Mathematical Foundations of Neuroscience - Lecture 1...

About the lectureThe brain

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Mathematical Foundations of Neuroscience -Lecture 1. Basic facts about the brain and its

analysis.

Filip Piękniewski

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,Toruń, Poland

Winter 2009/2010

Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 1 1/51

http://www.mat.umk.pl/~philip/


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PrerequisitesLecture contentsExamBibliography

About the lecture

The lecture is aimed at graduate (IV and V year) and PhDstudents of Mathematics and/or Computer ScienceThere will be some mathematics!There will be some programming!There will be many opportunities to learn fascinating stuff andrun interesting simulations!Materials and presentations will be available at my web pagehttp://www.mat.umk.pl/˜philip


http://www.mat.umk.pl/~philip


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Prerequisites

everybody needs their brainsproperties of matrices,eigenvalues, eigenvector decompositionnumerical solutions, problems of numerical stability,differential equations,schemes for solving differential equationselementary Matlab and/or C/C++ programming skills.

Most of the prerequisites are welcomed but not entirely necessary(except for the brain and an open mind), as I will briefly recall anymathematical facts before we need them.



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Lecture contents

Basic information about the brain - its large scale structure,consequences of various damages, methods of brain analysisand imaging. Brain/mind question.Neurons and their electrophysiological properties.Hodgkin-Huxley findings and formulation of their equations,action potentials and synapses.One dimensional dynamical systems, hysteresis, bistability,phase space analysis, bifurcations and their analysis, integrateand fire models.Two dimensional systems, vector fields, equilibria and locallinear analysis. Phase portraits and bifurcations.



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Lecture contents - continued

Conductance based models and their properties. Reductionsto simple models, FitzHugh-Nagumo model, E. Izhikevichsimple model.Bifurcations of two dimensional dynamical systems and theirsignificance in neuronal excitability.Simple neuronal models of various parts of the brain.Bursting, electrophysiology and mathematical interpretation insimplified models.Synchronization, modeling synapses, weak coupling, phaseresetting curves. Hebbian learning, Spike Timing DependentPlasticity and spike prop algorithm.Numerical simulations of large scale neuronal ensembles, goalsand obstacles.



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Exam

The lecture will end with an exam. An exam is a perfectopportunity to learn things we thought we knew...The exact form of the exam is not yet decided,Most probably the exam will be oralThe exam will be available in both Polish and English, thoughIt might be easier to pass English version due to insufficientvocabulary in Polish languageStudents performing exceptional projects in the Lab canexpect various benefits, including exempt from the exam!!!



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Dynamical Systems in Neuroscience

Eugene M. Izhikevich, MIT Press2007

Basic book supporting thelecture.http://izhikevich.org/publications/dsn/index.htm


http://izhikevich.org/publications/dsn/index.htm

http://izhikevich.org/publications/dsn/index.htm


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Weakly connected neural networks

Frank C. Hoppensteadt, EugeneM. Izhikevich, Springer 1997

More advanced book dealingwith more advanced subjects.



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Theoretical Neuroscience

Peter Dayan, L. F. Abbott, MITPress 2001

Quite useful book, with somemathematics.



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The quest for consciousness

Christof Koch, Roberts &Company Publishers 2004

Well written book onneuroscience, no mathematics.



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Spikes: Exploring the Neural Code

Fred Rieke, David Warland, Robde Ruyter van Steveninck,William Bialek, MIT Press 1999

Nice book on neural signaling.



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Scholarpedia

Scholarpedia - the peer-reviewedopen-access encyclopedia writtenby scholars from all around theworld.

www.scholarpedia.org


www.scholarpedia.org


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Basic anatomyHindbrainMidbrainForebrain

The brain

Human brain weights around 1.5kg, takes volume of about 1.2liters. Tan-gray on the outside,yellow-white on the inside.Contains between 50-100 billion(1011) neurons each having onaverage 10000 synapses, which intotal gives astronomical numberof 1015 synapses... The brain is amarvelous massively parallelbiological computer, the hardwarethat runs our software minds.



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Brain anatomy

The brain is far from beinguniform. One can easilydistinguish variousstructures.



