Equilibrium States and Their Stability in the Head-Direction Ring Network.

Caixia Wang,Kechen Zhang

doi:10.3389/fncom.2019.00096

Abstract

Head-direction cells have been found in several areas in the mammalian brains. The firing rate of an ideal head-direction cell reaches its peak value only when the animal's head points in a specific direction, and this preferred direction stays the same regardless of spatial location. In this paper we combine mathematical analytical techniques and numerical simulations to fully analyze the equilibrium states of a generic ring attractor network, which is a widely used modeling framework for the head-direction system. Under specific conditions, all solutions of the ring network are bounded, and there exists a Lyapunov function that guarantees the stability of the network for any given inputs, which may come from multiple sources in the biological system, including self-motion information for inertially based updating and landmark information for calibration. We focus on the first few terms of the Fourier series of the ring network to explicitly solve for all possible equilibrium states, followed by a stability analysis based on small perturbations. In particular, these equilibrium states include the standard single-peaked activity pattern as well as double-peaked activity pattern, whose existence is unknown but has testable experimental implications. To our surprise, we have also found an asymmetric equilibrium activity profile even when the network connectivity is strictly symmetric. Finally we examine how these different equilibrium solutions depend on the network parameters and obtain the phase diagrams in the parameter space of the ring network.

Highlights

Head-direction cells were first reported in several brain areas related to the limbic system in the rodents (Taube, 2007) and later in other mammalian species such as monkeys (Robertson et al, 1999) and bats (Finkelstein et al, 2015)
The equilibrium state of the head-direction ring network is often visualized as a single bump of activity whose peak position corresponds to the animal’s current heading direction (Figure 1, top and middle rows)
Imagine that the ring network can sustain two activity bumps instead of one (Figure 1, bottom row), if one records from an individual cell in the ring, one would still find a head-direction cell with a perfectly normal, single-peaked tuning curve, assuming that the activity bumps rotates at half of the speed as the single activity bump

Summary

Introduction

Head-direction cells were first reported in several brain areas related to the limbic system in the rodents (Taube, 2007) and later in other mammalian species such as monkeys (Robertson et al, 1999) and bats (Finkelstein et al, 2015). Besides head-direction cells, attractor networks have been used as a general theoretical framework for modeling other types of spatial cells in the hippocampus and related systems (Knierim and Zhang, 2012). The equilibrium state of the head-direction ring network is often visualized as a single bump of activity whose peak position corresponds to the animal’s current heading direction (Figure 1, top and middle rows). While this picture is compelling and highly intuitive, it is not the only theoretical possibility for explaining the experimental data. If we focus on a single cell corresponding to north, we see that in both situations, the cell fires at maximal rate only when the animal is facing north (N)

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Methods

Conclusion