Fourier Analysis: The Art and Science of Finding The Needle in a Haystack

Every time I listen to the heavy metal band Pantera my wife would invariably wonder aloud why I listen to all this ‘noise’. True, many music lovers would rather refer bands like Pantera as quintessential noise than music; there are persons like me who can dissect the melody from the apparent chaos of runaway frequencies of guitars, drums and so on. I can even analyze and follow individual instruments over time. This is what Fourier is about, or stated otherwise, my ear & brain can be said to be doing a Fourier transform on the said musical piece.

Joseph Fourier, a French mathematician, realized that all periodic waves could be ‘synthesized’ by mixing sine waves of right frequency, amplitude and phase. For example, a square wave could be prepared by ‘adding’ the fundamental frequency (the lowest frequency; say 70Hz) with an infinite number of its odd harmonics (e.g. 210Hz, 350Hz, 490Hz and so on. Harmonics are multiples of the fundamental frequency.) This is Fourier synthesis. Similarly, you could break down a periodic signal in which the amplitude varies over time into one of a frequency versus time graph. This is Fourier analysis, and it can be seen that here we are actually ‘decomposing’ the ‘signal’ into its frequency spectrum, over time. The process of decomposing a function into its constituent frequencies is known as Fourier transform. You can have a ‘hands on experience’ at what a square wave ‘looks’ or ‘sounds’ like and how a periodic wave is decomposed into its constituent parts here. Do experiment on the sine, cosine, triangle wave and square wave functions as well and turn on the sound of your PC while you are at this site!

While Fourier originally devised this to solve the problem of heat propagation, the impact of Fourier analysis can now be felt in almost every field of science, instrumentation, entertainment and telecommunications, and even arts. Whenever you use your audio graphic equalizer to suit a piece of musical performance to your taste, you are doing a Fourier. Here you are boosting some particular audio frequencies while suppressing others, obtained by a Fourier analysis of the audio signal. You are assigning relative weights to the frequencies by sliding those sliders. Likewise, when you compress a picture (graphic) file using software such as JPEG, an inbuilt program does a Fourier transform,--> eliminates the weaker components from the analysis and--> then saves the information in a compact way.

In Nuclear Magnetic Resonance Imaging (NMRI), the emitted radio frequency is Fourier transformed to give frequency versus time, throwing valuable information about nuclear spins. Fourier analysis may also be employed to remove mains AC hum frequencies, in mobile telephony and many other situations.

One day, we may expect, that Fourier analysis may be used to pick up the ‘right frequency’ in the brain EEG waves and may put the study of ‘mirror neuron’ and ‘thought controlled devices’ into a whole new domain.

Last modified: May15, 2013
Reference: Fourier analysis (Wikipedia),
Explained: The Discrete Fourier Transform

Do We Really Forget? Fathoming The Esoteric Realms of Memory

“Smells like teen spirit”, but it could be true that we never really loose any memory in our lifetime. Our memories are stored in the synapses (junction, more specifically, gaps between adjoining neurons) as a function of synaptic strength, in the nerve cells like dendrites as proteins, and some other processes which mostly encompasses a chemical interaction. I am excluding memories such as T cell or B cell memories here; memory, here, will refer to neural ones that occur in the CNS.

We know that in dementias such as global multi infarct dementia, Alzheimer’s disease; there are diffuse losses of neurons and losses of cholinergic neurons in particular, respectively. In surgical cases of epilepsy or brain tumor, there are losses of neurons too. In these cases, memory loss may be irrecoverable, though cases are on record which points to shifting of those memories into some other safe havens. But what about the rest of the population? Does an established long term memory vanish completely?

Let’s consider some facts. The numbers of synapses and their strengths are finite, though both can change in response to stimuli. Even the number of neuron themselves can increase, contrary to the belief held earlier. Neuronal stem cell pool has been identified in the brain. Memories stored in the brain are finite too. Memories are inherently dynamic in nature. Even long term memory stored in the neocortex (medial temporal lobe, on the other hand, stores memories as a buffer, like a D RAM chip, a temporary storage) can change location, as much as transferring itself to the other hemisphere (intercortical transfer), when needed; via the optic chiasm and corpus callosum. So, we see that the number of synapses, though finite, can rise to the demand of an enhanced input from sensory cues which are finite too, leading to memories that can jump across their own allocated territories. A finite brain capacity (say C) can certainly contain a finite memory (say M) as long as C is greater than/ equal to M. Certain computer softwares even trespass this limit; a zip file of 2 MB may deliver 3MB of contents on unzipping! Who knows if the brain isn't using this for the past thousand years.

