## Archive for the ‘mathematics’ category: Page 12

Database of 200,000 cell images yields new mathematical framework to understand our cellular building blocks.

A recent study from researchers at the University of California, Irvine found that the removal of cilia from the striatum region of the brain negatively impacted time perception and judgement, opening the possibility for new therapeutic targets for mental and neurological conditions such as schizophrenia, Parkinson’s and Huntington’s diseases, autism spectrum disorder.

Autism Spectrum Disorder (ASD) is a complex developmental disorder that affects how a person communicates and interacts with others. It is characterized by difficulty with social communication and interaction, as well as repetitive behaviors and interests. ASD can range from mild to severe, and individuals with ASD may have a wide range of abilities and challenges. It is a spectrum disorder because the symptoms and characteristics of ASD can vary widely from person to person. Some people with ASD are highly skilled in certain areas, such as music or math, while others may have significant learning disabilities.

Connectome harmonic decomposition (CHD) generalises the mathematics of the Fourier transform to the network structure of the human brain. The traditional Fourier transform operates in the temporal domain (Fig. 1a): decomposition into temporal harmonics quantifies to what extent the signal varies slowly (low-frequency temporal harmonics) or quickly (high-frequency temporal harmonics) over time (Fig. 1b). Analogously, CHD re-represents a spatial signal in terms of harmonic modes of the human connectome, so that the spatial frequency (granularity) of each connectome harmonic quantifies to what extent the organization of functional brain signals deviates from the organization of the underlying structural network (Fig. 1c, d). Therefore, CHD is fundamentally different from, and complementary to, traditional approaches to functional MRI data analysis. This is because CHD does not view functional brain activity as composed of signals from discrete spatial locations, but rather as composed of contributions from distinct spatial frequencies: each connectome harmonic is a whole-brain pattern with a characteristic spatial scale (granularity)—from an entire hemisphere to just a few millimetres.

On one hand, this means that CHD is unsuitable to address questions pertaining to spatial localisation and the involvement of specific neuroanatomical regions; such questions have been extensively investigated within the traditional framework of viewing brain activity in terms of spatially discrete regions, and several previous studies have implicated specific neuroanatomical regions in supporting consciousness33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49. On the other hand, CHD enables us to consider how brain activity across states of consciousness is shaped by the brain’s distributed network of structural connections, reflecting the contribution of global patterns at different spatial scales—each arising from the network topology of the human connectome. We emphasise that neither approach is inherently superior, but rather they each provide a unique perspective on brain function: one localised, the other distributed.

There are two aspects to a computer’s power: the number of operations its hardware can execute per second and the efficiency of the algorithms it runs. The hardware speed is limited by the laws of physics. Algorithms—basically sets of instructions —are written by humans and translated into a sequence of operations that computer hardware can execute. Even if a computer’s speed could reach the physical limit, computational hurdles remain due to the limits of algorithms.

These hurdles include problems that are impossible for computers to solve and problems that are theoretically solvable but in practice are beyond the capabilities of even the most powerful versions of today’s computers imaginable. Mathematicians and computer scientists attempt to determine whether a problem is solvable by trying them out on an imaginary machine.

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‘Spin’ is a fundamental quality of fundamental particles like the electron, invoking images of a tiny sphere revolving rapidly on its axis like a planet in a shrunken solar system.

Only it isn’t. It can’t. For one thing, electrons aren’t spheres of matter but points described by the mathematics of probability.

But California Institute of Technology philosopher of physics Charles T. Sebens argues such a particle-based approach to one of the most accurate theories in physics might be misleading us.

Year 2017 face_with_colon_three

A basic question [1] in the study of the gauge-gravity duality is this: which field theories have a gravity dual? In the case of applications to actual strongly coupled systems such as the Quark–Gluon Plasma [2], [3], [4], [5], [6], this question becomes: does every realistic strongly coupled system have such a dual? To settle this, one needs to examine the most extreme cases. The most extreme strongly-coupled systems currently accessible to experiment are probably (see below) the plasmas produced by collisions of heavy ions at the LHC [7], [8] ; so one needs to consider whether holography works in this case.

In [9] we adduced evidence suggesting that it does not. The problem is a very fundamental one: it appears that the purported gravity dual in some cases does not exist when one attempts to interpret it (as one ultimately must [10]) as a string-theoretic system.

A model for information storage in the brain reveals how memories decay with age.

Theoretical constructs called attractor networks provide a model for memory in the brain. A new study of such networks traces the route by which memories are stored and ultimately forgotten [1]. The mathematical model and simulations show that, as they age, memories recorded in patterns of neural activity become chaotic—impossible to predict—before disintegrating into random noise. Whether this behavior occurs in real brains remains to be seen, but the researchers propose looking for it by monitoring how neural activity changes over time in memory-retrieval tasks.

Memories in both artificial and biological neural networks are stored and retrieved as patterns in the way signals are passed among many nodes (neurons) in a network. In an artificial neural network, each node’s output value at any time is determined by the inputs it receives from the other nodes to which it’s connected. Analogously, the likelihood of a biological neuron “firing” (sending out an electrical pulse), as well as the frequency of firing, depends on its inputs. In another analogy with neurons, the links between nodes, which represent synapses, have “weights” that can amplify or reduce the signals they transmit. The weight of a given link is determined by the degree of synchronization of the two nodes that it connects and may be altered as new memories are stored.

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