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Brain anatomy

Medulla oblongata

Spinal cord

Pons

Cerebellum

Brain stem

Thalamus

Hypothalamus

Corpus callosum

Cerebrum

Basal ganglia

Pituitary gland

The most obvious isthe distinction of thespinal cord followed bythe brain stem andCerebellum. On top ofthose we find thalamusand basal ganglia. Aswe move upper weenter cerebrum dividedinto two hemispheresconnected via corpuscallosum, a bunch ofmillions of neural fibers.



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Divisions of the brain

For various functional, anatomical and developmental reasons thebrain is divided into three main parts:9/27/09 7:57 PM Brain_human_sagittal_section

Page 1 of 1 file:///Users/philip/Desktop/PdW/Lecture%201/Brain_human_sagittal_section.svg

Hindbrain

Midbrain

Forebrain

Hindbrain (Rhombencephalon)further subdivided intoMyelencephalon andMetencephalonMidbrain (Mesencephalon)Forebrain (Prosencephalon)subdivided into Diencephalon andTelencephalon



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HindbrainHindbrain is furthersubdivided intoMyelencephaloncontaining medullaoblongata (whichdirectly extends tothe spinal cord) andMetencephaloncomposed of thepons and thecerebellum.



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Midbrain

Medulla oblongata

Spinal cord

Pons

Brain stem

ThalamusHypothalamus

Midbrain

Basal ganglia

Tectum

Midbrain is placed on topof Pons, underneaththalamus. This part iscomposed of tectum (”topof the brain”) and cerebralpeduncle. Innon-mammalianvertebrates parts of tectum(in particular superiorcolliculus) play role similarto that of primary visualcortex in mammals.



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Forebrain

Frontal lobe

Parietal lobeTemporal lobeOccipital lobe

ThalamusHipocampus

Forebrain is the largestand the most complexpart of a mammalianbrain. It is exceptionallylarge in humans, thereforeit is believed to hosthigher cognitive functionsand consciousness. Thedivision into lobes is notbased on anatomicalreasons. It reflects thenames of skull bonescovering the cortex.



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Forebrain

Forebrain (composed of diencephalon and telencephalon) has impor-tant anatomical features:

There are two hemispheres, the structures are mirrorsymmetricInternals of the forebrain (white matter) are composed ofneuronal fibersThe external surface of the forebrain is called cerebral cortex.It is where most of information processing is done.The cortex is folded to fist large surface inside small cranium.There are ridges and fissures called gyrus (pl. gyri) and sulcus(pl. sulci) respectively.



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Forebrain

Thalamus and basal ganglia mediate in most of connections toand from the cortex to midbrain and nerves. Information fromsenses is also routed through thalamus except of olfactionwhich is connected to the cortex via the olfactory bulb.Cortex is mostly recurrently connected to itself. Bothhemispheres are connected via massive set of more than 200million neuronal fibers called the corpus callosum.Though strongly symmetric, hemispheres differ substantially.Most people have one dominating hemisphere (usually leftone) which is responsible for verbal thinking and speech.



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Forebrain

It seems that the two hemispheres are in fact two brains!Patients who had their corpus callosum cut (Corpuscallosotomy) in order to stop epileptic seizures, experiencestrange ”internal conflicts” (one hemisphere decides onsomething, but the other wants something else).Various parts of the cortex, perform different cognitive tasks,related to visual and auditory perception, motion, integrationof sensory stimuli, consciousness, speech, decision making,imagination, prediction etc.



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Post mortem studiesBrain damagesEEGCellular recordingsTomographyMagnetic Resonance Imaging (MRI)

Brain studySince the brain is such a magnificent organ, which hosts our mind it iswort every effort to study its function. Methodologies include:

Post mortem studies - most of our knowledge of brain anatomy isobtained that wayStudy of patients with brain damages caused by strokes oraccidentsBrain imaging - from X-ray to magnetic resonance, these methodsenable us to observe brain activity correlates on a living subjectReading out electric activity via EEG or more directly vieelectrodes placed directly inside the brainDirect stimulation via electrodes or indirect via strong magneticpulsesInjecting anesthetics and neuroactive substances (taking drugs!)



