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a Schematic diagram of a neural recurrent inhibitory feedback loop. Neuron E makes an excitatory synapse onto the inhibitory interneuron I, which in turn makes an inhibitory synapse back onto E. b The time course of the membrane potential v for the integrate-and-fire neuron E under time-delayed inhibitory feedback. The dashed line indicates the threshold. See text for details.
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The multistability that arises in delayed feedback control mechanisms has applications for dynamic short term memory storage. Here we investigate the effects of multiplicative, Gaussian-distributed white noise on an integrate-and-fire model of a recurrent inhibitory neural loop: when the neuron fires an inhibitory pulse decreases the membrane poten...
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... we study the noise-induced transitions that occur between coexistent limit cycle attractors in an integrate-and- fire model for a recurrent inhibitory neural loop Fig. 1a 2. In Sec. II we determine the number of attractors and their properties for this model in the absence of noise. In Sec. III we characterize the mechanism for noise-induced transi- tions between attractors. Finally, in Sec. IV we study the effects of noise on the memory storage capabilities of an electronic circuit analog of the ...
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... integrate-and-fire model for a recurrent inhibitory loop is presented in Fig. 1b. The membrane potential v of the neuron increases linearly at a rate A until it reaches the firing threshold . When v, the neuron fires and v is reset to its resting membrane potential v 0 . The firing of the neuron ex- cites the inhibitory interneuron such that at a time later the membrane potential of the excitatory neuron is ...
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... order to obtain a practical assessment of the impact of noise on memory storage by a multistable dynamical system, we constructed an electronic circuit that mimics the integrate-and-fire model discussed in Secs. II and III Fig. 10. This circuit is fully described in the Appendix. Briefly, neuron E Fig. 1a is represented by a capacitor that charges linearly. The role of the neuron I is played by a time-delay circuit bucket brigade device, which introduces an incremental reduction to the capacitor charging voltage at a time after the capacitor discharges. In order ...
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... order to obtain a practical assessment of the impact of noise on memory storage by a multistable dynamical system, we constructed an electronic circuit that mimics the integrate-and-fire model discussed in Secs. II and III Fig. 10. This circuit is fully described in the Appendix. Briefly, neuron E Fig. 1a is represented by a capacitor that charges linearly. The role of the neuron I is played by a time-delay circuit bucket brigade device, which introduces an incremental reduction to the capacitor charging voltage at a time after the capacitor discharges. In order to put noise on Sec. III we added noise to the discharging cur- rent ...
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... noise was generated by a random noise generator General Radio Company, Model 1390-B and was first passed through a filter circuit with a time constant of 1 ms. Figure 11 shows the distribution of interspike intervals for different injected noise levels measured as V rms , the root- mean-square voltage. In all cases the model was initialized to the regular spiking attractor Fig. 2a. ...
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... with a time constant of 1 ms. Figure 11 shows the distribution of interspike intervals for different injected noise levels measured as V rms , the root- mean-square voltage. In all cases the model was initialized to the regular spiking attractor Fig. 2a. As expected when V rms 0, the distribution of ISI is approximated by a single delta function Fig. 11a. As V rms increases, the distribution of ISI broadens Figs. 11b and 11c. This occurs because the interspike interval is equal to 1p and noise is injected through . Once V rms becomes large enough, the histogram becomes multimodal Fig. 11d. These modes appear be- cause once the noise level becomes sufficiently high, switches between ...
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... of interspike intervals for different injected noise levels measured as V rms , the root- mean-square voltage. In all cases the model was initialized to the regular spiking attractor Fig. 2a. As expected when V rms 0, the distribution of ISI is approximated by a single delta function Fig. 11a. As V rms increases, the distribution of ISI broadens Figs. 11b and 11c. This occurs because the interspike interval is equal to 1p and noise is injected through . Once V rms becomes large enough, the histogram becomes multimodal Fig. 11d. These modes appear be- cause once the noise level becomes sufficiently high, switches between basins of attraction frequently occur. The same experiment can be repeated ...
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... Fig. 2a. As expected when V rms 0, the distribution of ISI is approximated by a single delta function Fig. 11a. As V rms increases, the distribution of ISI broadens Figs. 11b and 11c. This occurs because the interspike interval is equal to 1p and noise is injected through . Once V rms becomes large enough, the histogram becomes multimodal Fig. 11d. These modes appear be- cause once the noise level becomes sufficiently high, switches between basins of attraction frequently occur. The same experiment can be repeated by starting the circuit with a different spiking pattern corresponding to a different attrac- tor. The amplitude of V rms necessary for switches to occur at a ...
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... schematic diagram of the electronic circuit of the integrate-and-fire model described in Secs. II and III is shown in Fig. 10. Neuron E Fig. 1a is represented by the capacitor C. This capacitor can be charged with the constant current source I c and discharged by the two switches S 1 and S 2 . Switch S 1 is closed on the leading edge of a pulse from the flip-flop circuit shown. The flip-flop is operated by com- paritors 1 and 2, which cause it to change state ...
