Canards of mixed type in a neural burster (original) (raw)
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Chaotically spiking canards in an excitable system with 2D inertial fast manifolds
Physical review letters, 2007
We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically but deterministically triggers excitable spikes. This scenario is also found in a model for a nonlinear Fabry-Perot cavity with one pendular mirror.
Regular & Chaotic Dynamics, 2004
The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. We demonstrate how two different codimension-one bifurcations of a saddle-node periodic orbit with homoclinic orbits can explain transitions between tonic spiking and bursting activities in neuron models following Hodgkin-Huxley formalism. In the first case, we argue that the Lukyanov-Shilnikov bifurcation of a saddle-node periodic orbit with non-central homoclinics is behind the phenomena of bi-stability observed in a model of a leech heart interneuron under defined pharmacological conditions. This model can exhibit two coexisting types of oscillations: tonic spiking and bursting. Moreover, the neuron model can also generate weakly chaotic trains of bursting when a control parameter is close to the bifurcation value. In the second case, the transition is continuous and reversible due to the blue sky catastrophe bifurcation. This bifurcation provides a plausible mechanism for the regulation of the burst duration which may increases with no bound as 1/ √ α − α0, where α0 is the transitional value, while the inter-burst interval remains nearly constant.
Canard Cycles in Global Dynamics
International Journal of Bifurcation and Chaos, 2012
Fast-slow systems are studied usually by "geometrical dissection" . The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems and with applications to physiology [10, 26], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered when the transition points are folds, in this article, we focus on the case where one or several transition points or "jumps" are instead transcritical. We present several new surprising effects like the "amplification of canards" or the "exceptionally fast recovery" on both (1+1)-systems and (2+1)-systems associated with tritrophic food chain dynamics. Finally, we also mention their possible relevance to the notion of resilience which has been coined out in ecology .
Neural Excitability and Singular Bifurcations
The Journal of Mathematical Neuroscience (JMN), 2015
We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.
Routes to chaos in a model of a bursting neuron
Biophysical Journal, 1990
Chaotic regimes have been observed experimentally in neurons as well as in deterministic neuronal models. The R 15 bursting cell in the abdominal ganglion of Aplysia has been the subject of extensive mathematical modeling. Previously, the model of Plant and Kim has been shown to exhibit both bursting and beating modes of electrical activity. In this report, we demonstrate (a) that a chaotic regime exists between the bursting and beating modes of the model, and (b) that the model approaches chaos from both modes by a period doubling cascade. The bifurcation parameter employed is the external stimulus current. In addition to the period doubling observed in the model-generated trajectories, a period three "window" was observed, power spectra that demonstrate the ap-proaches to chaos were generated, and the Lyaponov exponents and the fractal dimension of the chaotic attractors were calculated. Chaotic regimes have been observed in several similar models, which suggests that they are a general characteristic of cells that exhibit both bursting and beating modes.
Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh--Rose Burster
SIAM Journal on Applied Dynamical Systems, 2012
The well-studied Hindmarsh-Rose model of neural action potential is revisited from the point of view of global bifurcation analysis. This slow-fast system of three paremeterised differential equations is arguably the simplest reduction of Hodgkin-Huxley models capable of exhibiting all qualitatively important distinct kinds of spiking and bursting behaviour. First, keeping the singular perturbation parameter fixed, a comprehensive two-parameter bifurcation diagram is computed by brute force. Of particular concern is the parameter regime where lobe-shaped regions of irregular bursting undergo a transition to stripe-shaped regions of periodic bursting. The boundary of each stripe represents a fold bifurcation that causes a smooth spike-adding transition where the number of spikes in each burst is increased by one. Next, numerical continuation studies reveal that the global structure is organised by various curves of homoclinic bifurcations.
