Loop Bifurcation Research Articles

Qualitative switches in single-neuron spike dynamics on neuromorphic hardware: implementation, impact on network synchronization and relevance for plasticity

Abstract Most efforts on spike-based learning on neuromorphic hardware focus on synaptic plasticity and do not yet exploit the potential of altering the spike-generating dynamics themselves. Biological neurons show distinct mechanisms of spike generation, which affect single-neuron and network computations. Such a variety of spiking mechanisms can only be mimicked on chips with more advanced, nonlinear single-neuron dynamics than the commonly implemented leaky integrate-and-fire (LIF) neurons. Here, we demonstrate that neurons on the BrainScaleS-2 chip configured for exponential leaky integrate-and-fire (EIF) dynamics can be tuned to undergo a qualitative switch in spike generation via a modulation of the reset voltage. This switch is accompanied by altered synchronization properties of neurons in a network and thereby captures a main characteristic of the unfolding of the saddle-node loop (SNL) bifurcation - a qualitative transition that was recently demonstrated in biological neurons. Using this switch, cell-intrinsic properties alone provide a means to control whether small networks of all-to-all coupled neurons on the chip exhibit synchronized firing or splayed-out spiking patterns. We use an example from a central pattern generating circuit in the fruitfly to show that such dynamics can be induced and controlled on the chip. Our study thereby demonstrates the potential of neuromorphic chips with relatively complex and tunable single-neuron dynamics such as the BrainScaleS-2 chip, to generate computationally distinct single unit dynamics. We conclude with a discussion of the utility of versatile spike-generating mechanisms on neuromorphic chips.

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

Heteroclinic loop bifurcations by perturbing a class of Z2-equivariant quadratic switching Hamiltonian systems with nilpotent singular points

This paper investigates the problem of heteroclinic loop bifurcation by perturbing a class of Z2-equivariant quadratic switching systems with nilpotent singular points. Firstly, it provides sufficient and necessary conditions for the occurrence of a heteroclinic loop. Next, the system is perturbed by piecewise polynomial systems of degree n≥1. The paper then considers the lower bound for the maximum number of limit cycles that bifurcate from the generalized heteroclinic loop, finding at least n+[n2] limit cycles.

The discontinuous limit case of an archetypal oscillator with a constant excitation and van der Pol damping

We investigate global dynamics of the discontinuous limit case of an archetypal oscillator with a constant excitation that is a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. The discontinuous dynamical system can exhibit three limit cycles and rich nonlinear phenomena, including boundary equilibrium bifurcation, Hopf bifurcation, grazing bifurcation, pseudo saddle–node loop bifurcation and double crossing limit cycle bifurcation. Because of destroyed symmetry for this system, it exhibits more complex and abundant dynamics than the smooth oscillator with van der Pol damping and can appear new dynamical phenomena, such as coexistence of a crossing limit cycle and a pseudo homoclinic loop, coexistence of two crossing limit cycles and the existence of pseudo saddle–node loops. The bifurcation diagram of the discontinuous system is shown in parameter space and all global phase portraits in the Poincaré disk are sketched in the phase plane.

Global dynamics of a quintic Liénard system with -symmetry I: saddle case

In the paper we deal with a quintic Liénard system of the form with a -symmetry, where and a 3 b 2 ≠ 0. A complete study of this system with b 2 > 0, called the saddle case, is finished, showing that the system exhibits at most two limit cycles, and the necessary and sufficient conditions are obtained on the existence of two limit cycles and a two-saddle heteroclinic loop. We also present a global bifurcation diagram and the corresponding phase portraits of this system, including Hopf bifurcation, Bautin bifurcation, two-saddle heteroclinic loop bifurcation and double limit cycle bifurcation.

Global Dynamics of a Planar Filippov System with Symmetry

Chen [2016a, 2016b] studied global dynamics of the Filippov systems [Formula: see text], respectively. To study the global dynamics of [Formula: see text] completely, since the dynamics of [Formula: see text] is very simple, we are only interested in the global dynamics of [Formula: see text] in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system [Formula: see text] is totally different from systems [Formula: see text].

