^{1,a)}and Krasimira T. Tsaneva-Atanasova

^{2,b)}

### Abstract

We consider the effect of a brief stimulation from the rest state of a minimal neuronal model with multiple time scales. Such transient dynamics brings out the intrinsic bursting capabilities of the system. Our main goal is to show that a minimum of three dimensions is enough to generate spike-adding phenomena in transient responses, and that the onset of a new spike can be tracked using existing continuation packages. We take a geometric approach to illustrate how the underlying fast subsystem organises the spike adding in much the same way as for spike adding in periodic bursts, but the bifurcation analysis for spike onset is entirely different. By using a generic model, we further strengthen claims made in our earlier work that our numerical method for spike onset can be used for a broad class of systems.

Cellular responses are often characterised in terms of the sets of stimuli that increase or decrease their activity. The mathematical framework employed in studying such responses has put most emphasis on the stable states (point or periodic attractors) of a cellular system. In contrast, its transient (non-stationary) temporal behaviour has been given much less attention. Here, we focus on computational ideas that allow us to study the transient dynamics of cellular systems. Indeed, there are many situations in which the transient cellular behaviour manifests most of its functional significance, for example, while switching between different stable attractors. In order to understand better the relationship between response dynamics and system function, we need to investigate transient dynamics and not just the asymptotically stable states. In this paper, we focus on the generation of bursts, consisting of one or more spikes, as a result of a brief external stimulus. We use a simple polynomial model of bursting to illustrate our analysis. In particular, we show the geometry of the mathematical framework used to generate a new spike and we discuss how to detect numerically the onset of a new spike by continuation of a two-point boundary value problem (2PBVP). This means that we can compute and present transitions between topologically different transient responses in a bifurcation diagram.

We would like to thank the World-wide Universities Network scheme for facilitating our collaboration through research visits to the University of Bristol and The University of Auckland, respectively. The research of H.M.O. was supported by Grant No. UOA1113 of the Royal Society of New Zealand Marsden Fund. The research of K.T.-A. was supported by Grant No. EP/I018638/1 of the Engineering and Physical Sciences Research Council (EPSRC) and by AlterEgo, a project funded by the European Union Grant No. 600610.

I. INTRODUCTION

II. THE THREE-DIMENSIONAL POLYNOMIAL MODEL

A. Basic set-up of the 2PBVP in Auto

B. Computing spike-adding transitions

III. SPIKE ONSET ALONG SADDLE-TYPE SLOW MANIFOLD

A. Transition from one- to two-spike response

B. Transition from three- to four-spike response

IV. TWO-PARAMETER CURVES OF SPIKE ONSETS

V. DISCUSSION

### Key Topics

- Manifolds
- 46.0
- Bifurcations
- 37.0
- Time series analysis
- 10.0
- Nerve cells
- 9.0
- Boundary value problems
- 8.0

## Figures

Bifurcation diagram plotted in (z, y, x)-space of system (1) in the singular limit ε → 0. We used b = 0.75 and h = 1.0. The fast subsystem has a Z-shaped family of equilibria containing the saddle-node bifurcation points SN1 and SN2. The family also contains a subcritical Hopf bifurcation point H on the upper branch that gives rise to the family of periodic orbits that exhibit a SNP before it ends in a homolinic bifurcation.

Bifurcation diagram plotted in (z, y, x)-space of system (1) in the singular limit ε → 0. We used b = 0.75 and h = 1.0. The fast subsystem has a Z-shaped family of equilibria containing the saddle-node bifurcation points SN1 and SN2. The family also contains a subcritical Hopf bifurcation point H on the upper branch that gives rise to the family of periodic orbits that exhibit a SNP before it ends in a homolinic bifurcation.

Bifurcation diagram in the (b, h)-plane near b = 0. Additional equilibria of system (1) exist due to fold bifurcations, labelled SNFP and SNtop, which can become stable due to a Hopf bifurcation, labelled Htop.

Bifurcation diagram in the (b, h)-plane near b = 0. Additional equilibria of system (1) exist due to fold bifurcations, labelled SNFP and SNtop, which can become stable due to a Hopf bifurcation, labelled Htop.

Transient response generated for system (1) with b = 0.75 and h = 1.0. A perturbation with amplitude I app = 0.02 and duration T ON = 15 takes the system away from its equilibrium FP (grey segment). The relaxation back to FP (black segment) exhibits three additional spikes before reaching FP. The response is shown in (z, y, x)-space in panel (a), with the underlying bifurcation diagram of the fast subsystem for reference; see also Figure 1 . Panel (b) shows the corresponding time series for x with t [−50, 500].

