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mathematics

Heart rate

Heart rate 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 Heart rate rather than just read about it. In short: Heart rate is the frequency of the heartbeat measured by the number of contractions of the heart per minute (beats per minute or bpm). The heart rate varies according to the body's physical needs, including the need to absorb oxygen and excrete carbon dioxide.

Heart rate — main illustration
Heart rate — illustration

Key takeaways

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

Reference excerpt

Heart rate is the frequency of the heartbeat measured by the number of contractions of the heart per minute (beats per minute or bpm). The heart rate varies according to the body's physical needs, including the need to absorb oxygen and excrete carbon dioxide. It is also modulated by numerous factors, including genetics, physical fitness, stress or psychological status, diet, drugs, hormonal status, environment, and disease/illness, as well as the interaction between these factors. It is usually equal or close to the pulse rate measured at any peripheral point. The American Heart Association states the normal resting adult human heart rate is 60–100 bpm. An ultra-trained athlete may have a resting heart rate of 37–38 bpm. Tachycardia is a high heart rate, defined as above 100 bpm at rest. Bradycardia is a low heart rate, defined as below 60 bpm at rest. When a human sleeps, a heartbeat with rates around 40–50 bpm is common and considered normal. When the heart is not beating in a regular pattern, this is referred to as a arrhythmia. Abnormalities of heart rate sometimes indicate heart disease.

Physiology

While heart rhythm is regulated entirely by the sinoatrial node under normal conditions, heart rate is regulated by sympathetic and parasympathetic input to the sinoatrial node. The accelerans nerve provides sympathetic input to the heart by releasing norepinephrine onto the cells of the sinoatrial node (SA node), and the vagus nerve provides parasympathetic input to the heart by releasing acetylcholine onto sinoatrial node cells. Therefore, stimulation of the accelerans nerve increases heart rate, while stimulation of the vagus nerve decreases it.

Central nervous system stimulants such as substituted amphetamines increase heart rate. Central nervous system depressants or sedatives decrease the heart rate (apart from some particularly strange ones with equally strange effects, such as ketamine which can cause – amongst many other things – stimulant-like effects such as tachycardia). There are many ways in which the heart rate speeds up or slows down. Most involve stimulant-like endorphins and hormones being released in the brain, some of which are those that are 'forced'/'enticed' out by the ingestion and processing of drugs such as cocaine or atropine. This section discusses target heart rates for healthy persons, which would be inappropriately high for most persons with coronary artery disease.

Influences from the central nervous system

Cardiovascular centres The heart rate is rhythmically generated by the sinoatrial node. It is also influenced by central factors through sympathetic and parasympathetic nerves. Nervous influence over the heart rate is centralized within the two paired cardiovascular centres of the medulla oblongata. The cardioaccelerator regions stimulate activity via sympathetic stimulation of the cardioaccelerator nerves, and the cardioinhibitory centers decrease heart activity via parasympathetic stimulation as one component of the vagus nerve. During rest, both centers provide slight stimulation to the heart, contributing to autonomic tone. This is a similar concept to tone in skeletal muscles. Normally, vagal stimulation predominates as, left unregulated, the SA node would initiate a sinus rhythm of approximately 100 bpm. Both sympathetic and parasympathetic stimuli flow through the paired cardiac plexus near the base of the heart. The cardioaccelerator center also sends additional fibers, forming the cardiac nerves via sympathetic ganglia (the cervical ganglia plus superior thoracic ganglia T1–T4) to both the SA and AV nodes, plus additional fibers to the atria and ventricles. The ventricles are more richly innervated by sympathetic fibers than parasympathetic fibers. Sympathetic stimulation causes the release of the neurotransmitter norepinephrine (also known as noradrenaline) at the neuromuscular junction of the cardiac nerves. This shortens the repolarization period, thus speeding the rate of depolarization and contraction, which results in an increased heartrate. It opens chemical or ligand-gated sodium and calcium ion channels, allowing an influx of positively charged ions. Norepinephrine binds to the beta–1 receptor. High blood pressure medications are used to block these receptors and so reduce the heart rate.

Parasympathetic stimulation originates from the cardioinhibitory region of the brain with impulses traveling via the vagus nerve (cranial nerve X). The vagus nerve sends branches to both the SA and AV nodes, and to portions of both the atria and ventricles. Parasympathetic stimulation releases the neurotransmitter acetylcholine (ACh) at the neuromuscular junction. ACh slows HR by opening chemical- or ligand-gated potassium ion channels to slow the rate of spontaneous depolarization, which extends repolarization and increases the time before the next spontaneous depolarization occurs. Without any nervous stimulation, the SA node would establish a sinus rhythm of approximately 100 bpm. Since resting rates are considerably less than this, it becomes evident that parasympathetic stimulation normally slows HR. This is similar to an individual driving a car with one foot on the brake pedal. To speed up, one need merely remove one's foot from the brake and let the engine increase speed. In the case of the heart, decreasing parasympathetic stimulation decreases the release of ACh, which allows HR to increase up to approximately 100 bpm. Any increases beyond this rate would require sympathetic stimulation.

… excerpt ends here. Continue reading the full article.

Illustrations

Heart rate: A medical monitoring device displaying a normal human heart rate
A medical monitoring device displaying a normal human heart rate
Heart rate: The human heart
The human heart
Heart rate illustration
Heart rate: Autonomic innervation of the heart: Cardioaccelerator and cardioinhibitory areas are components of the paired cardiac centers located in the medulla oblongata of the brain. They innervate the heart via sympathetic cardiac nerves that increase cardiac activity and vagus (parasympathetic) nerves that slow cardiac activity.[12]
Autonomic innervation of the heart: Cardioaccelerator and cardioinhibitory areas are components of the paired cardiac centers located in the medulla oblongata of the brain. They innervate the heart via sympathetic cardiac nerves that increase cardiac activity and vagus (parasympathetic) nerves that slow cardiac activity.[12]
Heart rate: Effects of parasympathetic and sympathetic stimulation on normal sinus rhythm: The wave of depolarization in a normal sinus rhythm shows a stable resting HR. Following parasympathetic stimulation, HR slows. Following sympathetic stimulation, HR increases.[12]
Effects of parasympathetic and sympathetic stimulation on normal sinus rhythm: The wave of depolarization in a normal sinus rhythm shows a stable resting HR. Following parasympathetic stimulation, HR slows. Following sympathetic stimulation, HR increases.[12]

Worked examples

Example 1 — a first encounter with Heart rate

Start with the simplest possible case. Write down what Heart rate 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 Heart rate 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 Heart rate 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 Heart rate

In research
Heart rate 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 Heart rate 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
Heart rate is common in secondary-school and first-year university syllabi. It links to neighbouring topics Cardiovascular physiology, Mathematics in medicine, Medical signs, so understanding it makes those chapters shorter.
In everyday life
Look for Heart rate 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 Heart rate in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Heart rate 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 Heart rate out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Heart rate in simple terms?

Heart rate is the frequency of the heartbeat measured by the number of contractions of the heart per minute (beats per minute or bpm). The heart rate varies according to the body's physical needs, including the need to absorb oxygen and excrete carbon dioxide.

Why does Heart rate 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 Heart rate?

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 Heart rate.

Tags

  • Cardiovascular physiology
  • Mathematics in medicine
  • Medical signs
  • Temporal rates

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