Multistability and Delayed Recurrent Loops

Foss, Jennifer; Longtin, André; Mensour, Boualem; Milton, John

doi:10.1103/physrevlett.76.708

Cited by 312 publications

(180 citation statements)

References 27 publications

(16 reference statements)

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“…Moreover, the presence of delays often results in patterns of oscillatory behaviour that repeat from time to time and that strongly depend on the initial conditions, multi-stability being a wellknown feature in delayed systems [6,7]. In addition, time delays add new degrees of freedom, which can result in the delayed system having a high-dimensional dynamics similar to that of spatio-temporal systems [8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Quantifying the complexity of the delayed logistic map

Masoller

Rosso

2011

Phil. Trans. R. Soc. A.

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Statistical complexity measures are used to quantify the degree of complexity of the delayed logistic map, with linear and nonlinear feedback. We employ two methods for calculating the complexity measures, one with the 'histogram-based' probability distribution function and the other one with ordinal patterns. We show that these methods provide complementary information about the complexity of the delay-induced dynamics: there are parameter regions where the histogram-based complexity is zero while the ordinal pattern complexity is not, and vice versa. We also show that the time series generated from the nonlinear delayed logistic map can present zero missing or forbidden patterns, i.e. all possible ordinal patterns are realized into orbits.

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Section: Introductionmentioning

confidence: 99%

Quantifying the complexity of the delayed logistic map

Masoller

Rosso

2011

Phil. Trans. R. Soc. A.

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“…Multistability in a delayed neural network has been extensively studied in the literature, in particular for delayed neural recurrent loops [8,9], and experimentally in electrical circuits [9] and in recurrently clamped neurons [10]. Foss et al [8] studied neural recurrent inhibitory loops using the well-known HodgkinHuxley model and found three coexisting attracting periodic solutions by computer simulation. Ma and Wu [17,18] showed that the phenomenological spiking neuron model incorporating the firing process and the absolute refractory period can generate a large number of asymptotically stable periodic solutions with predictable patterns of oscillations.…”

Section: Introductionmentioning

confidence: 99%

“…The coexistence of multiple stable patterns in neural networks is the basis of the mechanism for (associative) content-addressable memory storage and retrieval [8,13,14,20,22] where each stable equilibrium is identified with a static memory, while stable periodic orbits are associated with temporally patterned spike trains [4,8,9]. Periodic patterns exhibited in neural networks have been linked to a variety of rhythms, which are associated with important behavioral and cognitive states in the nervous system, including attention, working memory, associative memory, object recognition, sensory motor integration and perception processing [5,7,15].…”

Section: Introductionmentioning

confidence: 99%

Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

2010

Math. Model. Nat. Phenom.

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Abstract. We study the coexistence of multiple periodic solutions for an analogue of the integrateand-fire neuron model of two-neuron recurrent inhibitory loops with delayed feedback, which incorporates the firing process and absolute refractory period. Upon receiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits a spike with a pattern-related delay, in addition to the synaptic delay. We present a theoretical framework to view the inhibitory signal from the inhibitory neuron as a self-feedback of the excitatory neuron with this additional delay. Our analysis shows that the inhibitory feedbacks with firing and the absolute refractory period can generate four basic types of oscillations, and the complicated interaction among these basic oscillations leads to a large class of periodic patterns and the occurrence of multistability in the recurrent inhibitory loop. We also introduce the average time of convergence to a periodic pattern to determine which periodic patterns have the potential to be used for neural information transmission and cognition processing in the nervous system.

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“…At the single neuron level and in ensembles of coupled neurons, delays have been shown to generate instabilities, multi-stability, chaotic oscillations and oscillation death (Foss et al 1996;Pakdaman et al 1996;Vibert et al 1998;Foss & Milton 2000;Sainz-Trapaga et al 2004;Masoller et al 2008). When the neurons are coupled through both the instantaneous and the delayed collective mean field, by varying the delay one can control (i.e.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of globally delay-coupled neurons displaying subthreshold oscillations

Masoller

Torrent

García‐Ojalvo

2009

Phil. Trans. R. Soc. A.

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We study an ensemble of neurons that are coupled through their time-delayed collective mean field. The individual neuron is modelled using a Hodgkin-Huxley-type conductance model with parameters chosen such that the uncoupled neuron displays autonomous subthreshold oscillations of the membrane potential. We find that the ensemble generates a rich variety of oscillatory activities that are mainly controlled by two time scales: the natural period of oscillation at the single neuron level and the delay time of the global coupling. When the neuronal oscillations are synchronized, they can be either in-phase or out-of-phase. The phase-shifted activity is interpreted as the result of a phase-flip bifurcation, also occurring in a set of globally delay-coupled limit cycle oscillators. At the bifurcation point, there is a transition from in-phase to out-of-phase (or vice versa) synchronized oscillations, which is accompanied by an abrupt change in the common oscillation frequency. This phase-flip bifurcation was recently investigated in two mutually delay-coupled oscillators and can play a role in the mechanisms by which the neurons switch among different firing patterns.

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Multistability and Delayed Recurrent Loops

Cited by 312 publications

References 27 publications

Quantifying the complexity of the delayed logistic map

Quantifying the complexity of the delayed logistic map

Patterns, Memory and Periodicity in Two-Neuron Delayed Recurrent Inhibitory Loops

Dynamics of globally delay-coupled neurons displaying subthreshold oscillations

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