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Post mortem studies

Autopsies are a main source ofknowledge about human anatomy.The brain is no exception, and hasbeen studied extensively for manyyears. This is the only source ofinformation about micro anatomy,types of neurons, structure of thecortex. Special staining techniquesalso allow for tracking wires(myelinated axonal fibers).



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Brain damages

Crude as it may seem, brain damages are an essential source ofknowledge about functional properties of the brain. Of course no-body is damaging brain purposefully (unless it is necessary for med-ical reasons, like stopping epileptic seizures). Quick facts:

A human can live (in a deep irreversible coma) with hiscerebral cortex totally removed.Cutting corpus callosum (Corpus callosotomy) results insurprisingly faint cognitive deficitsDamages to primary visual cortex (V1) introduce blindness,though the patient can still unconsciously ”see” via othervisual pathways (the condition is called blindsight)



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The story of dr Jill Bolte TaylorDr Jill Bolte Taylor is a trained andpublished neuroanatomist. Sheexperienced a severe stroke in the lefthemisphere of her brain in 1996. It isworth to have a look at her book MyStroke of Insight: A Brain Scientist’sPersonal Journey (published in 2008 byViking Penguin), as well as her talk atTED conference:http://www.ted.com/talks/lang/eng/jill_bolte_taylor_s_powerful_stroke_of_insight.html


http://www.ted.com/talks/lang/eng/jill_bolte_taylor_s_powerful_stroke_of_insight.html




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Electroencephalography (EEG)

The brain is an electric device, and consequently it is a sourceof electromagnetic fieldThe field can be recorded by a set of electrodes sticked to thehead in certain placesThe recorded signal is greatly averaged, there is now way totrack a single neuron with EEG!Nevertheless it turns out, that the averaged brain activity hasperiodic components. Depending on the frequency of thedominating waveform the brains mode of operation can beinferred.EEG is therefore very useful in monitoring unconsciouspatients, tracking sleep phases etc. Flat EEG is a signal ofbrain death.



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Brain rythms

Delta < 4 Hz slow wave (NREM) sleepTheta 4–7 Hz drowsiness, idlingAlpha 8–12 Hz relax, closed eyes, daydreamingBeta 12–30 Hz anxious thinking, active concentration

Gamma 30–100 Hz The most mysterious brainwaves,”Gamma rhythms are thought to rep-resent binding of different populationsof neurons together into a network forthe purpose of carrying out a certaincognitive or motor function.”

Table: Brain rythms



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Cellular recordings

A set of tiny electrodesare implanted into arodents brain. Slowlythe wires are insertedinto deeper layers(avoiding anysignificant damages)until they reach sayhippocampus or brainstem.



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Figure: A rat with a bunch of its neurons monitored in real time.Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 1 30/51


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Tomography

Tomography is one of the mostuseful medical imaging techniques.It allows to gather a number of slicesthrough the body. But how does itwork?



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Radon transform

The essential fact underlyingtomography is that a certainimage transform (RadonTransform) is reversible.

The object is being x-rayedfrom every angle in (−π2 ,

π2 )

As an output we get the socalled sinogramSuch a sinogram has to becleverly untwined to get theactual slice Figure: Sketch of the tomographic

procedure.



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Figure: A sample image and its sinogram.



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Untwining the Radon Transform - Projection-slice theorem

Theorem (Projection-slice)Fourier transform of a projection of 2d image on a 1d line is equalto the 1d slice of the Fourier transform of the complete image(under the same angle).

Therefore having a sufficiently dense sinogram one can reconstructthe Fourier transform of the original image! Then using inverseFourier transform one can get back the image itself.



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Projection-slice theorem - sketch proof

Proof. Without loosing generality we may assume to project onthe x axis. Then:

p(x) =∫∞−∞ f (x , y)dy (1)

The Fourier transform of f (x , y) is:

F (ωx ,ωy ) =

∫∞−∞

∫∞−∞ f (x , y)e−2iπ(xωx+yωy)dxdy (2)



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Projection-slice theorem - sketch proof

Proof. Consequently the slice of the transform:

s(ωx ) =F (ωx , 0) =∫∞−∞

∫∞−∞ f (x , y)e−2iπxωx dxdy =

=

∫∞−∞[∫∞

−∞ f (x , y)dy]

e−2iπxωx dx =

=

∫∞−∞ p(x)e−2iπxωx dx = p̂(ωx )

