Elsevier

Neurocomputing

Volumes 26–27, June 1999, Pages 533-539
Neurocomputing

Traveling waves in a ring of three inhibitory coupled model neurons

https://doi.org/10.1016/S0925-2312(99)00031-4Get rights and content

Abstract

The pyloric network of the stomatogastric ganglion of the lobster generates motor patterns with specific phase-lags between single neurons. This network inspired us to investigate a simplified model consisting of three mono-directionally coupled Morris–Lecar Oscillators. We have systematically analyzed the high-dimensional space of the synaptic parameters and identified parameter combinations which lead to biologically plausible phase-lags that exist even in a network with identical cells in the absence of an intrinsic burster. The dependence of the phase lags on the synaptic parameters was also explored.

Introduction

A network of three model neurons, coupled with mono-directional inhibition could, in principal, be used to control various asynchronous movement sequences by producing patterns with different phase lags. This makes its behavior interesting with respect to biological as well as artificial movement control. A related system is the pyloric network in the stomatogastric (STG) ganglion of the lobster (Jasus lalandii), which consists of 14 STG neuron and controls rhythmic contractions of the pyloric region of the foregut (Fig. 1A, B) [2], [3]. The dilator group consists of pyloric dilator neurons (PD) and an anterior burster (AB). The anterior constrictor group consists of a lateral pyloric neuron (LP) and the posterior constrictor group is formed by pyloric neurons (PY). The neurons within each group tend to be synchronously active due to electric coupling. The synapses are inhibitory. This network generates oscillations which are characterized by the phase lags between the different groups, defined as: x-phase=(time between onset of PD-burst and x-burst)/(pyloric period), where x stands for either LP or PY and the pyloric period is the time between onsets of two adjacent PD-bursts (Fig. 1A). Typical values are: LP-phase=0.4, PY-phase=0.7 and pyloric frequency=1 Hz [4]. A hybrid network was constructed by replacing PD with a model neuron, implemented on a computer and interfaced with artificial synapses [3]. The network architecture was reduced to a ring of mono-directionally coupled neurons (Fig. 1C, D). The typical phase lags of the hybrid network are 0.55–0.6 for LP and 0.76–0.8 for PY. We consider a model (Fig. 1D) of the same architecture that additionally produces patterns like the pyloric network. Examination of the synaptic properties of this model enabled us to investigate under which circumstances the model can produce a pyloric pattern, and furthermore to find out how the phase lags of the asynchronous traveling waves that the model produces depend on the synaptic parameters.

Section snippets

Methods

Each neuron is modeled by the reduced Morris–Lecar equations (Appendix). They are a set of two differential equations originally formulated to model the barnacle giant muscle fiber [5] and represent an abstraction rather than a detailed biophysical model of the STG neuron groups. The voltage variable V (Appendix) models the envelope of the spiking. Therefore, an oscillation of V indicates a burst. This simplification is justified, because the envelope expresses the features of the pattern which

Results

First, the PD-cell is modeled as an intrinsic burster, while the other two cells are modeled as tonic firing cells, when uncoupled and the synapses are not identical (Fig. 3.1–3 and Fig. 4, Fig. 5). The model reproduces a pattern with phase lags that match the ones of the hybrid network (Fig. 3.1). Stable solutions with such phase lags occur in an extended region in parameter space (Fig. 4). In this configuration the network can produce a variety of asynchronous traveling waves, the phase lags

Conclusion

A ring of three model neurons, coupled with mono-directional inhibition produces traveling waves with phase lags typical of those measured in the pyloric network of the lobster STG, only when the synaptic coupling is not identical. This is independent of whether or not the ring contains an intrinsically bursting cell. In the presence of an intrinsically bursting cell and non-identical synapses the phase lags of the asynchronous traveling waves depend smoothly on the parameters gsyn and τs of

Susanne Still studied physics at the University of Hannover, Germany and at the ETH Zürich, Switzerland. She received her degree (Dipl. Phys.) in 1995. Since then she works as a doctoral student at the Institute of Neuroinformatics in Zürich.

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Susanne Still studied physics at the University of Hannover, Germany and at the ETH Zürich, Switzerland. She received her degree (Dipl. Phys.) in 1995. Since then she works as a doctoral student at the Institute of Neuroinformatics in Zürich.

G. Le Masson was trained as a neurologist in Bordeaux, France and did his Ph.D. on Neural Network Dynamics using both experimental and theoretical tools followed by a post-doc at Brandeis University, USA. He is now working on sensory integration in spinal cord and thalamic relay nuclei using hybrid networks of interconnected real neurons and models in the National Institute for Medical Research (INSERM) in Bordeaux.

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Supported by the CSEM and the SNSF SPP.

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