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# Bioelectricity BIOEN 6460

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This 40 page Class Notes was uploaded by Kaylee Schowalter on Monday October 26, 2015. The Class Notes belongs to BIOEN 6460 at University of Utah taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/229916/bioen-6460-university-of-utah in Bioengineering at University of Utah.

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Date Created: 10/26/15

EIDEN 6460 Lecture 3 HodgkinHuxley HodgktneHuxtey Equattons t Ouantttauve Descnptton ot Conductance Assutnpttons Scurrems due to pumps ate negttgtoty stnatt ounngthe actton potenttat 2 Na and K ow through sepatate channets mdependen y 3 Each type ot channet can be tn one ottwo states OPEN ot CLOSED 4 Each channet ts conttotteo by one or tnote tnoepenoent gates 5 A channet ts ctoseo tt at teast one ot the gates ts ctoseo A channet ts open tt aH gates ate open 6 Each Change tn the onentatton ot the charged group Form ot Equattons K the tt VVVKgK VVVNagNa VVVLgL L tsthe teak Whtch We tht constoet negttgtote From aSSumonnS 3 and 6 Open W E Closed Closed 1 Open By By 80 the Fractton ot gates tn the Open state Anothe Fractton ot gates tn the ctoseo state 17y Thetetote the net change tn the ttactton ot open gates dyttvndt ownmey e mey where ameY ey ts the rate of opentng and VMN ts the rate of opentng Thts ts aftrstorder kmettCS This first order differential equation with constant coef cients has the solution of NF yoo yoo yo BAHt where yoo ayOcy y yo W 1 1 Xy y Still ocy and y depend on V voltate since depolarization opens the channels ocy increases with V and y decreases 9 y Closed 0pm 3 Open pr W lt E Wquot Closed Ymay y b l Wquot mama Time From assumptions 3 4 and 5 g gbar yC gbar maximum conductance cofgates ForK Here 5 the shape ofthe conductance fordwferem step poterma s m m thh dryd1 an meny Bn n And therefore nD VWMBn For Na Here are me conductances for dnerent step potenua a to dose and ms pararneter 5 nrne dependent 6 quotEn M actwauon Vamab e h nacwauon Vamab e n fact an eventofprobabmty17hcau5esmacwauon Tne probabmty offacmtauon and no b ockmg 5 therefore mah m39 We eAHL n hm39 Ha eAHnquot n 397 11owbn Pumng m 5H togetnen men gNat gbarNamw3 ho1etrm e39mh This makes sense because at the steady state after activation gNa is approximately 0 because hoe is approximately 0 complete inactivation At resting membrane potential RMP m0 is approximately 0 completely deactivated channel If you try to solve for ocnyocm ochyand n m h We get for Potassium 00110 e um 3H ex 10 Um I J p 10 it Jn01256x 39 I p 80 For Sodium 0125 um um am ex 25 vm 1 m 4explt7 gt 1 10 ll 1 y I 139 I30 Aquot It 1 Llj lcxplj 7 m 39 ll y L 20339 l p When you put these equations together you get an action potential that looks like this 2 t rm um Mode Let s ook at an easter prob em Whtch WtH be used for your homework A cnannet ts etner open wtn conductance 91 arottrary vatue or A cnannet ts dosed wtn conductance 90 A cnannet ts open wttn Some probabmty denned by Pt open probabmty vatue Tnts ts t e c annet tnne tedt and Pt lt pco prooaomty ot gotng from dosed to open Where 0ltpc0lt1 tnen tne cnannet opens ttkewtse tt p p to dosed tnen tne cnannet doses Fora stngte cnannet We can catcutate tne conductance gtt uSmg tne above rutes p eacn tnne step Tne purpose ot tne Mattao exerctse ts to get you corntortaote uSmg Mattao Tne Monte o ake rnacroscoptc cunents Fonunate y Hodgkm Hux ey ktneucs can atso be sowed nurnencauy uSmg parttat dttterenttat equauons Bioelectric Sources Rob MacLeod BIOEN 6460 Bioengineean 6460 W ElectmpWsioogy and Bioelectric Sources Bloelecmclly Monopole Field y Writing a flux equation for current IO r dA Jr4nr2 And then apply the definition of current under assumptions of radial symmetry JroEro 6 6r Equating the two we get a l O X 6 r 47wr2 With solution 0 7 C 4nor True monopoles never arise in biology but that does not stop us from using them Bioengineean 6460 W ElectmpWsioogy and Bioelectric Sources Enosecsz can write Dipole Field For the potential gradient from displacing the monopolar source 10 we 2 86 475 1 6 Now if we de ne a dipole moment as j Z 06 Assuming we take the limits for 60 and 10 p 1 vlt gt 47m 7 1 1 A Which reduces to V 270 p 1 A 7quot 7 z I 2T Note allfor infinite 47139039 7 homogeneous Bioelectric Sources volume 35 Represent bioelectric sources Membrane currents Coupled cells Activation wavefront Whole heart Why Dipoles 39h gt Bioengineering 6460 Electmphysiology and Bioelectric Sources Bioelectricin Equivalent Sources Match celltissue structure to current sources Multiple models possible depending on formulation and assumptions Typical assumptions uniform characteristics of tissue simple geometries oeqbe ltlt 011131 so