Synapses, simplistically, may be thought of in binary terms: 1, when it is on; 0, when it is off. Both 1 and 0 is a bit in Boolean terms. We leave aside the synaptic strength part here for the sake of simplicity. In addition, memories may shuttle between synapses in such a way so that it is present in the brain, but not represented by any synapse. I will explain. We all have seen those jugglers juggling those colorful balls too many at a time using only their two hands. A similar thing like dipole dynamics may occur in the brain. Added to this is quantum superposition, which allows the situation of BOTH 1 and 0 state at the same time at the synapse. That the brain can be in a quantum state at the core body temperature and the brain can effectively avoid ‘decoherence’ in the background thermal noise has been discussed by Roger Penrose and Stuart Hameroff. We also know that memories aren’t kept as such, but they are fragmented into individual elements, which are mostly matched to existing elements and are associated. This is economic as it saves space, and useful for indexing and contextual retrieval.

Thus it seems that we ought to have immense memory storage. Haven’t we encountered long forgotten memories in our dreams? Electrical stimulations in some parts of the hippocampus (deep brain stimulation or DBS, figure shown) during routine surgical procedures have given rise to ‘deja vu’ phenomena. The patients remembered things considered long forgotten. We may not be aware of the vast database of memories and are liable to infer that we have “killed ‘em all”, but in reality this may not be the case as Norio Ota et al clearly points it out in their paper. It smells like another chapter from your favorite science fiction novel, but it could be true.

Last modified: never; N.B.There is a substantial amount of speculation in this paper. Please exercise your own judgment and enlighten me about any possible error.
Reference: hyper-links, unless specifically mentioned.

Math Flaw Allows The Proverbial Tortoise Beat The Hare

How fast can a man run? More importantly, what is the speed limit for us human beings? The mere answer of 'the speed of light' (for bodies having a rest mass) isn't enough. You might also suggest that going at sound's speed ain't a good idea too, due to the 'sonic boom' that happens at 1 Mach. Recently,Usain Bolt took 9.69 seconds in 100 meter dash in Beijing 2008 Olympics. He broke his own previous record. What if someone bettered him or even someone surpassed this 'bettering man'?

Speed will certainly depend on various biomechanical and metabolic factors, and will be limited by some constraints. The height and weight of the individual, the efficiency of anaerobic glycolysis, ATP and phosphoryl creatine stores, the training, which induces certain (fast twitch fibers, for sprint) are important. The strength of bones and the Young's modiolus, governing the stress-strain relationship (also of cartilages) are important factors of biomechanics of sports.

Nerve conduction velocities, refractory period (the time during which an excitable tissue like the nerve or muscle is NOT responsive to stimuli) of nerves and muscles are determining factors too, as is the pumping of calcium ions from muscle fibers back into the sarcoplasmic reticulum. Here I am disregarding external factors like friction, wind drag; and other physiological parameters like metabolic build-up like lactates or temperature, for the sake of lucidity.

So how much is the maximum possible speed? We are yet to decide using the physical and physiological attributes. But how about 'running' slowly and claiming a trophy? The bottom line is: you can even beat Bold, boldly! Just run the 100 meter dash in 100 seconds. A flaw in the mathematical calculation system allows you to do the trick. Read about it at 'cosmic variance'.

Last modified: never
Reference: hyper-links, unless specifically mentioned

Logarithm In Medicine

Mathematics plays a very critical role in all the branches of medicine. Calculus, exponentials and logarithms are no exceptions. They make calculations less cumbersome and easy in many circumstances. Take for example, the plight of little Johnny. He wants to plot the intensity of various sounds that we hear, in a graphical scale. He knows that a barely audible whisper (threshold of hearing) is 10^-12 watts/m² and that of a very loud sound, for example, a jet plane taking off is 10¹³ times this! Oh, what an exceptional range of hearing this is. Even when Johnny is taking the height of this barely audible sound as 1 mm (above the baseline, in the Y axis), do you think he can manage the jet engines' roar in the graph paper? No, not in a linear scale.
Joanna is in a fix too. She fights hard to memorize the number of hydrogen ions present in one liter of pure water (at 25 degree Celsius), in moles per liter. It is 10^-7 moles per liter (or a meager 0.0000001 mole in a liter!) She can't even dare to memorize the concentration of hydrogen ions in human arterial blood. Can you help her?

John Napier of Scotland, in 1614, lent a helping hand to the humanity by formulating his famous logarithms. Now Johnny can easily depict the ear's sensitivity in this whole new log scale, or Joanna can remember the number of moles of H+ per liter (concentration) easily, in terms of pH. The graph is squeezed, shrunk-fit into the graph paper now, with the region of interest expanded and highlighted; but extremes not much decompressed.

The log scale can be thought of as the reverse of exponential scale. It is denoted as following: log_a^b=x, means a^x=b. Thus log₁₀¹⁰⁰=2. In this notation, 'a' is called the 'base', and the equation is pronounced as logarithm of 'b' to the base 'a' is 'x'. Normally, in logarithms, bases used are 10 and 'e'.