(3)

QED. �



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Untwining the Radon Transform - Filtered Backprojection

The projection slice theorem gives quick and elegant proofthat the Radon transform is in fact reversible. Unfortunatelyapplying this technique to real images gives very poor resultsdue to numerical instabilities.There is another clever technique to untwine the transform,called the Filtered Backprojection. This algorithm isnumerically stable and efficient, but its mathematicalfoundations are much more delicate (though very elegant).Following few slides give a short explanation, I encourage toread through and understand the principles, though foryounger students this might require considerable effort.



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Untwining the Radon Transform - Filtered Backprojection

Recall that the Radon transform is defined:

g(s, θ) = R(f ) =∫∞−∞

∫∞−∞ f (x , y)δ(x cos θ+ y sin θ− s)dxdy (4)

We shall define the backprojection operator as follows

B(g(s, θ))[x , y ] =∫π

0g(x cos θ+ y sin θ, θ)dθ (5)

The operator spreads all of the projections back on the plain. Theresulting image would obviously be very blurry.



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Untwining the Radon Transform - Filtered BackprojectionConsider the following composition:

B(g)[x , y ] = B(R(f ))[x , y ] =

=

∫π0

∫∞−∞

∫∞−∞ f (u, v)δ(u cos θ+ v sin θ− x cos θ− y sin θ)dudvdθ =

=

∫∞−∞

∫∞−∞ f (u, v)

(∫π0δ((u − x) cos θ+ (v − y) sin θ)dθ

)dudv =

(6)

Now lets examine the integral in parenthesis (variables have been substituted forclarity):

d(r ,w) =

∫π0δ(r cos θ+ w sin θ)dθ (7)

This integral operator acts like a radial broom, sweeping the wholeplain in search for non zero values (δ is Dirac delta). But for anyfixed r and w this integral seems to be zero! Shouldn’t it followthat the whole composition is zero?



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The problem is, that operator d is not a typical function but rather a distribution

d(r ,w) =

∫π0δ(r cos θ+ w sin θ)dθ (8)

Observe how d acts inside an integral, multiplied by a characteristic function IAof some planar subset:∫

IA(r ,w) ∗ d(r ,w)drdw =

∫A

∫π0δ(r cos θ+ w sin θ)dθdrdw (9)

This integral will compute the measure of set A multiplied by a fraction of”time” (angle) the ”broom” spends in A while sweeping the plain! By samplingwith infinitesimally small sets of measure ε the outcome will always be inverselyproportional to the distance of sampled set from the origin. For A whose distancefrom [0, 0] is l =

√l2r + l2

w the outcome will be∫IA(r ,w) ∗ d(r ,w)drdw ≈ ε

l =ε√

l2r + l2

w(10)



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Getting back to the original composition:

B(g)[x , y ] = B(R(f ))[x , y ] =

=

∫∞−∞

∫∞−∞ f (u, v)


)dudv =

(11)

since the distribution in parenthesis is inside the integral, it can be substitutedwith its density (as estimated on previous slide):

=

∫∞−∞

∫∞−∞ f (u, v)


)dudv =

=

∫∞−∞

∫∞−∞ f (u, v) 1√

(u − x)2 + (v − y)2dudv =

= f (x , y) ∗ ∗ 1√x2 + y2

(12)

We get the original function (image) convoluted with a known ker-nel. We can now construct a sharpening filter by finding inverse of√

x2 + y2−1 in the frequency domain.Filip Piękniewski, NCU Toruń, Poland Mathematical Foundations of Neuroscience - Lecture 1 41/51


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Positron emission tomography (PET)

In this technique, a short-livedradioactive tracer isotope, is injectedinto the living subject. As theradioisotope undergoes beta decay itemits a positron, an antiparticle ofthe electron. Shortly the positronhits an electron (there are plentyelectrons in the body), and theyannihilate producing a pair ofgamma photons moving in (nearlyexactly) opposite directions. Thesepairs of photons are detected andvisualized.