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... schematic diagram of the electronic circuit of the integrate-and-fire model described in Secs. II and III is shown in Fig. 10. Neuron E Fig. 1a is represented by the capacitor C. This capacitor can be charged with the constant current source I c and discharged by the two switches S 1 and S 2 . Switch S 1 is closed on the leading edge of a pulse from the flip-flop circuit shown. The flip-flop is operated by com- paritors 1 and 2, which cause it to change state when the voltage ...
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... PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0151697 make the transition from one attractor to another [33][34][35][36] . The control of bistability was proposed through a linear augmentation in Refs. ...
Birhythmicity is evident in many nonlinear systems, which include physical and biological systems. In some living systems, birhythmicity is necessary for response to the varying environment while unnecessary in some physical systems as it limits their efficiency. Therefore, its control is an important area of research. This paper proposes a space-dependent intermittent control scheme capable of controlling birhythmicity in various dynamical systems. We apply the proposed control scheme in five nonlinear systems from diverse branches of natural science and demonstrate that the scheme is efficient enough to control the birhythmic oscillations in all the systems. We derive the analytical condition for controlling birhythmicity by applying harmonic decomposition and energy balance methods in a birhythmic van der Pol oscillator. Further, the efficacy of the control scheme is investigated through numerical and bifurcation analyses in a wide parameter space. Since the proposed control scheme is general and efficient, it may be employed to control birhythmicity in several dynamical systems.
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... (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Two-parameter bifurcation diagram for system(2) in the ( 0 , ) plane. Black lines denote fold bifurcations of limit cycles LPC 1,2,3 and CPC denotes the global cusp bifurcation of cycles. ...
... Piecewise smooth retarded delay differential equations can exhibit a rich range of dynamical behaviors ranging from limit cycle oscillations to multistability and chaotic fluctuations [9,10,20,21,89,144,169,247]. It has been possible to observe the dynamics in electronic circuits [98,100,202] and in the clamped human pupil light reflex [189,229,232]. In our experiments we have been able to observe three types of dynamical behaviors: complex oscillations, power law behaviors and microchaos. ...
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... That is, the disappearance of seizures in adolescence could be attributed to a loss of multistability in the network due the reduction of conduction time delays. This is reminiscent of the properties exhibited by neural oscillators that are influenced by timedelayed pulsatile coupling (Foss et al. 1996(Foss et al. , 1997Foss and Milton 2000;Ma and Wu 2007). In these examples, multistability becomes possible when the time delay, τ , becomes longer than the intrinsic period, T , of the neural oscillator. ...
... As a consequence, the neurons in the epileptic system are pulse-coupled, namely the interactions between them take the form of discrete synaptic potentials driven by neural spikes. One paradigm which is simple enough for mathematical insight into these cross-scale neurodynamical systems is the delayed recurrent loop control of an oscillator (Foss et al. 1996(Foss et al. , 1997Foss and Milton 2000;Ma and Wu 2007). Here we have shown that the 3neuron motif for CAE is equivalent to an oscillator formed by the interaction between intrathalamic neurons (RT and TC) and a delayed recurrent excitatory loop that involves corticothalamic interactions. ...
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... Systems, where the rate of change of state depends not only on the present state of the system but also on the past states are described by delay-differential equations. Delayed equations arise in several fields of science: in the digital control of robotics applications with information delay [16,17], in neural network models, where the interactions of the neurons are delayed [18], or in delayed feedback control mechanism of human balancing [19,20]. The time delay is also a key element of the explanation of machine tool chatter, more and more sophisticated models have appeared for turning and milling applications in the last decade [21][22][23][24][25][26][27]. ...
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... In particular, when delays are sufficiently large compared to intrinsic neural response times (typically when the ratio of these time scales exceeds 1-see [27]), multistability can arise in the sense that different initial conditions can lead to different steady state solutions [32]. This is apparent in the presence of a mild amount of noise or even in the absence of input [33]. From those results, it is expected that forcing a network such as the one considered here could further be sustained by multistability, beyond the basic effect described in our paper. ...
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... However, biology and electronics are two of the most recurrent areas. A few examples considering the former area are described in the multiple behavioral patterns in neural dynamics or the multistable coordination dynamics [19,20]; in the dynamics of some biological central pattern generators through neurons connected in rings [21]; in the integrate-and-fire model of the neurons affected by white noise [22]. Now, consider the case of the area of electronics, in the circuit implementation of the digital selection of an active set of neurons or in the use of nonlinear reactances or negative resistors [23,24]. ...
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A switching dynamical system by means of piecewise linear systems in R^3 that presents multistability is presented. The flow of the system displays multiple scroll attractors due to the unstable hyperbolic focus-saddle equilibria with stability index of type I, i.e., a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. This class of systems is constructed by a discrete control mode changing the equilibrium point regarding the location of their states. The scrolls are generated when the stable and unstable eigenspaces of each adjacent equilibrium point generate the stretching and folding mechanisms to generate chaos, i.e., the unstable manifold in the first subsystem carry the trajectory towards the stable manifold of the immediate adjacent subsystem.
