are well suited to describe the spatiotemporal evolution of cortical activations on multiple scales. Here we develop a multi-resolution approximation (MRA) framework for the integro-difference equation (IDE) neural field model based on semi-orthogonal cardinal B-spline wavelets. In this way, a flexible framework is created, whereby both macroscopic and microscopic behavior of the system can be represented simultaneously. State and parameter estimation is performed using the expectation maximization (EM) algorithm. A synthetic example is provided to demonstrate the framework.

show abstract

Stability and Bifurcations in Neural Fields with Finite Propagation Speed and General Connectivity

Cited by 101 publications

References 44 publications

Effects of the anesthetic agent propofol on neural populations

Effects of the anesthetic agent propofol on neural populations

Delays, Connection Topology, and Synchronization of Coupled Chaotic Maps

Spatiotemporal multi-resolution approximation of the Amari type neural field model

Contact Info

Product

Resources

About