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Positron emission tomography (PET)

The spatial resolution o PET is limited. Firstly a positron canfly an order of millimeters before it hits an electron. Secondlyonly small portion of gamma photons is detected. Theoccurrences are averaged and the image can be reconstructedvia backprojection similarly to x-ray tomography.The PET scan is usually done together with x-ray tomographyin the same device, to map positron emission sites onanatomic background.The use of short-lived radionuclides (such as carbon-11 ( 20min), nitrogen-13 ( 10 min), oxygen-15 ( 2 min), andfluorine-18 ( 110 min)) requires a nearby cyclotron.The exposure of a patient to the radiation is strictly limited!



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Magnetic resonance

X-ray tomography is very helpful, but lacks contrast for softtissues (x rays are mostly absorbed by bones).To overcome these limitations another, much more complexand technologically demanding technique has been devisedThe so called magnetic resonance produces images much likea tomogram, but with tremendous soft tissue contrastThe device requires superconducting helium cooledmagnets and much more sophisticated imaging algorithms...



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Magnetic resonance

MRI delivers much more detailedimages of soft tissues (in particularthe brain) compared to x-raytomography. But how does it work?



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Magnetic resonance - working principles

The patient is placed in a strong magnetic field (1.5-3 Tesla)The field causes the water molecules to align with thedirection of the fieldA radio frequency electromagnetic field is then briefly turnedon, causing the molecules to alter their alignment relative tothe field.When this field is turned off the molecules return to theoriginal magnetization alignment, emitting electromagneticsignal which is picked up by receiver.The response (resonant) frequency depends on the strength ofthe magnetic field, which is varied by gradient coils. That waythe position of respective molecules can be inferred.



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Functional Magnetic Resonance Imaging (fMRI)

The MRI scanner can receive andprocess many other signals. Onesuch signal is related to hemoglobinwhich is diamagnetic whenoxygenated but paramagnetic whendeoxygenated. By tracking thechanges in blood oxidization, thescanner can pick up sites consumingmuch oxygen. In general oxygenconsumption in neural tissuecorrelates with neural activity...



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Difusion tensor imaging (DTI)MRI is also capable ofmeasuring local diffusionproperties of water.Fortunately the diffusion ofwater in neuronal fibers isanisotropic (it is much fasteralong the direction of thetracts). MRI can extract thewhole diffusion displacementtensor, but in particular thedirection of largestcomponent correspond to thedirection of the nerve fibre.



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Difusion tensor imaging (DTI)

The effect of a DTI scan is a vector field with millimeterresolutionThe field can the be used to track individual fibers ofthalamo-cortical and cortico-cortical connections!Of course a single nerve fibre is a lot smaller and could not betracked individually in a living brain, but at least DTI givessome average connectivity information. Furthermore the datais global!This method is very promising for diagnosing medical disordersand creating large scale computer models of human brain!



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DictionaryReview of important concepts

Dictionary (some of the terminology in Polish)

Medulla oblongata - rdzeńprzedłużonyPons - mostThalamus - wzgórzeSpinal cord - rdzeń kręgowyCerebellum - móżdżekBasal ganglia - jądra podstawneCorpus callosum - ciałomodzelowate, spoidło wielkieMesencephalon - śródmózgowieDiencephalon - międzymózgowieTelencephalon - kresomózgowieMetencephalon - tyłomózgowie

Brain stem - pień mózguPituitary gland - przysadkamózgowaHypothalamus - podwzgórzeCranium - czaszkaOccipital lobe - płat potylicznyTemporal lobe - płat skroniowyParietal lobe - płat ciemieniowyFrontal lobe - płat czołowyOlfactory bulb - opuszkawęchowavertebrate - kręgowiec



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DictionaryReview of important concepts

Brain is an enormously complex organ whose wiring looksmuch like those found modern data centers. A human at theage of 20 has between 150000-180000 km (!) of neuronalfibers in his brain...Nevertheless the brain is being extensively studied onmolecular, cellular and anatomic scale. Clever methods ofbrain imaging have been developed. These tools are used on adaily basis in hospitals all over the world to help diagnosepatients with tumors, hemorrhages and other brain damages.Everything in the world is telling that the brain is the host forour minds, our consciousness. Damages to the brain causeinevitable damages to our perception, self-awareness, cognitiveabilities, our mind...


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