Elsevier

Physics Letters A

Volume 328, Issues 2–3, 26 July 2004, Pages 177-184
Physics Letters A

Subthreshold oscillations in a map-based neuron model

https://doi.org/10.1016/j.physleta.2004.05.062Get rights and content

Abstract

Self-sustained subthreshold oscillations in a discrete-time model of neuronal behavior are considered. We discuss bifurcation scenarios explaining the birth of these oscillations and their transformation into tonic spikes. Specific features of these transitions caused by the discrete-time dynamics of the model and the influence of external noise are discussed.

Introduction

Studies of dynamical behavior of biological networks require numerical simulations of arrays containing a very large number of neurons. Despite the variety of physiological processes involved in the formation of neuron activity, the thorough studies of the large-scale networks need simple phenomenological models that can replicate the dynamics of individual neurons. Various suggestions for the design of low-dimensional maps for modeling the neurons' behavior have been proposed, see, for example, [1], [2], [3], [4], [5], [6] and references therein. Most of them were focused on the replication of either fast spikes or relatively slow bursts while the mechanisms for generation of specific footprints of spikes were neglected.

A simple discrete-time model replicating the spiking-bursting neural activity has been suggested recently in [7]. This model is a 2D map that mimics rather realistically various types of transitions that occur in biological neurons. These transitions include routes between silence and tonic spiking as well as a triplet: silence ↔ bursts of spikes ↔ tonic spiking. Such simple phenomenological models bear a high potential for further developments of computationally efficient methods for studies of functional behavior in large-scale neurobiological networks [8].

The bifurcation analysis of the map model carried out in [9] has shown that the transition from silence (a stable fixed point) to generation of action potentials is characterized by a sub-critical Andronov–Hopf bifurcation when an unstable invariant closed curve collapses into the stable fixed point. Therefore, the original map-model [7] provides only an abrupt transition from silence to spiking as a control parameter (e.g., the depolarization current) passes the excitability threshold. This scenario is quite typical for most types of biological neurons. However, experimental studies suggest that some neurons may come out of the silence softly through the regime of small oscillations below the threshold of the spike excitation [10]. These subthreshold oscillations of almost sinusoidal form facilitate the generation of spike oscillations when the membrane gets depolarized or hyperpolarized [11], [12]. These small oscillations can play an important role in shaping specific forms of rhythmic activity that are vulnerable to the noise in the network dynamics [15], [16].

In this Letter we modify the map model so that it can generate stable subthreshold oscillations. We start the discussion of the model dynamics with the analysis of local bifurcations of a fixed point of the map. We show that the loss of stability of the fixed point is accompanied by the birth of the stable invariant circle which initiate a family of canards in the map. Further evolution of the circle leads to a breakdown of the invariant circle that gives rise to chaos. We also elaborate on the mechanism of the onset of irregular spiking which is due to heteroclinic-like crossings between the stable and unstable invariant sets, which are the images of the slow motion “surfaces” in the unperturbed map. The role played by small subthreshold oscillations in the responsiveness of the map to external noise is considered.

Section snippets

Map-based model with stable subthreshold oscillations

The map-based model of spiking-bursting neuron oscillations, following [7], can be written in the form of the two-dimensional map

x̄=fα(x,y+β),ȳ=y−μ(x+1−σ), where the x-variable replicates the dynamics of the membrane potential, the parameters α, σ and 0< μ⪡1 control individual dynamics of the system. Some input parameters β and σ are employed to provide coupling with other such models afterwards; both stand for injected currents. The principal distinction of the original map analyzed in [7],

Tangles of critical curves and chaos

In the numerical simulations of map , we found that subthreshold oscillations may be interrupted by irregular spiking even in the absence of external noise. An example of such a behavior is presented in Fig. 6. This intermittent dynamics is observed only within a rather thin parameter interval at the border between the regimes of continuous subthreshold oscillations and tonic spike generation. To understand the dynamical mechanisms behind this sporadic spiking we numerically studied the

Conclusion

It is shown that a simple map can be employed to replicate the behavior of neurons with self-sustained subthreshold oscillations. These oscillations are achieved by a special selection of a non-linear function in the fast subsystem. The dynamical mechanisms behind the transitions from silence to subthreshold oscillations of small amplitude and then to spiking activity are explained using the bifurcation approach.

Here we focused mostly on the individual dynamics of the map-based model. As a

Acknowledgements

N.R. was supported in part by US Department of Energy (grant DE-FG03-95ER14516). A.S. acknowledges the RFBR grants Nos. 02-01-00273 and 01-01-00975.

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