Transient response generated for system (1) with b = 0.75 and h = 1.0. A perturbation with amplitude I app = 0.02 and duration T ON = 15 takes the system away from its equilibrium FP (grey segment). The relaxation back to FP (black segment) exhibits three additional spikes before reaching FP. The response is shown in (z, y, x)-space in panel (a), with the underlying bifurcation diagram of the fast subsystem for reference; see also Figure 1 . Panel (b) shows the corresponding time series for x with t [−50, 500].

As b decreases, the response changes from 1 spike to more and more spikes, with each spike-adding transition characterised by a strong increase in the L 2 integral norm. Responses for h = 1 and b = 1.15 (one spike), b = 1.0 (two spikes), b = 0.85 (three spikes) and b = 0.43 (nine spikes) are highlighted and their corresponding time series for x are shown in panels (b)–(e), respectively.

As b decreases, the response changes from 1 spike to more and more spikes, with each spike-adding transition characterised by a strong increase in the L 2 integral norm. Responses for h = 1 and b = 1.15 (one spike), b = 1.0 (two spikes), b = 0.85 (three spikes) and b = 0.43 (nine spikes) are highlighted and their corresponding time series for x are shown in panels (b)–(e), respectively.

Two viewpoints in panels (a) and (b) of the transient response at the onset from one to two spikes; we also show the bifurcation diagram of the fast subsystem, along with a subset of the family W ^{ s }(e M) of stable manifolds associated with saddle equilibria on the middle branch e M in between SN1 and the homoclinic bifurcation (not labelled). The response for b 1.07256 traces e M up to the fold point at SN2; panel (c) shows the corresponding time series for x with t [−50, 500].

Two viewpoints in panels (a) and (b) of the transient response at the onset from one to two spikes; we also show the bifurcation diagram of the fast subsystem, along with a subset of the family W ^{ s }(e M) of stable manifolds associated with saddle equilibria on the middle branch e M in between SN1 and the homoclinic bifurcation (not labelled). The response for b 1.07256 traces e M up to the fold point at SN2; panel (c) shows the corresponding time series for x with t [−50, 500].

Two viewpoints in panels (a) and (b) of the transient response at the onset from three to four spikes; we also show the bifurcation diagram of the fast subsystem, along with a subset of the family W^{s} (e M) of stable manifolds associated with saddle equilibria on the middle branch e M in between SN1 and just past the homoclinic bifurcation (not labelled). The response for b 0.778355 traces e M up to the fold point at SN2; panel (c) shows the corresponding time series for x with t [−50, 500].

Two viewpoints in panels (a) and (b) of the transient response at the onset from three to four spikes; we also show the bifurcation diagram of the fast subsystem, along with a subset of the family W^{s} (e M) of stable manifolds associated with saddle equilibria on the middle branch e M in between SN1 and just past the homoclinic bifurcation (not labelled). The response for b 0.778355 traces e M up to the fold point at SN2; panel (c) shows the corresponding time series for x with t [−50, 500].

Deformation of an orbit segment pair (u ON, u OFF) through the moment of the second spike onset. Here, h = 1 is fixed and b = 1 at the start of the continuation. Note how the end point transforms from a local maximum into a local minimum, which is detected as a fold bifurcation. Panel (a) shows a waterfall diagram of the orbit segments computed as part of the continuation; panel (b) shows the variation of T OFF (black) and b (grey), while panel (c) plots T OFF (black) and the end point z end (grey) as a function of N. The spike onset is detected at LP.

Deformation of an orbit segment pair (u ON, u OFF) through the moment of the second spike onset. Here, h = 1 is fixed and b = 1 at the start of the continuation. Note how the end point transforms from a local maximum into a local minimum, which is detected as a fold bifurcation. Panel (a) shows a waterfall diagram of the orbit segments computed as part of the continuation; panel (b) shows the variation of T OFF (black) and b (grey), while panel (c) plots T OFF (black) and the end point z end (grey) as a function of N. The spike onset is detected at LP.

Two-parameter bifurcation diagram indicating the regions in the (b,h)-plane with different numbers of additional spikes.

Two-parameter bifurcation diagram indicating the regions in the (b,h)-plane with different numbers of additional spikes.

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