Logarithms are used in measuring sound intensity, in a relative scale. If one asks you how much loud is John's voice over Joanna's? You wouldn't say that it was louder by 50 (or 17), would you? Rather you would compare them and are more likely would say that it was double or triple that of Joanna's. Our ears indeed work in a 'multiplying scale' rather than simple 'interval scale' or arithmetic scale. The volume control 'variable resistors' of electronic devices such as transistor radios are logarithmic potentiometers to suit our 'log' loving ears and each successive keys of piano keyboards have frequencies that are multiples of their previous keys (hence the concept of octaves: there are 12 keys and the frequency is increased by twelfth root of 2, in each successive step; thus the frequency of the twelfth pitch following a given pitch will be exactly double = octave). Sound intensity is compared with another sound (threshold of hearing) in a logarithmic scale and the intensity (relative, of course!) of the test sound is given in 'decibel'. The intensities of earthquakes (in Richter scale, as is found by seismograph activities) are also relative ones, an earthquake measuring '5' on Richter scale is 10 times the intensity of '4' (and 1/10th that of '6') in the same scale. Likewise, the hydrogen ion concentration at pH 5 is 100 times than that of pH 7. Situation looks apparently opposite due to the fact that pH is -log (negative logarithm) of hydrogen ion concentration, which is expressed as 10 raised to the power -n. 10 raised to the power -3 is MORE than 10^-5. It must be remembered that though there is zero ('0') in the log scale; you can NOT calculate the logarithm of zero, as 10 raised to the power of anything will NOT give you "0".

Ever looked at your transistor radio's dial? Frequencies of radio stations are plotted in a log scale too. When you want to hear your rock music louder, you turn the volume control knob. This knob is actually a logarithmic potentiometer which allows equal change in loudness for equal rotation, using log rule for the track.

If you slapped a person real hard, he would certainly feel pain! But why should you hurt someone anyway! Weber Fechner law states that the magnitude of pain felt was proportional to the log of intensity of the stimulus (your slap). However, recently power functions are deemed more fit to calculate the magnitude of sensation felt.

In pharmacology too, log-dose response curve is a valuable tool in calculating the concentration of chemicals in bio-assays. During the exponential phase of bacterial cell division, their numbers double after every step. Logarithms can be used to plot the number of cells, in this log phase, in Microbiology. Physiology has plenty of its applications in the Nernst Equation (which describes the potential across the cell membrane for a given ion), Goldmann equation (calculates the potential for all the ions present), Henderson Hasselbalch Equation (governing the relationship between pH and pK) and many many others.

Thus, even in biological systems we can see how logarithms abound and how we can use these concepts in instrumentation or in counting bacterial colonies.

Last modified: Mar10,2014
Reference: hyper-links, unless specifically mentioned

Toward An Objective Correlate Of Pain

While taking a hot water bag to find relief for a severe spondylosis pain, I wondered why pain could not be expressed in a way different from the generally used Visual analog scale. The patient is asked to look at a chart (shown in the figure) and told to rate his sensation, that tallies most well with his pain. If that hot bag were to be applied to a person not having any pain, he would jump almost instantly. The fact that I tolerated it so well (and benefited from it), only shows its countering (counter irritant property) property, which should be somewhat proportional to the severity of one's pain. The relationship of a counter-irritant to pain severity, whether linear, logarithmic or exponential, needs to be established and quantified.

A patient's own account of pain may be subjectively modified according to the personality of the patient and many other factors. As such, this type of quantification is liable to be erroneous. Pain should better be measured in an objective manner, free of bias. In this instance cited above, one could use the formula: Q(heat)=m(mass) x s(specific heat of the substance) x t(temperature), to know the amount of heat energy transferred to the patient. I am assuming that the pain relieving techniques will, kind of, obey Newton's Third law. Amount (intensity) of counterirritant that just suffices pain relief will be equal to the degree of pain. But it is not actually so, as we will see later.

By noting the difference in local temperatures before and after the procedure, one could get "t".
Mass or "m" could be measured by estimating the volume of the body tissue that was actually heated by infrared mapping, for example; and the expected density of that area. Specific heat for the tissue in question could be easily known and standardized using some cross-sectional studies. A suitable nomogram may later be drawn by plotting values obtained from such observations, for quick estimates. It seems logical that pain so measured, will have its units in British Thermal Units (btu), calories or their work (mechanical) equivalents like ergs, Joules, foot-pounds etc..

