Bifurcations in Nonsmooth Dynamical Systems

Bernardo, Mario di; Budd, C. J.; Champneys, Alan R; Kowalczyk, Piotr; Nordmark, Arne; Tost, Gerard Olivar; Piiroinen, Petri T.

doi:10.1137/050625060

Cited by 339 publications

(221 citation statements)

References 109 publications

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“…Recently we have shown how spatially structured oscillations can also occur in neural fields with synaptic depression, provided that the firing rate function has finite gain [19,20]. It would be interesting to explore scenarios where oscillations arise in neural fields with synaptic depression and Heaviside nonlinearities via some form of generalized Hopf bifurcation, along lines analogous to recent studies of nonsmooth dynamical systems [10,44]. One recent example occurs in a competitive neural network model of binocular rivalry [22].…”

Section: Discussionmentioning

confidence: 92%

Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression

Bressloff¹,

Kilpatrick²

2011

SIAM J. Appl. Math.

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Abstract. We analyze radially symmetric bumps in a two-dimensional piecewise-smooth neural field model with synaptic depression. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Synaptic depression dynamically reduces the strength of synaptic weights in response to increases in activity. We show that in the case of a Mexican hat weight distribution, sufficiently strong synaptic depression can destabilize a stationary bump solution that would be stable in the absence of depression. Numerically it is found that the resulting instability leads to the formation of a traveling spot. The local stability of a bump is determined by solutions to a system of pseudolinear equations that take into account the sign of perturbations around the circular bump boundary.Key words. neural fields, synaptic depression, piecewise-smooth dynamics AMS subject classification. 92C20 DOI. 10.1137/100799423 1. Introduction. Continuum neural field models provide an important example of spatially extended excitable systems with nonlocal interactions. These models represent the large-scale dynamics of populations of neurons in terms of nonlinear integrodifferential equations, whose associated integral kernels represent the spatial distribution of neuronal synaptic connections [2,3,5,11,40]. As in the case of nonlinear PDE models of diffusively coupled excitable systems [17], neural field models can exhibit a variety of coherent pulse-like structures, including both stationary and traveling solitary pulses. Traveling pulses tend to occur when synaptic connections are predominantly excitatory and there is some form of slow local adaptation or recovery [30], whereas stationary pulses (activity bumps) occur in the presence of lateral inhibition [2,31]. The formation of localized activity states can be used to model a number of neurobiological phenomena. For example, traveling pulses have been observed in disinhibited slice preparations [6,41,42] using voltage sensitive dyes and multiple electrodes. A second example is given by a delayed response task in which an animal is required to retain information of a sensory cue across a delay period between the stimulus and behavioral response. Physiological recordings in the prefrontal cortex have shown that spatially localized groups of neurons fire during the recall task and then stop firing once the task has finished [38]. Thus persistent localized states of activity are thought to be neural correlates of spatial working memory.

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Section: Discussionmentioning

confidence: 92%

Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression

Bressloff¹,

Kilpatrick²

2011

SIAM J. Appl. Math.

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“…The following definitions of all types of equilibria of Filippov system (2.3) are necessary throughout the paper [19,26,31,32].…”

Section: Sliding Mode Dynamics and Existence Of The Equilibriamentioning

confidence: 99%

“…In this case the models (1.2) and (1.3) can be rewritten as a Filippov system, a model which has been applied widely in many fields of science and engineering. Furthermore, the theory of Filippov systems is being recognized as not only richer than the corresponding theory of continuous systems, but also as representing a more natural framework for the mathematical modelling of real-world phenomena [18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Ag(t)(c + Mi(t)/(n + I(t))mentioning

confidence: 99%

Modelling the regulatory system for diabetes mellitus with a threshold window

Tang

Cheke

2015

Communications in Nonlinear Science and Numerical Simulation

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Please cite this article as: Yang, J., Tang, S., Cheke, R.A., Modelling the regulatory system for diabetes mellitus with a threshold window, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns. 2014.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.Modelling the regulatory system for diabetes mellitus with a threshold window AbstractPiecewise (or non-smooth) glucose-insulin models with threshold windows for type 1 and type 2 diabetes mellitus are proposed and analyzed with a view to improving understanding of the glucose-insulin regulatory system. For glucose-insulin models with a single threshold, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between regular equilibria and a pseudo-equilibrium are studied. Furthermore, the sufficient and necessary conditions for the global stability of regular equilibria and the pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Sliding bifurcations related to boundary node bifurcations were investigated with theoretical and numerical techniques, and insulin clinical therapies are discussed. For glucose-insulin models with a threshold window, the effects of glucose thresholds or the widths of threshold windows on the durations of insulin therapy and glucose infusion were addressed. The duration of the effects of an insulin injection is sensitive to the variation of thresholds. Our results indicate that blood glucose level can be maintained within a normal range using piecewise glucose-insulin models with a single threshold or a threshold window. Moreover, our findings suggest that it is critical to individualize insulin therapy for each patient separately, based on initial blood glucose levels.

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“…Due to the difference in allowed perturbations, the scenarios observed in the present paper are not considered in [25]. In [17,21], the authors study bifurcations of equilibria in continuous systems, that are not differentiable. Due to the assumptions posed in these papers, all equilibria are isolated points.…”

Section: Introductionmentioning

confidence: 99%

“…in [16][17][18][19][20][21][22][23][24][25]. Bifurcations of limit cycles of discontinuous systems are studied using a return map; see [16,17,22,24].…”

Section: Introductionmentioning

confidence: 99%