Global dynamics of the Josephson equation in TS1

The Josephson equation ϕ˙=y,y˙=−sin⁡ϕ+ϵ(a−(1+γcos⁡ϕ)y) was researched by Sanders and Cushman (1986) [12] for its phase portraits when ϵ>0 is small by applying the averaging method. The parameter ϵ can actually be large or even any real number in the practical application of this model. When |ϵ| is not small, we cannot apply the averaging method because the system is not near-Hamiltonian. For general ϵ∈R, we present complete dynamics and more complex bifurcations of the Josephson equation in TS1, including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic loop bifurcation, two-saddle heteroclinic loop bifurcation, upper saddle connection bifurcation and lower saddle connection bifurcation. Moreover, we prove the monotonicity of bifurcation functions with respect to parameters and the nonexistence of a two-saddle heteroclinic loop for all a≠0.

Steady flow of a viscoelastic film over an inclined plane featuring periodic slits

We consider the steady flow of a viscoelastic film over an inclined plane featuring a periodic arrangement of slits, which are oriented normal to the main direction of flow. The film creates a second gas-liquid interface connecting the two hydrophilic sidewalls of a slit. This interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. We develop a computational model and carry out detailed numerical simulations, based on the finite element method to investigate this flow. To this end, we solve the two-dimensional momentum and mass conservation equations, while employing the Phan-Thien-Tanner (PTT) constitutive model to account for the rheology of the viscoelastic material. An elliptic grid generation scheme is used to follow the large deformation of the film shape. We perform a thorough parametric analysis to investigate the combined effects of elastic, inertia, capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that increasing fluid elasticity decreases the two wetting lengths along each side wall of a slit. On the contrary, the wetting of the slit is enhanced by the shear-thinning of the fluid. Multiple steady solutions connected by turning points forming a hysteresis loop and transcritical bifurcations as well as isolated solution branches are revealed by pseudo-arc-length continuation. In particular, it is predicted that under certain conditions, the transition from the capillary to the inertia regime is not smooth; instead a hysteresis loop arises. This is the signature of an abrupt decrease of the film penetration with increasing flow rate since higher deformations cannot be sustained. Additionally, we have performed calculations for a wide range of the geometrical characteristics of the substrate. We find that the elastic effects of the fluid become more pronounced when the slits are closer together in their periodic arrangement and of width decreasing to become almost comparable to the capillary length of the liquid. These can lead to almost no wetting of the slit, i.e. the Cassie-Baxter state, even in a hydrophilic surface.

Versal unfolding of a nilpotent Liénard equilibrium within the odd Liénard family

Recently attentions were paid to versal unfolding of degenerate equilibria within a restricted family of vector fields for a practical sense. The family of Liénard systems is such a family of significant physical sense but only within the family of even Liénard systems there were found results for a nilpotent degenerate equilibrium. In this paper we discuss versal unfolding of a nilpotent Liénard equilibrium within the family of odd Liénard systems. As the well-known Bogdanov-Takens normal form is not available for the odevity, we prove that the degeneracy within the family is of codimension 2 and find its versal unfolding. We further discuss the unfolding system, displaying all its bifurcations such as pitchfork bifurcation, center-saddle bifurcation, center bifurcation and homoclinic (heteroclinic) loop bifurcation, which show how a single homoclinic loop, a twin homoclinic loop and a heteroclinic loop arise from a center or a (degenerate) saddle.

