The mathematics of knowledge networks in the brain

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This article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

Understanding Knowledge Networks in the Brain

For SIAM News:

One strength of the human mind is its ability to find patterns and draw connections between disparate concepts, a trait that often enables science, poetry, visual art, and a myriad of other human endeavors. In a more concrete sense, the brain assembles acquired knowledge and links pieces of information into a network. Knowledge networks also seem to have a physical aspect in the form of interconnected neuron pathways in the brain.

During her invited address at the 2018 SIAM Annual Meeting, held in Portland Ore., last July, Danielle Bassett of the University of Pennsylvania illustrated how brains construct knowledge networks. Citing early 20th century progressive educational reformer John Dewey, she explained that the goal of a talk—and learning in general—is to map concepts from the speaker/teacher’s mind to those of his or her listeners. When the presenter is successful, the audience gains new conceptual networks.

More generally, Bassett explored how humans acquire knowledge networks, whether that process can be modeled mathematically, and how such models may be tested experimentally. Fundamental research on brain networks can potentially facilitate the understanding and treatment of conditions as diverse as schizophrenia and Parkinson’s disease.

[Read the rest at SIAM News…]

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The secret to good digital animation is physics

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

This article is a little different from the fare you’re used to getting from me: it’s for SIAM News, which is the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

The Serious Mathematics of Digital Animation

For SIAM News:

While computer simulations have a wide range of uses, their goals are generally similar: find the simplest model that recreates the properties of the system under investigation. For scientific systems, this involves matching observed or experimental phenomena as precisely as necessary.

But what about movie simulations? Should they match the processes they replicate so closely? Computer-generated imagery (CGI) is a common feature in both animated and live-action films. For these CGI systems, creating visuals that look right is an important task. However, Joseph Teran of the University of California, Los Angeles believes that starting from physical models is still a good idea.

During his invited address at the 2018 SIAM Annual Meeting, held in Portland, Ore., this July, Teran pointed out that beginning with a mathematical system is often easier than drawing from real life. Many movies model a system’s various forces and internal structures with partial differential equations (PDEs) for this reason. While solving these equations to produce CGI is computationally expensive, such methods have become powerful tools for creating realistic visual cinematic effects.

[Read the rest at SIAM News]

The math behind leopard spots and chemical waves

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

This article is a little different from the fare you’re used to getting from me: it’s for SIAM News, which is the glossy magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

Leopard Spots, Frog Eggs, and the Spectrum of Nonlinear Diffusion Processes

For SIAM News:

Stripes, spots, or a mix of both appear on the skin of many animals — from tigers to beetles to whale sharks. These patterns are typically unique to individual creatures, and biologists often use them for identification. While distinct patterns may seem random, they obey certain rules that suggest a common underlying description. Striping and spotting occur in many unrelated species, implying that both evolutionary advantages and simple biochemical mechanisms drive such patterns.

As Björn Sandstede of Brown University noted during his invited address at the 2018 SIAM Annual Meeting, held in Portland, Ore., this July, similar patterns appear in certain chemical reactions and granular material under vibration. Nonlinear reactions and diffusion describe biological and non-biological patterns, producing stable concentrations in this space.

Alan Turing—best known for his work in computer science and cryptography—first made the mathematical connection between nonlinear diffusion processes and animal stripes in the 1950s. Many researchers have applied the resulting model to demonstrate how various species get their spots and describe nonlinear waves in chemical reactions.

[Read the rest at SIAM News]

Using math to understand why species don’t out-eat each other

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

This article is a little different from the fare you’re used to getting from me: it’s for SIAM News, which is the glossy magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

Competitive Adaptation Prevents Species from Eradicating Each Other

For SIAM News:

Evolution is frequently rough and unforgiving; individuals within a species compete for food, reproductive partners, or other resources. Species fight each other for survival, especially when preying on one another.

Mathematical biologists often simplify these dynamics to predator versus prey. Real-world populations of predator and prey species within a given ecosystem cycle between booms and busts. In various cases, multiple species—including both predators and prey—coexist with similar diets. For example, a cubic meter of seawater can harbor several species of plankton, consisting of tiny plants and animals (see Figure 1).

One would naively expect reproductive success (more offspring) or competitive performance (eating more than your neighbor) to lead to one species’ domination. But that does not occur. While many of these organisms consume the same food, one species does not out-eat the others; the plankton swarm’s overall diversity remains fairly constant. Biologists refer to this phenomenon as the “paradox of the plankton” or the “biodiversity paradox,” among similar terms.

[read the rest at SIAM News]

Swarming in time, synchronizing in space

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

This article is a little different from the fare you’re used to getting from me: it’s for SIAM News, which is the glossy magazine for members of the Society for Industrial and Applied Mathematics. The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. In this case, I covered research by mathematicians looking at a type of system that occurs in biology and materials science. While the article contains equations, I wrote it to be understandable if you skim that part.

Self-organization in Space and Time

For SIAM News:

Self-organization is an important topic across scientific disciplines. Be it the spontaneous flocking of birds or dramatic phase transitions like superconductivity in materials, collective behavior without underlying intelligence occurs everywhere.

Many of these behaviors involve synchronization, or self-organization in time, such as activation in heart cells or the simultaneous blinking of certain firefly species. Others are aggregations, or self-organization in space, like swarming insects, flocking birds, or the alignment of electron spins in magnetic material.

Despite their conceptual similarity, self-organization in space and time have largely been treated separately. “I was curious about whether the two fields had been wedded, and it turns out they hadn’t, at least not fully,” Kevin O’Keeffe, a postdoctoral researcher at the Massachusetts Institute of Technology, said. “I knew all these tricks and mathematical tools from synchronization, and I was looking to cross-fertilize them into the swarming world.”

[Read the rest at SIAM News]