the topology of neuronal structures exposed to cosmic ... · ual big data analytic and machine...

The topology of neuronal structures exposed to cosmic radiation

ARMEL MAGANGA MIHINDOUFaculty of Sciences, Rabat

Department of Mathematics4 Avenue Ibn Battouta B.P. 1014 RP, Rabat

Rabat, [email protected]

MY ISMAIL MAMOUNICRMEF Rabat, Morocco

Department of MathematicsAv. Allal Al Fassi, Madinat Al Irfane

Rabat, [email protected]

Abstract: In this paper, we focus on some leader NASA experiences to explore how cosmic radiation causedsignificant reductions in dendrite and spine complexity. We adopt a topological data analysis approach and extractmore information then the classical methods. Our key idea is to use the NASA images of the neural networks ofsome mouses that were exposed 12 weeks to cosmic radiation. We associate to this neural network code bares thatgive us more information, that that given by the original experiences.

Key–Words: topological data analysis, persistent homology, cosmic radiation, image analysis, segmentation.Typing manuscripts, LATEX

Received: January 28, 2020. Revised: May 6, 2020. Accepted: May 18, 2020. Published: May 28, 2020.

1 Introduction

Since human has gave himself the mission to conquerthe space, and then inevitably be exposed to cosmicrays, the cosmic radiation became a hot topic thatcatches the attention of the medias, politicians, sci-entists, ... Due to the increased use of the technol-ogy supports (as smart phones), families are more andmore interested in rays negative effects on their chil-dren, and are asking a lot of questions, like : How toprotect efficiently the neural network of our navigatorsagents from cosmic ray? How to predict the impact ofcosmic radiation on neural network

Cosmic rays are high energy radiation particles,mainly originating outside the Solar System. Com-posed primarily of high energy protons and atomicnuclei, they are of mysterious origin. The origin ofthe cosmic radiation is an old subject of discussion: E. Fermi (see [7]) proposed a theory according towhich cosmic rays are originated and accelerated pri-marily in the interstellar space of the galaxy by col-lisions against moving magmetic fields. R. D. Richt-myer and E. Teller (see [15]) has advocated the view-point that the cosmic rays are of solar origin and arekept relatively near the sun by the action of magneticfields.

NASA hopes to send the first round-trip, mannedspaceflight to Mars by the 2030s. Studies in mice sug-gest that these particles could alter the shape of neu-rons, impairing astronauts memories and other cogni-tive abilities. Indeed, nervous system risks which in-clude during space missions and lifetime risks due tospace radiation exposure are of concern for long-term

exploration missions to Mars or other destinations.Possible risks during a mission are altered cognitivefunction, including detriments in short-term mem-ory, reduced motor function, and behavioral changes,which may affect performance and human health.

So, should we worry? The answer is yes : en-ergies of the most energetic ultra high energy cosmicrays have been observed to approach 3× 1020eV andabout 40 million times the energy of particles accel-erated by the Large Hadron Collider.1 Thus and evenafter 54 years of sending astronauts into space, NASAis still learning about how space conditions affect thehuman body. The agency is studying the effects oflong-duration space flight on human vision, with itsunique twins study, and researchers continue to anal-yse how micro gravity wears away at the musculo-skeletal system.

In this paper, we focus on some leader experi-ences (e.g. [12], [13], [14]), after what authors areconvinced that irradiation caused significant reduc-tions in dendrite complexity, spine density and alteredspine morphology along medial prefrontal corticalneurons known to mediate neuro transmission inter-rogated by our behavioural tasks. Behavioural deficitsfor individual animals correlated significantly with re-duced spine density and increased synaptic puncta,providing quantitative measures of risk for develop-ing cognitive impairment.

Thanks to the persistence homology, as an alge-braic topology tool used in the topological data anal-ysis approach, we propose to issue a barcode of the

1The worlds largest and most powerful particle accelerator.

WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.26 Armel Maganga Mihindou, My Ismail Mamouni

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results of these experiences. The only data we willuse is high-resolution imaging of brain tissue takenby the authors for rodents’ brain before and after hav-ing bombarded them with some cosmic radiation. Wewill implement an algorithm that store regions adja-cency under topology signature to evaluate the den-sity of dendrites and spines. Moreover, we will designa software which issue from any humain IRM’s brainan alert about the neuronal density. This can be useful,for example, to alarm the parents whose kids are ad-dict of video games, they only need brain scan to putin the software. An alert will be issued according tothe dendrite complexity and spine density readed bythe software, based only on brain scan. Indeed, per-sistent homology persistence allowed us to interpretgeometrical constitution shape, and then can be usedto make shape recognition.

