# Ring Attractors as the Basis of a Biomimetic Navigation System

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ring Attractor Model

#### 2.1.1. Direct Inhibition of Conjunctive Input

^{−1}would cause the bump to collapse. Clearly, it would be detrimental to lose path integration state every time the animal stops, which suggests that the ring attractor requires a ‘floor’ of constant velocity input to maintain its activity pattern. Recordings in vivo support this; cells that display Conjunctive Cell-like activity show a rate of spiking that, though proportional to velocity, does not diminish entirely when the animal is at rest [32]. This forms one half of the inspiration for the development of antagonistic input signals discussed in Section 2.1.2.

#### 2.1.2. Antagonistic Input Signals

- At rest, a baseline level of excitability is provided, with each Conjunctive population trying to push the bump left or right with equal intensity. This holds the bump ‘in tension’; ready to move, yet in balance.
- Changes in incoming velocity proportionally increase input to the population associated with its sign, with a equal and proportional reduction to the opposing population. Baseline input is increased proportional to velocity magnitude; speed coding [15].
- When a change in velocity direction occurs, the respective Conjunctive Cells are already close to the required spike rate to accurately convey the velocity information to the bump and induce appropriate movement.

#### 2.1.3. Multiple Rings for Multiple Components

#### 2.2. Predictive Coding Network

#### 2.3. Associative Memory

#### 2.3.1. Deriving Position from Ring Attractor State

#### 2.4. Compensating for Drift in the System

#### 2.5. Variable Power Consumption

#### 2.6. Introducing Variance

#### 2.7. Confidence

#### 2.8. Influence of Allothetic Sensory Modality

## 3. Results

#### 3.1. Spiking Ring Attractors Can Track Position

#### 3.2. Spiking Ring Attractors Benefit from Corrective Input

#### 3.3. Higher Power Consumption Reduces Error

- A certain amount of current is required to establish the attractor dynamics and generate any bump at all; currents below this fall into the ‘dead zone’
- The error declines proportional to input current for a certain range of values only
- Past a certain value, extra input current has an inconsistent effect on performance

#### 3.4. Correction Intensity as a Proxy for Confidence

#### 3.5. Sensory Modality of Memories Affects Corrective Performance

## 4. Discussion

- The corrective influences being injected as step currents, rather than spike trains
- A bio-plausible model of memory that can be used to trigger corrective inputs.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SNN | Spiking Neural Network |

PCN | Predictive Coding Network |

## References

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**Figure 1.**WhiskEye is a biomimetic robot platform that mimics the sensory systems and behaviour of rats and shrews. It has been recreated in simulation along with rich visuo-tactile environments. Exploring these environments autonomously, it gathers multisensory datasets along with odometry data describing its trajectory. These ‘experiences’ of the environment are later used as corrective inputs.

**Figure 2.**(

**A**) The ring model is composed of repeating units of Excitatory, Inhibitory and Conjunctive Cells that work together to integrate an input into a unitary ring state. (

**B**) A single ring integrating a component of a trajectory (blue trace, from [15]) will have its Excitatory Cells spike (orange) in regular intervals along an axis, similar to recorded Stripe Cells [31]. A cell taking input from three rings, each integrating a different component of the same trajectory, produces Grid Cell-like firing patterns, suggesting that three rings are collectively able to track position in a 2D plane.

**Figure 3.**An example of the visualisation used for the ring tuning and evaluation process, with ring principle axes being aligned to ${0}^{\circ}$, ${60}^{\circ}$ and ${120}^{\circ}$ offsets from the reference axis as described in Section 2.1.3. Polar plots represent the three ring attractors with ring activity composed of spikes gathered over a 100 ms window, with their unitary activity bumps clearly visible. Trajectory plots show the ground truth trajectory the rings are trying to track, with the Cartesian transformation of the ring states overlaid as per Section 2.3.1. The error plot shows the cumulative error of the uncorrected and corrected ring attractor estimates, with each timestamp representing 20 ms of simulation time. An animated version of this figure is provided at https://github.com/TomKnowles1994/Biomimetics-Ring-Attractors.

**Figure 4.**The effect of ‘confidence scaling’ the injected currents on mean error. When enabled, the injected current will be proportional to the Pearson’s Correlation between the incoming sensory view and best-matching recalled memory. F-statistic and p-value taken from an F-test ($n=100$, $\alpha =0.05$). ${C}_{\mathrm{min}}$ was set to 2700.

**Figure 5.**The effect of varying baseline current input ${C}_{\mathrm{min}}$, a proxy for power consumption, on mean error. Test values increment by 10 $\mathrm{p}$$\mathrm{A}$ with 10 runs conducted for each value. Error is given as the cumulative Euclidean distance between the ground truth trajectory and the corresponding ring state-estimated trajectory. The dead zone represents no activity in the ring, due to a lack of sufficient input current. Confidence Scaling applied as per Section 3.4.

**Figure 6.**The effect of correction modality—the sensory data used to form the experience memories—on performance, benchmarked against the uncorrected run. Confidence Scaling applied as per Section 3.4. F-statistic and p-value taken from an F-test ($n=100$, $\alpha =0.05$). ${C}_{\mathrm{min}}$ was set to 2700.

**Table 1.**The parameters for the Random Number Generators used in the simulations. After each condition was tested for its requisite number of runs, the seed would be reset to its Seed Start value.

RNG | Seed Start | Distribution | Mean | Variance |
---|---|---|---|---|

Master | 9032867582 | - | - | - |

Membrane | 2390786556 | Uniform | - | [−70, +55] |

Trajectory | 6983476394 | Normal | 0 | 0.1 |

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**MDPI and ACS Style**

Knowles, T.C.; Summerton, A.G.; Whiting, J.G.H.; Pearson, M.J.
Ring Attractors as the Basis of a Biomimetic Navigation System. *Biomimetics* **2023**, *8*, 399.
https://doi.org/10.3390/biomimetics8050399

**AMA Style**

Knowles TC, Summerton AG, Whiting JGH, Pearson MJ.
Ring Attractors as the Basis of a Biomimetic Navigation System. *Biomimetics*. 2023; 8(5):399.
https://doi.org/10.3390/biomimetics8050399

**Chicago/Turabian Style**

Knowles, Thomas C., Anna G. Summerton, James G. H. Whiting, and Martin J. Pearson.
2023. "Ring Attractors as the Basis of a Biomimetic Navigation System" *Biomimetics* 8, no. 5: 399.
https://doi.org/10.3390/biomimetics8050399