In pain therapies using mechanical energies (Ultrasound), electromagnetic devices (laser, short wave diathermy, high frequency photons such as X rays) , a similar formula may be used to obtain the pain equivalent. For example, in laser or short wave therapy, we may design a device that will measure the amount of energy in Watt.seconds/Joules the given area of tissue is supposed to absorb, over a given period of time. The chemical analgesics (pharmaceuticals e.g. Non Steroidal Anti Inflammatory Drugs or steroids; counter irritants such as capsaicin) may be quantified using Scoville scale or by evaluation on the degree of relief from algesia.

Calculating pain may be quite painful in itself. Pain sensation does not tally linearly with noxious stimulus. Rather, a logarithmic relationship was proposed in the Weber Fechner law, which held that the magnitude of pain (or a sensation) felt, was proportional to the log of the intensity of a stimulus. In other words, to feel twice as much pain, you needed to hurt 10 times! To complicate matters further, our present knowledge suggests that the magnitude of a sensation is related by a power factor to the intensity of that sensation. R=KS^A; where R is the sensation felt, S the intensity of stimulus, K and A are constants for that particular tissue. The brighter side is, we get a preformulated relationship for pain calculation.

Pain (musculo skeletal/ visceral, exogenous/endogenous) is generally of two types: fast pain and slow pain. Fast pains such as sharp pain of pricks, stabs are usually carried by Ad (A delta) nerve fibres, while slow aching pains are carried by type C nerve fibers. Ad fibres can carry impulses rapidly as these fibers are myelinated and are of large caliber. C fibers, on the other hand, are unmyelinated and narrow. A-delta fibres release glutamate and C fibers secrete substance-P. A way to measure these chemicals could be a step closer to quantifying pain.

The signals from these fibers travel to the thalamus, a part of the lower brain, on their way to the cerebrum for the localization of pain. Measuring the metabolic activity in thalamus, arising out of increased neural discharges there, by fMRI or PET scan may also shed some light on the intensity of pain stimuli (the stimulus at this level is unmodified by the higher brain) that reaches thalamus. We can also measure the blood levels of endogenous opioids (enkephalins, endorphins) which are secreted in body's response to the pain and adrenaline, secreted in response to increased sympathetic discharge, which is an usual accompaniment of pain. Their blood/plasma levels may correspond with pain severity. Other pain markers like bradykinin, histamine, potassium ions and proteolytic enzymes could be probed too.

True, that the patient may or may not feel as much pain as has been measured this way, because pain perception may not be proportional to the physical/chemical parameters thus described and it is not uniform in all subjects. The brain sees pain in its own mathematical terms, and everyones' brain is different in this regard. A soldier may overlook his gushing wounds, whereas pampered girls of rich persons may feel a lot of pain from an apparently trivial injury. But, quantification of pain in this way (by measuring the physical/chemical yardsticks) may correctly establish the severity of pain in silent myocardial infarction of neuropathy (pain sensation is dulled here due to neural malfunction), decubitus ulcers, trophic ulcers, malingering and in similar situations. In this way we may be able to find a better and objective correlate of pain in clinical practice and develop more efficient analgesics.

P.S. In a recent development, some objective physiological correlates of pain has been tracked. These include measurements from the nonlinear composite of heart rate, heart rate variability, amplitude of the photoplethysmogram, skin conductance, fluctuations in skin conductance, and their time derivatives.  Algorithms can then convert the data into a real-time, continuous index on a bedside monitor. This has resulted in the fabrication of a wearable sensing device can be mounted on a finger.

Last modified: Nov 28, 2015
Reference: hyper-links, unless specifically mentioned

Maverick Moebius

In my article on 'the universe must expand' I mentioned a certain Mobius strip (also known as Moebius strip).

The Mobius strip, which is really interesting and thought provoking, can be prepared in a very simple way. What you have to do is, to take a rectangular piece of paper, say about 30 cm x 3 cm. Paint one surface green and the other surface red, for example. Now, obviously, the piece of paper has two surfaces and four sides. Hold the sheet longitudinally (lengthwise) and twist it a little and meet the ends together so that the red surface will meet the green one. Congratulation, you are done!

Now, how many surfaces/sides does it have? Run your finger along one surface and continue. You'll find that there's only one surface and not 2 as our intuition suggests. Now you may trace the line (margin/border) and similarly be prepared for another surprise. You'll end up in the same point you started with. It also has just one border. If that's not enough, cut the strip in half along its length. You wont get 2 strips, do it yourself!

Doesn't it give the feeling of infiniteness? The recursive pattern gives a pseudo-sense of infinity. A similar topological model called the Klein bottle illustrates this apparent fallacious of infinity. The universe which is infinite (is it?) could be thought of in this perspective.

P.S. The step marked in red may be omitted, it was invoked for sake of lucidity. Please wear your thinking cap for sometime and ask yourself if you could find any similarity of this with the 'infinity'.