The paper is organized as follows : In section 2we will briefly describe the neural network and howcosmic rays can damage it. Section 3 is the theoret-ical part devoted to summarize the persistent homol-ogy concept, while in the section 4 we will present ourideas to analyse topologically some NASA’s experi-ences on the possible negative impacts of the cosmicradiations on some behavior tasks. Our contribution isa shape recognition algorithm based on the persistenthomology viewpoint that analyzes the geometry andthe topology of the spines and dendrites distribution.Finally, in the section 5, we will present a softwarewhich can be used by doctors to predict the dendritecomplexity and spine density based (for example) on abrain IREM image. This can be useful to alert parentswhose children are addict to high resolution radiationvideo games.

2 The Human brain and cosmic ra-diation

The neural networks are a neural arrangement, eachneural is extended by a dendrite topped with thornscalled spines. These spines play an important role inthe communication between neural.

Figure 1: Structure of a typical dendrite

Dendrites are the branches of neurons that receive sig-nals from other neurons and pass the signals into thecell body. The axon is the part of a neuron that sendsthe signal. Axons that seel like a highly developeddendrite are long and thin. The axon carries an elec-trical signal from the cell body to the synapse : thestructure that permits a neuron to pass an electrical orchemical signal to another neuron. Synapses are es-sential to neuronal function: neurons are cells that arespecialized to pass signals to individual target cells,and synapses are the means by which they do so : Adendrite from one neuron and an axon from anotherneuron meet at a synapse.

Figure 2: Communication process between two neu-rons

The cosmic radiation is a phenomena caused by theunstable atom kernels that are spread in the space, it isunavoidable that these displacements are not withoutrisk especially for human being, since they can affectthe neural network. The well known Ch. L. Limoliteam have to make an important breakthrough : to se-lect the best candidates for a one-way trip to Mars.For this purpose, the molecular biologist at the Uni-versity of California, Irvine, and his colleagues tookmale mice to a particle accelerator at the NASA SpaceRadiation Laboratory in Upton, New York. There,they catapulted oxygen and titanium ions down a 100-meter transport tunnel and into the restrained rodents’sbrains at roughly two-thirds the speed of light.

Six weeks after exposure, researchers at UCIrvine noticed distinct changes to the brains of the


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mice notably inflammation that disrupted communi-cation between the neurons. According to the study,the particles acted like tiny bullets, flying into thebrain and breaking off neuronal structures known asdendrites. The loss of these branch-like synapses,which carry electrochemical signals between neurons,are often associated with cognitive impairments andAlzheimers disease.

Going one step further, the researchers did somebehavioral experiments with the exposed mice to seehow these brain changes might affect their memoryand learning abilities. Sure enough, the mice exhib-ited less curiosity and seemed more confused thanmice who had not been exposed to space-like radi-ation. These symptoms are similar to the cognitivechanges brain cancer patients experience when under-going radiation treatments.

3 Topological data analysisTopological Data Analysis (TDA) is an emergingtrend in exploratory data analysis and data mining.It has known a growing interest and some notablesuccesses (such as the identification of a new typeof breast cancer, or the classification of NBA play-ers, or the prediction of the future USA president) inthe recent years. Indeed, with the explosion in theamount and variety of available data, identifying, ex-tracting and exploiting their underlying structure hasbecome a problem of fundamental importance. Manysuch data come in the form of point clouds, sittingin potentially high-dimensional spaces, yet concen-trated around low-dimensional geometric structuresthat need to be uncovered.

The non-trivial topology of these structures ischallenging for classical exploration techniques suchas dimensionality reduction. The goal is therefore todevelop novel methods that can reliably capture geo-metric or topological information (connectivity, loops,holes, curvature, etc) from the data without the needfor an explicit mapping to lower-dimensional space.Persistent homology is the main tool of the TDA, itconsists to represent any shape under a barcode. Asthe saying goes, ”every data have a shape, and anyshape have a meaning”. Thus the key idea of TDA isto represent a data as a shape (a point cloud for ex-ample), and the issue its barcode (the meaning of thedata).

