Gottfried Leibniz, Carl Gauss and Neural Coding

Outline of talk to students of German language classes at Bozeman High School May 9, 2002

Gottfried Wilhelm Leibniz

Daddy of Calculus 1646-1716 • • • • • Selected Chronology of Leibniz’s Life: Born in Leipzig , Germany Dad: Friedrich Mom: Catherine Schmuck Education – At 14: attended University of Leipzig (1661) – At 17: Degree (undergrad) in philosophy and mathematics (UL) (1663) – Masters in Philosophy (UL) (1663 or 1664) – At 20: Doctorate from University of Altdorf (1667) Vocation: Diplomat for Germany traveling to the capitols of Europe.

Daddy of Calculus – Discovered in 1676 – Published “A New Method for Maxima and Minima and also for tangents which is not obstructed by Irrational Quantities” in 1684 – Controversy with Sir Isaac Newton (the other Daddy of Calculus) discovered the calculus in 1665 and published his version of it in his

Philosophae Naturalis Mathematica

economics, biology, sociology, psycology, engineering … in 1687.

– Profound discovery with much generality! Today it is used profusely in statistics, business,

• • • • •

What is Calculus?

Calculus deals with extremely small (large) things, a concept that has eluded mathematicians since the 6 th century BC: – Pythagoreans (550 BC) • Right triangles and the Pythagorean Theorem.

– 3 2 + 4 2 = 5 2 !

– 3 2 + 5 2 = sqrt(34) 2 . Irrational quantities from square roots • Pythagoreans believed that there was a fraction which was

equal

to sqrt(34). The fact is that

sqrt(34) is NOT a fraction,

but you can get as (

infinitesimally

) close to sqrt(34) as you’d like with some fraction.

• The ancient Greeks could not deal (rigorously) with the concept of the infinitesimal (or of the infinite).

– Paradoxes of Zeno (450 BC) • Achilles and the Tortoise – a paradox of the infinitesimal!

Mr. J. drives to Sun Valley. – Distance curve – Steepness of the curve = how fast Mr. J. is deiving – Quantify this!! Steepness of curve = steepness (slope) of the tangent line to the curve – Where’s the infinitesimal? Ans: How do you get the tangent line?

Theme: Fractions get close to square roots, Achilles gets close to the tortoise, secants get close to the tangent.

POINT: Calculus is about the steepness of tangent lines.

These kids know about simple polynomials like lines and parabolas

• • • •

Johann Carl Friedrich Gauss

Prince of Mathematics

“Math is the queen of the sciences” 1777-1885 At 10, he summed 1, 2, 3, … 100 using n/2(n+1).

Education: – At 11: attended Gymnasium funded by the Duke of Brunswick, Wolfenbottel (1788) – Degree (undergrad) at Brunswick Collegium Carolinum (1799) – Doctorate at University of Helmstedt (1800-1801?) • Fundamental Theorem of Algebra (thesis): A polynomial of degree n has n roots . Vocation: director of Go ..

ttingen .

– Accurately predicted where Ceres, a new “small planet” (asteroid) discovered on Jan 1, 1801 by Italian G. Piazzi, would reappear the next year.

Prince of Mathematics: – Gaussian: unit of magnetic flux – Gaussian Numbers (“complex integers”) – Gauss’s Lemmas (Algebra) and Gauss’s Theorems (Complex) and Gaussian Plane and Gaussian Distribution

Gaussian Distribution • Weights of students in the class follow a bell shaped curve!

• Lots of data follow this distribution!

• CLT

Neural Coding

• How does the nervous system communicate information about the environment?

– You are (maybe) looking and listening to me!

– Luke Skywalker gets his hand cut off in Empire Strikes Back, Princess Lea rescues him, then a robot sews a new hand on then pricks his prosthetic with a needle. Luke says: “Ow!” • This is possible only if the prosthetic can successfully communicate pain (and heat and tactile info) with the nervous system.

• How do the great German mathematicians Leibniz and Gauss help neurobiologists try to answer this question?

German Math Guys and Neural Coding

• Start with a model (like a line or parabola!) • Leibniz’s Calculus helps us find where the steepness of the curve is “flat” – Like finding the vertex of a parabola – These are Max’s and Min’s!

– This is precisely what we do with our model for neural coding!

• Gaussian Distribution: Given a stimulus, there are lots of responses of the nervous system. We order the responses using Gauss’s bell shaped curve!