Phase‐response curves of ion channel gating kinetics

Reduced models of single‐neuron dynamics are widely used to characterize neuronal properties such as stimulus encoding and a cell's contribution to network function. Low‐dimensional phase oscillator models based on phase‐response curves, in particular, permit to capture the essence of a neuron's input‐output dynamics at the level of spike times from high‐dimensional conductance‐based neuron models. In principle, these phase models allow for a direct translation of perturbations in the state variables (such as changes in specific ion channel conductances and kinetics) to shifts in the timing of spikes. Previous mathematical work, however, has predominantly focused on voltage and current perturbations and less on perturbations occurring in any of the involved kinetic equations of ionic channels. Here, we provide the means to mathematically relate properties in these variables to the phase‐dependent voltage dynamics. Specifically, we derive the vector of phase‐response curves near two different onset bifurcations: the physiologically prominent saddle node on invariant cycle bifurcation as well as the saddle‐node loop bifurcation. We demonstrate that, locally around the saddle node, the tangent space of the isochrons is spanned by the strongly stable manifold of the saddle node. This finding enables us to quantitatively relate perturbations in the gating kinetics of ion channels to the resulting changes in spike timing. These results lay the methodological foundation for future quantitative analyses of the impact of specific biophysical channel properties on spike timing, including, but not limited to, channel noise.

The saddle case of Rayleigh–Duffing oscillators

The global dynamics of Rayleigh–Duffing oscillators $$\ddot{x}+a\dot{x}+b\dot{x}^3+cx+dx^3=0$$ , where $$(a,b,c,d)\in \{(a,b,c,d)\in \mathbb {R}^4: b\ne 0,~d>0\}$$ , have been investigated in Chen and Zou (J Phys A 49:165202, 2016). In this paper, the complement case $$(a,b,c,d)\in \{(a,b,c,d)\in \mathbb {R}^4: b\ne 0,~d<0\}$$ will be completely studied, where the bifurcation diagram includes pitchfork bifurcation, Hopf bifurcation, and heteroclinic bifurcation. Meanwhile, the global phase portraits in the Poincare disc are given. The system has at most one limit cycle. Moreover, when the limit cycle exists, its corresponding parameter region lies between Hopf and heteroclinic loop bifurcation curves in the parametric space. In addition, the analytic properties of the heteroclinic loop bifurcation curve are also analyzed. Finally, a few numerical examples are presented to verify our theoretical results.

Qualitative changes in phase-response curve and synchronization at the saddle-node-loop bifurcation.

Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several parameters, including capacitance, leak conductance, and temperature. This bifurcation turns out to induce even more drastic changes in synchronization than the BT transition. This result arises from a direct effect of the saddle-node-loop bifurcation on the limit cycle and hence spike dynamics. In contrast, the BT bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking. We specifically demonstrate that the saddle-node-loop bifurcation (i) ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation, and (ii) results in a symmetry breaking of the system's phase-response curve. The latter entails an increase in synchronization range in pulse-coupled oscillators, such as neurons. The derived bifurcation structure is of interest in any system for which a relaxation limit is admissible, such as Josephson junctions and chemical oscillators.

A proof of Wang–Kooij's conjectures for a cubic Liénard system with a cusp

In this paper the global dynamics of a cubic Liénard system with a cusp is studied to follow Wang and Kooij (1992) [13], who proved that the maximum number of limit cycles is 2 and stated two conjectures about the curves of the cuspidal loop bifurcation and the double limit cycle bifurcation. We give positive answers to those two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness. Finally, associated with previous results we obtain the complete bifurcation diagram and all phase portraits, and demonstrate some numerical examples.

Complex bifurcations in the oscillatory reaction model

The mixing of different types of bifurcations, i.e. supercritical Andronov–Hopf (SAH), double loop (DL) and saddle-loop (SL) bifurcations in the vicinity of their total annihilation, is examined on the highly nonlinear six-variable model for the Bray–Liebhafsky (BL) oscillatory reaction under continuously well-stirred tank reactor (CSTR) conditions. For this kind of the reaction system where the law of mass conservation is additional constraint that must be satisfied and where because of that, some simple bifurcations cannot be formed independently to the others, the considered transformations of the bifurcations are particularly important. That is why as the control parameters for bifurcation analysis, the specific flow rate (j0), as well as the inflow hydrogen peroxide concentration (h=[H2O2]in), were used. The complex bifurcations obtained from numerical simulations are compared with some experimental results. It was shown that these complex bifurcations cannot be easily recognized in experimental investigations without knowing their evolution.

Dynamic transitions in a model of the hypothalamic-pituitary-adrenal axis.