In the TDA community we agree, without formal-izing it, that H. Edelsbrunner is the founder of this the-ory Indeed, the fast algorithm described in his leaderpaper [4] triggered the explosion of interest we cur-rently observe because its availability as software fa-cilitates the application to a broad collection of prob-

lems and datasets.We agree also that the G. Carlsson research works

(e.g., [2], [3], [17]), and that the software platformand applications offered by its machine intelligencesoftware company Ayasdi, are the precursors of thecurrent popularity of TDA both in the scientific andindustrial communities. For example, one widely re-ported top five global systemically important bankwas that to build models required for the annual Com-prehensive Capital Analysis and Review (CCAR) pro-cess took 1,800 person-months with traditional man-ual big data analytic and machine learning tools, buttook 6 person-months with Ayasdi. Now, Ayasdi,founded in 2008, is considered now as ”A Big DataStart-Up With a Long History”2, and recently an-nounced a new 55 million USD round of Series Cfunding, led by Kleiner Perkins Caufield & Byers(KPCB), and joined by existing investors, InstitutionalVenture Partners (IVP), Khosla Ventures, FLOOD-GATE, Citi Ventures, and new investors, CenterviewCapital Technology and Draper Nexus.

3.1 Simplicial homology

Simplicial homology refers to two concepts, that ofabstract simplicies, which form a family of sets that isclosed under deletion of elements, and that of geomet-ric simplicies, which are geometric object of variousdimensions (points, line segments, triangles, tetrahe-dra, and so on), glued together according to certainrules. As we will see in a moment, the two conceptsare in fact closely related and can give some kind ofinformation about the Euclidean space considered.

Definition 1. Given some fixed points (ei)0≤i≤n inRm, their associated n-simplex is the convex hull

[e0, . . . , en] :=

{n∑

i=0

tiei, ti ≥ 0,n∑

i=0

ti = 1

}.

Hence, a 0-simplex is a vertex, a 1-simplex is anedge, a 2-simplex is a full triangle, while a 3-simplexis a full pyramide.

Figure 3: Examples of simplices

The convex hull of any non empty subset of this n+1points define an simplex, called a face of the simplex.

2The New York Times. January 16, 2013.


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Note that faces are simplices themselves, and that theempty set is a face of every simplex.

Definition 2. A simplicial complex K is a set of sim-plices that satisfies the following conditions:

1. Every face of a simplex from K is also in K;

2. The intersection of any two simplices σ1, σ2 ∈ Kis a face of both σ1 and σ2.

Figure 4: Simplicial complex or not?

A n-chain in K with coefficients in Z2 is any for-mal sum

∑niσi, where ni ∈ {0, 1} and σi is a n-

simplex in K. The subset of all this n-chains will bedenoted Cn(K), with the convention that C−1(K) = ∅.

Definition 3. The boundary operator is the Z2-linearmap defined by :

∂n : Cn(K) → Cn−1(K)σ := [e0, . . . , en] 7→ ∂nσ

,

where, ∂nσ :=n∑

i=0

(−1)i[e0, . . . , ei, . . . , en] and that

ei means omitted.

For example ∂0A = 0, ∂1[A,B] = B − A, while∂2[A,B,C] = [B,C] − [A,C] + [A,B], which welljustifies the appellation boundary. It is easy to showthat

Theorem 1.∂n−1 ◦ ∂n = 0.

In particular we have

Im∂n ⊂ ker ∂n−1.

This yields the chain complex

0i↪→ Cn(K)

∂n−→ Cn−1(K)∂n−1−→ ...

∂1−→ C0(K)∂0−→ 0,

which is an algebraic structure that consists of a se-quence of abelian groups (or modules) and a sequenceof homomorphisms between consecutive groups suchthat the image of each homomorphism is included inthe kernel of the next. Elements of Im∂k are calledboundaries, those of ker ∂k−1 are called cycles. Thusany boundary is a cycle, the inverse is not always true.

Definition 4. The k-th simplicial homology group ofK, is defined to be the quotient group

Hk(K) = ker ∂k−1�Im∂k.

Its rank, denoted βp(K), is called the k-th Betti num-ber of K.

Hk(K) represents the obstruction of a cycle to bea boundary, and βp(K) represents the number of thehomologus k-dimensional holes in a shape. Since theinterior of a circle is a disc, which is a variety of di-mension 1, one may consider a circle to have a one-dimensional hole. In particular β0 is the number ofthe path-conected components of a shape, since twopoints are homotopic if and only if they live in thesame path-conected component.

Figure 5: Betti numbers of some shapes

For further and deep details on the notion of homol-ogy, we recommand to the reader this standard refer-ences : [8] and [9].

