Dynamical systems analysis of spike-adding mechanisms in transient bursts

Jakub Nowacki 0 Hinke M Osinga 0 Krasimira Tsaneva-Atanasova 0 0 HM Osinga ( ) Department of Mathematics, The University of Auckland , Private Bag 92019, Auckland, 1142, New Zealand Transient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using oneparameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spikeadding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism. 1 Introduction How a single spike or a burst of spikes is generated and regulated for neuron cells is one of the most fundamental questions in neuroscience [1]. Spike generation is closely related to neuronal excitability, which is the ability of the cells membrane potential to undergo a large excursion, called an action potential or a spike, when subjected to a sufficiently strong stimulus [13]. An excitable cell either responds in full to such a stimulus or not at all, which allows for a reliable transmission of information. Therefore, the different mechanisms for excitability and bursting have been widely studied; we refer to Izhikevich [4] for a comprehensive overview of mechanisms for neurons. Excitable behaviour has also been reported to occur in many other types of cells [13] as well as in physical systems, such as lasers [5, 6], electronic circuits [710] and chemical reactions [11]. Neuronal excitability can be very sensitive to even relatively small changes in, for example, the biophysical properties of a neuron [1214] or its morphology [15]. This parameter sensitivity indicates that dynamical systems theory is particularly suited for explaining the rich dynamics found in excitable systems. The classification of different bursting mechanisms was pioneered by Rinzel [16], who used a system decomposition into slow and fast subsystems. He showed that the burst can be divided into active (spiking) and silent phases, which follow different types of attractors of the fast subsystem. Hence, a classification of the bursting oscillators is provided by the structure of the bifurcation diagram of the fast subsystem. Rinzels classification was extended in Izhikevichs studies [1, 4]. As an alternative approach, Golubitsky et al. [17] used singularity theory to classify the bifurcation diagrams of the fast subsystem and, hence, the different bursting mechanisms; see also [18]. These studies are primarily for systems with one slow variable; see Smolen et al. [19] for an extension to two slow variables and Ermentrout and Terman [3] for a summary of these ideas along with new results. A classification of bursting mechanisms, however, does not answer questions about the number of spikes in a particular burst of the same type nor does it explain possible transitions between spiking and bursting. While the latter has received considerable attention over the years, the former has hardly been addressed. Terman [20] analysed transitions between bursting and tonic (continuous) spiking in a pancreatic -cell model. He recognised the importance of connecting classical slow-fast analysis with full system bifurcation analysis and identified bifurcations of the periodic bursting solutions that organise the transitions between different parameter regimes. He further studied chaotic spiking that can arise in between such transitions [21]. Recently, Benes et al. [22] and Kramer et al. [23] reported that a new type of torus canard can play a role in the transition from spiking to bursting in a model of Purkinje cells. Other recent studies particularly focus on spike-adding in a periodic bursting oscillator; for example, see Govaerts and Dhooge [24], Guckenheimer and Kuehn [25], Tsaneva-Atanasova et al. [26] and Linaro et al. [27]. These studies show that the spike-adding mechanism is formed by a pair of saddle-node bifurcations of periodic orbits of the full system; bursts with different numbers of spikes are, in fact, different periodic attractors of the full system that may coexist only if the number of spikes differs by one [20, 26]. The recent study by Teka et al. [28] explains how one may predict the precise number of spikes in a burst; here, the use of two slow variables is essential. Methods to regulate the number of spikes are reported by Ghigliazza and Holmes [29], where a minimal Hodgkin-Huxley model of bursting is proposed to analyse spike-adding and transitions between bursting and tonic spiking in a more general context. As discussed by Izhikevich [4], the mechanisms for generating spikes do not depend on whether the neuron exhibits spiking only as a transient phenomenon when subjected to a strong enough stimulus or whether it is spiking or bursting continuously. This is indeed the case if transient bursting is organised by the applied stimulus. Applying a stimulus has the effect of changing the right-hand side of the underlying system of ODEs. The bifurcation diagram of the corresponding fast subsystem typically no longer has a stable equilibrium that corresponds to the resting potential, and a spike or burst arises from new attractors that exist only when the stimulus is on. As soon as the stimulus is switched off, the system relaxes back to the resting potential. Therefore, the mechanism is due to a change in the structure of the bifurcation diagram, which depends on the strength of the applied stimulus. Studies of this nature, where the type of burst is studied in dependence on the strength or duration of the stimulus, have been done, for example, by Tran et al. [30], Kim et al. [31] and Stern et al. [32]. In this paper, we investigate spike-adding in a transient burst in the model of hippocampal pyramidal neurons from Nowacki et al. [13]. In contrast to the above studies, the spike-adding occurs after the applied stimulus has been switched off. Hence, the bursting behaviour is gov (...truncated)