Mechanism of bistability: Tonic spiking and bursting in a neuron model
Physical Review E, 2005
Neurons can demonstrate various types of activity; tonically spiking, bursting as well as silent neurons are frequently observed in electrophysiological experiments. The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. Here we demonstrate that a bifurcation of a codimension one can explain a transition between tonic spiking behavior and bursting behavior. Namely, we argue that the Lukyanov-Shilnikov bifurcation of a saddle-node periodic orbit with noncentral homoclinics may initiate a bistability observed in a model of a leech heart interneuron under defined pharmacological conditions. This model can exhibit two coexisting types of oscillations: tonic spiking and bursting, depending on the initial state of the neuron model. Moreover, the neuron model also generates weakly chaotic bursts when a control parameter is close to the bifurcation values that correspond to homoclinic bifurcations of a saddle or a saddle-node periodic orbit.
Strange Nonchaotic Bursting in a Quasiperiodically-forced Hindmarsh-Rose Neuron
Journal of the Korean Physical Society, 2010
We study the transition from a silent state to a bursting state by varying the dc stimulus in the Hindmarsh-Rose neuron under quasiperiodic stimulation. For this quasiperiodically-forced case, a new type of strange nonchaotic (SN) bursting state is found to occur between the silent state and the chaotic bursting state. This is in contrast to the periodically-forced case where the silent state transforms directly to a chaotic bursting state. Using a rational approximation to the quasiperiodic forcing, we investigate the mechanism for the appearance of such an SN bursting state. Thus, a smooth torus (corresponding to a silent state) is found to transform to an SN bursting attractor through a phase-dependent subcritical period-doubling bifurcation. These SN bursting states, together with chaotic bursting states, are characterized in terms of the interburst interval, the bursting length, and the number of spikes in each burst. Both bursting states are found to be aperiodic complex ones. Consequently, aperiodic complex burstings may result from two dynamically different states with strange geometry (one is chaotic and the other is nonchaotic). Thus, in addition to chaotic burstings, SN burstings may become a dynamical origin for the complex physiological rhythms that are ubiquitous in organisms.
2016
Several experimental and numerical studies have observed that populations of oscillatory neurons can synchronize and converge to an dynamical attractor composed of periodic bursts which involve the whole network. We show here that a sufficient condition for this phenomenon to occur is the presence of an excitatory population of oscillatory neurons displaying spike-driven adaptation, which was observed for cultured pyramidal neurons from the cortex or the hippocampus. More than just synchronization – which takes place even for weak coupling – bursting occurs for stronger coupling as the adaptive neurons switch from their initial adaptive spiking to an intermittent burst-like behavior. Addition of inhibitory neurons or plastic synapses can then modulate this dynamics in many ways but is not necessary for its appearance. After discussing the origin of the bursting behavior, we provide a detailed analytic and numerical description of such dynamics, as well as the evolution of its proper...
The dynamics of individual neurons are crucial for producing functional activity in neuronal networks. An open question is how temporal characteristics can be controlled in bursting activity and in transient neuronal responses to synaptic input. Bifurcation theory provides a framework to discover generic mechanisms addressing this question. We present a family of mechanisms organized around a global codimension-2 bifurcation. The cornerstone bifurcation is located at the intersection of the border between bursting and spiking and the border between bursting and silence. These borders correspond to the blue sky catastrophe bifurcation and the saddle-node bifurcation on an invariant circle (SNIC) curves, respectively. The cornerstone bifurcation satisfies the conditions for both the blue sky catastrophe and SNIC. The burst duration and interburst interval increase as the inverse of the square root of the difference between the corresponding bifurcation parameter and its bifurcation value. For a given set of burst duration and interburst interval, one can find the parameter values supporting these temporal characteristics. The cornerstone bifurcation also determines the responses of silent and spiking neurons. In a silent neuron with parameters close to the SNIC, a pulse of current triggers a single burst. In a spiking neuron with parameters close to the blue sky catastrophe, a pulse of current temporarily silences the neuron. These responses are stereotypical: the durations of the transient intervals-the duration of the burst and the duration of latency to spiking-are governed by the inverse-square-root laws. The mechanisms described here could be used to coordinate neuromuscular control in central pattern generators. As proof of principle, we construct small networks that control metachronal-wave motor pattern exhibited in locomotion. This pattern is determined by the phase relations of bursting neurons in a simple central pattern generator modeled by a chain of oscillators.