3.2 Persistent homology

Persistent homology is a kind of a homology theorybased on the notion of filtration.

Definition 5. Given a simplicial complex K, a filtra-tion of K is any nested sequence of sub-complexes de-fined by:

1. ∅ = K0 ⊂ K1 ⊂ · · · ⊂ Kn = K;

2. Ki+1 = Ki∪σi+1, where σi+1 is a simplex ofK.

Let Cpk denote the set of the k-chains (with co-efficients in Z2) of Kp, then the restriction of theboundary operator is ∂ : Cpk −→ C

p−1k−1. Let Zp

k andBp

k denote respectively the sets of k-cycles and k-boundaries of Kp, then one may check obviously thefollowing inclusions

Z0k ⊂ Z1

k ⊂ · · · ⊂ Zpk ⊂ . . . Z

nk ,

B0k ⊂ B1

k ⊂ · · · ⊂ Bpk ⊂ . . . B

nk .


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Definition 6. For some fixed p ∈ {0, . . . , n}, q ∈{0, . . . , n− p}, the k-th persistent homology group ofKq, denoted Hq,p

k is the quotient group defined by

Hq,pk := Zp

k�(Bp+qk ∩ Zp

k).

Its rank is called the k-th persistent Betti number ofKq.

The k-th persistent Betti number ofKq representsthe number of the independent homology classes of kcycles in Kq that are not boundaries in Kq+p. Intu-itively, a k-cycle in Kq generating a non zero elementin Hq,p

k is a cycle that has appeared in the filtrationbefore the step q+1 and that is still not a boundary atstep q + p.

To understand how the topology of the filtrationevolves each time we add a simplex, we say that thesimplex σp is positive if it is contained in a (k + 1)-cycle inKp, which is necessarily not a boundary inKp

and negative otherwise. With the above definition thek- th Betti number of Kp is equal to the difference be-tween the number of positive k-simplices (which arecreating k-cycles) and the number of negative (k+1)-simplices (which are killing k-cycles).

Thus, a homology class is created when a posi-tive simplex is added in the filtration and that a ho-mology class is destroyed when a negative simplexis added. Topological persistence provides a naturalway to pair positive and negative simplices such thatwhenever a positive simplex is added to the filtrationit creates a homology class and a corresponding cy-cle that becomes a boundary when its paired negativesimplex is added.

A nice way to represent this pairing is to considerthe couple (σj , σ`(j)), where σj is a negative simplexand `(j) the largest index of the positive k-simplicesassociated to ∂σj . Drawing this pairs, we get the per-sistent diagram, whose horizontal axis indicates thebirth of a cycle, and whose vertical one indicates itsdeath. The ”life” of a cycle, which represents the per-sistent of a given data, is equal to the height of itsassociated pair computed from the diagonal (since thediagonal interpret the birth time of a cycle).

Figure 6: Example of a persistent diagram

Another equivalent way to represent this pairingbirth-death is the barcode, which is a graphical rep-resentation of the evolution of the homology of agiven simplicial complex following the chosen filtra-tion. That is a collection of horizontal line segmentsin a plane whose horizontal axis corresponds to thelifetime (the persistence) of a homology generator andwhose vertical axis represents an (arbitrary) orderingof homology generators.

Figure 7: Example of a filtration, and its associatedbarcode

The attentive reader will certainly infer that for afixed simplicial complex (a data represented as a pointcloud), the barecode issued depend on the choosenfiltration. There are several approach (VietorisRips,Cesh, alpha, weak witnes, Delaunay, ...), and one haveto choose the most adapted to the context of his prob-lematic. For more and further details, we refer the in-terested reader to the successful comparison [11]. Formore general information on persistent homology weadvise this standard references : [5], [6].


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4 Topological data analysis of someNASA experiences

Nervous system risks during space missions due tospace radiation exposure, are of concern for manny re-search (e.g. [1], [13] and [14]). Possible damage (see[1]) during a space mission are altered cognitive func-tion, including detriments in short-term memory, re-duced motor function, and behavioral changes, whichmay affect performance and human health. This ra-diation induces (see [13]) changes in synaptic plastic-ity underlie many neuro degenerative conditions thatcorrelate to specific structural alterations in neuronsthat are believed to be morphologic determinants oflearning and memory. The authors in [14] exposedsome rodents in separate groups to 5cGy48Ti or 16Oor 30cGy16O during 12 and 24 weeks. Irradiationcaused significant reductions in dendrite complexity,disrupted synaptic integrity and increased neurons in-flammation. This cosmic radiation causes some cog-nitive dysfunction like in the novel object recognition(NOR) task, the object in place (OiP) task or the tem-poral order (TO) task (see [12], [13], [14]),.

