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Path integration

Path integration is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Path integration rather than just read about it. In short: Path integration is the method thought to be used by animals for dead reckoning. History Charles Darwin first postulated an inertially-based navigation system in animals in 1873.

Path integration — main illustration
Path integration — illustration

Key takeaways

  • Path integration belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Path integration to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Path integration from memory before moving on to harder problems.

Reference excerpt

Path integration is the method thought to be used by animals for dead reckoning.

History Charles Darwin first postulated an inertially-based navigation system in animals in 1873. Studies beginning in the middle of the 20th century confirmed that animals could return directly to a starting point, such as a nest, in the absence of vision and having taken a circuitous outwards journey. This shows that they can use cues to track distance and direction in order to estimate their position, and hence how to get home. This process was named path integration to capture the concept of continuous integration of movement cues over the journey. Manipulation of inertial cues confirmed that at least one of these movement (or idiothetic) cues is information from the vestibular organs, which detect movement in the three dimensions. Other cues probably include proprioception (information from muscles and joints about limb position), motor efference (information from the motor system telling the rest of the brain what movements were commanded and executed), and optic flow (information from the visual system signaling how fast the visual world is moving past the eyes). Together, these sources of information can tell the animal which direction it is moving, at what speed, and for how long. In addition, sensitivity to the Earth's magnetic field for underground animals (e.g., mole rat) can give path integration.

Mechanism

Studies in arthropods, most notably in the Sahara desert ant (Cataglyphis bicolor), reveal the existence of highly effective path integration mechanisms that depend on determination of directional heading (by polarized light or sun position) and distance computations (by monitoring leg movement or optical flow). In mammals, three important discoveries shed light on this. The first, in the early 1970s, is that neurons in the hippocampal formation, called place cells, respond to the position of the animal. The second, in the early 1990s, is that neurons in neighboring regions (including anterior thalamus and post-subiculum), called head direction cells, respond to the head direction of the animal. This enables a much more fine-grained study of path integration since it is possible to manipulate movement information and see how place and head direction cells respond (a much simpler procedure than training an animal, which is very slow). The third finding was that neurons in the dorso-medial entorhinal cortex, which feeds information to the place cells in the hippocampus, fire in a metrically regular way across the whole surface of a given environment. The activity patterns of these grid cells looks very much like a hexagonally organized sheet of graph paper, and suggest a possible metric system that place cells can use to compute distances. Whether place and grid cells actually compute a path integration signal remains to be seen, but computational models exist suggesting this is plausible. Certainly, brain damage to these regions seems to impair the ability of animals to path integrate. David Redish states that "The carefully controlled experiments of Mittelstaedt and Mittelstaedt (1980) and Etienne (1987) have demonstrated conclusively that this ability [path integration in mammals] is a consequence of integrating internal cues from vestibular signals and motor efferent copy".

See also Cognitive map Motion perception

References

Bibliography Best PJ, White AM, Minai A (2001). "Spatial processing in the brain: the activity of hippocampal place cells". Annu Rev Neurosci. 24: 459–486. doi:10.1146/annurev.neuro.24.1.459. PMID 11283318. Etienne AS, Jeffery KJ (2004). "Path integration in mammals" (PDF). Hippocampus. 14 (2): 180–192. doi:10.1002/hipo.10173. PMID 15098724. S2CID 1646974. Archived from the original (PDF) on 2017-08-13. Retrieved 2019-09-11. McNaughton BL, Battaglia FP, Jensen O, Moser EI, Moser MB (2006). "Path integration and the neural basis of the 'cognitive map'". Nat Rev Neurosci. 7 (8): 663–678. doi:10.1038/nrn1932. PMID 16858394. S2CID 16928213. Redish, A David (1999). Beyond the Cognitive Map. MIT Press. PDF Taube JS (1998). "Head direction cells and the neurophysiological basis for a sense of direction". Prog Neurobiol. 55 (3): 225–256. doi:10.1016/S0301-0082(98)00004-5. PMID 9643555. S2CID 38083940.

Illustrations

Path integration: Path integration sums the vectors of distance and direction traveled from a start point to estimate current position, and so the path back to the start.
Path integration sums the vectors of distance and direction traveled from a start point to estimate current position, and so the path back to the start.

Worked examples

Example 1 — a first encounter with Path integration

Start with the simplest possible case. Write down what Path integration claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Path integration before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Path integration ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Path integration

In research
Path integration appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Path integration in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Path integration is common in secondary-school and first-year university syllabi. It links to neighbouring topics Animal migration, Cognitive neuroscience, Orientation (geometry), so understanding it makes those chapters shorter.
In everyday life
Look for Path integration outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Path integration in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Path integration means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Path integration out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Path integration in simple terms?

Path integration is the method thought to be used by animals for dead reckoning. History Charles Darwin first postulated an inertially-based navigation system in animals in 1873.

Why does Path integration matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Path integration?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Path integration.

Tags

  • Animal migration
  • Cognitive neuroscience
  • Orientation (geometry)
  • Spatial cognition

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