Connections Between Mathematics and Biology Carl Cowen Purdue University and the Mathematical Biosciences Institute

Connections Between Connections Between Mathematics and BiologyMathematics and Biology

Carl CowenCarl Cowen Purdue UniversityPurdue University and the and theMathematical Biosciences InstituteMathematical Biosciences Institute

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IntroductionIntroduction

Some areas of applicationSome areas of application

Example from neuroscience:Example from neuroscience: the Pulfrich Effect the Pulfrich Effect

IntroductionIntroduction• Explosion in biological research and Explosion in biological research and

progressprogress

• The mathematical sciences will be a partThe mathematical sciences will be a part

• Opportunity: Opportunity: few mathematical scientists are few mathematical scientists are biologically educated biologically educated few biological scientists are few biological scientists are mathematically educated mathematically educated

Colwell: “We're not near the fulfillment of biotechnology's promise. We're just on the cusp of it…”


progressprogress



Report Bio2010: “How biologists design, perform, and analyze experiments is changing swiftly. Biological concepts and models are becoming more quantitative…”


progressprogress



NSF/NIH Challenges: “Emerging areas transcend traditional academic boundaries and require interdisciplinary approaches that integrate biology, mathematics, and computer science.”

Some areas of application of Some areas of application of math in the biosciencesmath in the biosciences• Genomics and proteomicsGenomics and proteomics

• Description of intra- and inter-cellular Description of intra- and inter-cellular processesprocesses

• Growth and morphologyGrowth and morphology

• Epidemiology and population dynamicsEpidemiology and population dynamics

• NeuroscienceNeuroscience

Poincare: “Mathematics is the art of giving the same name to different things.”

























The Pulfrich EffectThe Pulfrich Effect

An experiment!An experiment!

Carl Pulfrich (1858-1927)Carl Pulfrich (1858-1927) reported effect and gave explanation reported effect and gave explanation in 1922 in 1922

F. Fertsch experimented, showedF. Fertsch experimented, showed Pulfrich why it happened, and was Pulfrich why it happened, and was given the credit for it by Pulfrich given the credit for it by Pulfrich


• The brain processes signals together that arrive from the two eyes at the same time

• The signal from a darker image is sent later than the signal from a brighter image, that is, signals from darker images are delayed

Hypothesis suggested Hypothesis suggested by neuro-physiologists: by neuro-physiologists:


filter


filter

s

d

x

s

• x, d, , and are all functions of time, but we’ll skip that for now

• s is fixed: you can’t move your eyeballs further apart •The brain “knows” the values of , , and s

• The brain “wants to calculate” the values of x and d

s

d

x

s

• x + s = tan d

s

d

x

s

• x + s = tan d

• x - s = tan d

s

d

x

s

• x + s = tan d

• x - s = tan d

• 2s = tan d - tan d

• d = 2s/(tan - tan )

• 2x = tan d + tan d

• x = d(tan + tan )/2

• x = s(tan + tan ) / (tan - tan )

s

d

x

s

• x + s = tan d

• x - s = tan d

• tan d = x + s

• tan = (x + s)/d

• = arctan( (x + s)/d )

• = arctan( (x - s)/d )

s

d

x(t)

s

• = arctan( (x(t-) + s)/d )

• = arctan( (x(t) - s)/d )

x(t-)

• x(t),d = actual position at time t

• x(t-),d = actual position

at earlier time t-

s

d

y(t)

s

• = arctan( (x(t-) + s)/d )

• = arctan( (x(t) - s)/d )

• e(t) = 2s / (tan - tan )

• y(t) = s(tan + tan ) / (tan - tan )

• x(t),d = actual position at time t

• x(t-),d = actual position at earlier time t-• y(t),e(t) = apparent position at time t

e(t)

s

d

y(t)

s

• e(t) = 2s / (tan - tan ) = 2sd / (x(t-) - x(t) + 2s)

• y(t) = s(tan + tan ) / (tan - tan ) = s(x(t-) + x(t)) / (x(t-) - x(t) + 2s)

• y(t),e(t) = apparent position at time t

• = arctan( (x(t-) + s)/d )

• = arctan( (x(t) - s)/d )

e(t)

s

d

y(t)

s

• The predicted curve traversed by the apparent position is approximately an ellipse

• The more the delay (darker filter), the greater the apparent difference in depth

• If the moving object is the bob on a swinging pendulum x(t) = sin(t)

• y(t),e(t) = apparent position at time t

e(t)

ConclusionsConclusions• Mathematical models can be useful Mathematical models can be useful

descriptions of biological phenomenadescriptions of biological phenomena

• Models can be used as evidence to Models can be used as evidence to support or refute biological hypothesessupport or refute biological hypotheses

• Models can suggest new experiments, Models can suggest new experiments, simulate experiments or treatments that simulate experiments or treatments that have not yet been carried out, orhave not yet been carried out, orestimate parameters that are estimate parameters that are experimentally inaccessibleexperimentally inaccessible

ConclusionsConclusions

Working together, biologists Working together, biologists and mathematicians can and mathematicians can

contribute more to science than contribute more to science than either group can contribute either group can contribute

separately.separately.

ReferenceReference

• ““Seeing in Depth, Volume 2: Depth Seeing in Depth, Volume 2: Depth Perception” by Ian P. Howard and Perception” by Ian P. Howard and Brian J. Rogers, I Porteus, 2002.Brian J. Rogers, I Porteus, 2002.Chapter 28: The Pulfrich effectChapter 28: The Pulfrich effect

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Connections Between Mathematics and Biology Carl Cowen Purdue University and the Mathematical Biosciences Institute