Our topological data analysis will focus on the re-sults of [14], however it can be extended to other ones.For 12 weeks, some mouses were exposed to cosmicradiation in the form of Titanium and Oxygen iso-tope, especially 5 or 30 cGy48Ti or 16O or 30cGy16O.We choose to restrict our analysis to the results givenby the Titanium irradiation, because that the damagecaused by the Titanium irradiation is more importantthan any other cosmic radiation (see the figures herebelow from [14]). One other reason, to defend ourchoice, is that the Titanium isotope is the only one wecan find in space.

Figure 8: Cognitive deficits evaluated 12 weeks aftercosmic radiation exposure.

Hence, this figure shows that a 30 cGy48Ti particleirradiation significantly reduces the recognition mem-ory, that it reduces the preference to explore an objectfound in a novel location and that it significantly im-pairs the memory by a reduced preference for the less

recently explored object in the Temporal Order task(TO).

Figure 9: Reduced dendrite complexity of neurons af-ter 12 weeks of cosmic radiation.

This quantification of the dendrite parameters, as barcharts, shows that the dendrite branching and lengthare significantly reduced 12 weeks after exposure to 5or 30 cGy48Ti or 16O or 30cGy16O particles.

Figure 10: Reduced spine density of neurons after 12weeks of cosmic radiation, see [12], [13], [14]

This representative digital images of 3D reconstructeddendrite segments (green) containing spines (red) inunirradiated (top left panel) and irradiated (bottompanels) brains, show that total spine numbers (left barchart) and spine density (right bar chart) are signifi-cantly reduced after exposure to 5 or 30 cGy48Ti or16O or 30cGy16O particles.

Our approach is the following : We use this im-ages to regionalize dendrites and spines and get, byusing the Voronoi diagrams approach, some adjacentareas where spines and dendrites are concentrated.Once, the centers of the adjacent areas are connected,we get a Delaunay diagram which yields to a Ripscomplex on which TDA theory can be applied to getbarecodes.

Firstly, to get the complex of Rips, we choose athreshold distance between the dendrites and spinesregion, that is the average distance between these re-gions for a graph exposed to 0cGy. With this thresh-old distance, we get a filtered graph which leads to aboundary matrix.


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Secondly, we implement a matrix reduction algo-rithm to pairs the persistent data, and represent Thesepairs represent the focus of cosmic radiation exposure.Finally, we calculate the diameter between two distantareas and compare the that obtained for another expo-sure to get prediction tool of the different cosmic ra-diations from the different obtained diameters of twofocus of dendrites-spines cliques). The figure here be-low describes the pipe line of our approach.

Figure 11: The pipe line to get persistent diagramsfrom digital images of a cosmic radiation.

In what follows, we more explain our approach stepby step :

Step 1 : Get the adjacency matrix from an image(as explained by the image here below).

For this purpose we implemented the following algo-rithm :

Algorithm 1 Adjacency matrix

Input: imageOutput: matrix pixel tresholdi, j=size of imagefor for u in i do

for for v in j dopixel=python image pixelif pixel == green or pink, then

matrix pixel(i,j)=1else

matrix pixel(i,j)=0end if

end forend for

Step 2 : It consists to create the associatedVoronoi diagrams by inserting the sites events in a file

Algorithm 2 Voronoi diagramswhile while E is no empty do

withdraw a p eventif this event is a site then

create a new parabolic arccreate the Voronoi edgedelete the depraved circle eventinsert circle events

end ifif this event is a circle then

delete parabolic arccreate the Vorono vertexdelete the depraved circle eventinsert circle events

end ifend while

Here above some images of the dendrite andspines repartition after a cosmic radiation and their as-sociated Voronoi diagrams.

Figure 12: Voronoi diagram issued from a 5 cGy48Tiradiation.


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Figure 13: Voronoi diagram issued from a 30 cGy48Tiradiation.