Dynamic properties of a nonlinear five-dimensional stoichiometric model of the hypothalamic-pituitary-adrenal (HPA) axis were systematically investigated. Conditions under which qualitative transitions between dynamic states occur are determined by independently varying the rate constants of all reactions that constitute the model. Bifurcation types were further characterized using continuation algorithms and scale factor methods. Regions of bistability and transitions through supercritical Andronov-Hopf and saddle loop bifurcations were identified. Dynamic state analysis predicts that the HPA axis operates under basal (healthy) physiological conditions close to an Andronov-Hopf bifurcation. Dynamic properties of the stress-control axis have not been characterized experimentally, but modelling suggests that the proximity to a supercritical Andronov-Hopf bifurcation can give the HPA axis both, flexibility to respond to external stimuli and adjust to new conditions and stability, i.e., the capacity to return to the original dynamic state afterwards, which is essential for maintaining homeostasis. The analysis presented here reflects the properties of a low-dimensional model that succinctly describes neurochemical transformations underlying the HPA axis. However, the model accounts correctly for a number of experimentally observed properties of the stress-response axis. We therefore regard that the presented analysis is meaningful, showing how in silico investigations can be used to guide the experimentalists in understanding how the HPA axis activity changes under chronic disease and/or specific pharmacological manipulations.

On the Limit Cycles of a Perturbed Z3-Equivariant Planar Quintic Vector Field

In this paper, the number and distributions of limit cycles in a Z3-equivariant quintic planar polynomial system are studied. 24 limit cycles with two different configurations are shown in this quintic planar vector field by combining the methods of double homoclinic loops bifurcation, heteroclinic loop bifurcation and Poincaré–Bendixson Theorem. The results obtained are useful to the study of weakened 16th Hilbert problem.

Analysis of chaotic oscillations induced in two coupled Wilson-Cowan models.

Although it is known that two coupled Wilson-Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson-Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson-Cowan oscillators.

Global qualitative analysis of a predator–prey system with Allee effect on the prey species

In this paper, the Allee effect is incorporated into a predator–prey model with linear functional response. Compared with the predator–prey model without the Allee effect, it is found that the Allee effect of the prey species increases the extinction risk of both the prey and predator. If the Allee effect of the prey species is strong and the mortality of the predator species is relatively low, then the prey and predator cannot coexist after the predator invasion. Moreover, it is shown that the model with Allee effect undergoes the heteroclinic loop bifurcation and subcritical and supercritical Hopf bifurcations. With the brokenness of the heteroclinic loop, a stable or unstable limit cycle will appear. The Allee effect of the prey species can lead to unstable or stable periodic fluctuations. It is also found that the positive equilibrium of the model could change from stable to unstable, and then disappear when the strength of Allee effect increases continuously from zero.

BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS

In this paper, heteroclinic loop bifurcations with double orbit flips are investigated in four-dimensional vector fields. We obtain the bifurcation equations by setting up a local coordinate system near the rough heteroclinic orbit and establishing the Poincaré map. By means of the bifurcation equations, we investigate the existence, coexistence and noncoexistence of periodic orbit, homoclinic loop and heteroclinic loop under some nongeneric conditions. The approximate expressions of corresponding bifurcation curves (or surfaces) are also given. An example of application is also given to demonstrate the existence of the heteroclinic loop with double orbit flips.

Active spike transmission in the neuron model with a winding threshold manifold

We analyze spiking responses of excitable neuron model with a winding threshold manifold on a pulse stimulation. The model is stimulated with external pulse stimuli and can generate nonlinear integrate-and-fire and resonant responses typical for excitable neuronal cells (all-or-none). In addition we show that for certain parameter range there is a possibility to trigger a spiking sequence with a finite number of spikes (a spiking message) in the response on a short stimulus pulse. So active transformation of N incoming pulses to M (with M>N) outgoing spikes is possible. At the level of single neuron computations such property can provide an active “spike source” compensating “spike dissipation” due to the integrate-and-fire N to 1 response. We delineate the dynamical mechanism for the N to M transformation based on the winding threshold manifold in the neighborhood of big saddle loop bifurcation. Based on the theoretical predictions, a nonlinear electronic circuit is designed implementing the active transmission in physical conditions.