Step 3 : Build a Deluanay complex from the re-cently constructed Voronoi diagram as outlined by thefollowing picture :

Figure 14: Examples of Delaunay complexes issuedfrom Voronoi diagrams by using the Algorithm 3

Algorithm 3 Delaunay diagramInput : Voronoi pointsOutput : Delaunay tesselationfor u from 0 to n− 1 do

mnb=mostnearneigbor(u)rn=rightneigboor()while rn!=mnb and rn!=-1 do

if rn==-1 thenconvexe[u]=1lneig=leftneigbor()if lneig!=-1 then

get neighborend ifwhile lneig!=-1 do

delaunay tesselation= neighborend while

end ifend while

end for

Step 4 : Its aim is to build a Rips complex fromthe obtained Delaunay triangulation. For this purposewe consider the Wasserstein distance in the followingalgorithm :

Algorithm 4 Rips complexInput : Delaunay GraphOutput : Rips complexj = 0, wass = ri = 0, Ball(a[i]) == a[i]bool receives truefor i from 0 to n− 1 do

while bool doif d(a[i], a[i+ 1]) ≤ r then

add a[i+ 1] to Ball(a[i], r)elser == d(a[i], a[i+ 1])bool receives false

end ifend while

end forRips complex receives Ball

Step 5 : The final step is to get the bound-ary matrix. The idea is that since for all r0 ≥ r,we have Rips(N, r) ⊂ Rips(N, r0), then F ={Rips(N, r), 0 ≤ r ≤ rmax} is a filtration. ThisRips filtration induces a boundary matrix as stated bythe following algorithm :

Algorithm 5 Boundary matrixInput : Rips complexOutput : boundary matrixfor i from 0 to n− 1 do

for j from 0 to n− 1 doif a[j] ==face of a[i] thenM [i, j] receives 1

elseM [i, j] receives 0

end ifend for

end for

Finally to pair birth and death of cycles as men-tioned in the subsection 3.2, we index columns by j,and i = `(j) denotes the line that contains the lowestone in the column j, denoted Cj . A matrix is calledreduced, when no two different columns have theirlower at the same level. To get a reduced boundarymatrix, we appeal this algorithm:


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Algorithm 6 Boundary matrixInput : Boundary MatrixOutput : pairs (i, j)for j = 0..m do

for j < j dowhile `(j) == `(j) doC[j] = C[j] + C[j] mod(2)

end whileend fordraw (`(j), j)

end for

The fruit of this implementation is the following per-sistent diagrams

Figure 15: Persistent diagram that represents the den-drite and spines density under 5 cGy48Ti radiation.

Figure 16: Persistent diagram that represents the den-drite and spines density under 30 cGy48Ti radiation.

5 ConclusionOur persistent diagrams show clearly how much the30 cGy48Ti radiations damage deeply neuronal tissuemore than the 5 cGy48Ti radiations. This shows thepowerful of the topology approach in the data analy-sis. Indeed, the experiences leaded by the the Limoli

staff (as outlined here below) were unable to differen-tiate the negative effects on the neuronal tissue afterradiation 5 cGy48T and 30 cGy48T.

Figure 17: Dendrite density under different radiations

Figure 18: Spines density under different radiations

6 ConclusionThe cosmic ray is classical scientific subject, in thepast many results have been obtained in laboratories,or by some empiric other ones. We propose an algo-rithm that will explore the data from images and getdeep results by using persistent homology. For ourknowledge, this is the first persistent homology ap-proach to identify the intensity of cosmic ray of oneisotop by regarding his damage throughout the den-sity of spine.

The impact in neuroscience of our algorithm, ofcomplexity 2n ln(n), is to get a ready tool that canidentify the intensity and type of isotop in a cosmicradiation, it is fast recognition process of pathologyrelated to cosmic ray. In fact, since our algorithm en-code the intensity of a cosmic ray into a barre code,the further application we work on is to develop asoftware to identify cosmic radiation by analysis somebrain images to localise damaged spines and dendritesdamaged regions: the output will be barcodes whichencode this information.

The reader can see how mathematical abstractionsseem so often to be so powerfully predictive in thereal world, and how pure mathematics (topology) andapplied one (algorithmic) can be married in persistenthomology to get kind of applications in real life.

An unavoidable question that doctors and patientswith brain tumours could not ignore any longer is:What really happens to your brain when you blastit with radiation? They found that their once-bushyneurons looked like they were curbside hedges after


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the township tree had gotten through them, they wereemasculated from their branches and leaves. Based onthis topological approach, we are working on imple-menting a software to predict from any IREM imageof the neuronal tissue, the negative possible impact ofaddict use of the electronic devices.

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