that szqbi Primary versus secondary sources Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Equations for Distributed Sources For dipole source not at the origin w I I 0 pxyz O source I i l Lina39r becomes 10 I m C 47ror r l and 17 1 17 1 A Imv quot mm 47m 47TH 27 becomes 13 1 1 7 p i m v m 47m Ir M 47T0 F F Fr I 391 1 1 MP l y 1 t 3 32 Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Volume Dipole Sources If we assume there is a volume dipole density we can write for the extracellular field 1 1 ltIgt mo vltgt W 47rd Or more generally for sources at p not on the origin Note the equation I Z 1 I V L dV becomes invalid 47m V v 77 when the pointp 1 moves inside the 37 F a r 7 1V39 volume 47W V U 7 F li Bioengineering 6460 Electmphysiology and Bioelectric Sources Biosadmit Surface Dipole Sources If the dipoles are distributed on a surface we can write for the potential 1 1 1 I Z 4 H l I r 47m QspSO 7 m2 r ndS which we can rewrite as lt1gt rs 9774 7 4770511 By defining the differential solid angle as 7 7quot 77 77 d zw 39 dS vhf392 n FFll3 ndS Bioengineering 6460 Electmphysiology and Bioelectric Sources Biosadmit Surface Dipole Sources 1 4 cm 2 Ami psr 39 dS W gt If the dipole r 39 7 Pixyz distribution is uniform we can further simplify to p59 gt I 7 n 4W0 rjiquot 77 6 NW Bioelectric Sources Bioengi ee ing 6460 Electmphysiology and Bioelec tricity Solid Angle Area of unit sphere 7dsxyz T T F F do d39 39 77 7112 3 lie m d3 NV Bioengineering 6460 Electmphysiology and Bioelectric Sources Bioelec tricity Potential from a Dipole Layer Assume a disk of radius a and uniform dipole distribution ps y We can write for this case 1 a 2 I 2 Vl th solution Z 2 i Z Z 9 q p8 Z 20 a2z212 Note This solution suggests a jump in potential ofpSo at the disk And at F the solution is zero as expected P3 I Z E gilvea int surface p8 01 0 p urce Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Potential from a Single Cell The boundary conditions at the cell membrane are Oeqe e Vm W 08qu 0 ENDS J S 1 8n 8 an m If we de ne a new variable as 11 01 And write the boundary associated boundary conditions g we V aw at 02 06 an an Note that now only the value of is discontinuous not its derivative Thus we can define an equivalent dipole source in terms of 1P Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Equivalent Source Single Cell Recall that for a dipole surface we had 1 lt1gt 7 77 do 4770 six M And for a potential step of q we had equivalent dipole source of p3 01 This allows us to write forthe single cell case 1 1 R 5 1 e I rzWer Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Equivalent Source Single Cell From the equation 1 Ir 74 me e an 7T0 S We can substitute back form and write 1 T 06136 O39Z Qid9rrl 4770 3 Note that this assumes qgte evaluated at the membrane and q outside the cell Vl th the usual simplification 06De ltlt 0i1i we can write 1 7139 10 de rr 477 06 3 Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Plxyyyz nfinite Cable Source We can use the same approximate equation for the cell to an infinite cable with conductivity 0i in an infinite medium of conductivity 0e Starting again from the equation 1 0i 7 477 039 And assuming that potential is constant over the ber cross section can expand the solid angle and apply the divergence theorem to get P zei anAV dV 4w 06 V on 5 Vim Bioengineering 6460 Electmphysiology and Bioelectric Sources Biosadmit Vm dQM S Infinite Cable as Volume Dipole a 1 1 7 MT E V WM dl We can compare this equation to 1 lt1gt w v 2 Abram F r Which indicates that the spatial gradient of potential can be an equivalent volume dipole density Bioengineering 6460 Electmphysiology and Bioelectric Sources Biosadmit dV Cable Example Space Signals 3V Time Signals Depolarization 2 an Repolarization 1I39 21 1 2 6 6239 z39 i Sum lt lt 4 39 gt gt DU an e lt lt lt lt gtgt Gigz7ez lt lt lt lt gt39gt lt lt lt lt lt i gt I g 1 0i avm l 1 r e x V g g diz 47 09 V 61139 r r AJ Bioengineering 6460 Electrophysiology and Bioelectric Sources Bioelec tricity Dipole Characteristics Surface dipoles when the surface is Closed the solid angle goes to zero and so does potential potential always jumps as we cross the dipole surface by a value proportional to the dipole strength Cable models derivation for cable based on the unnecessary assumption of constant potential Vm across the extended membrane surface integration only necessary over active region of the cable 2 different equivalent models for the same thing Bioengineering 6460 Electrophysiology and Bioelectric Sources Biosadmit Application to Fiber Bundles Syncitial tissue Treat a bundle of cells as a single fiber Requires that cells connect longitudinally and transversely Example Purkinje fibers in the heart Spach 1972 Excised nerve bundle Cells are not homogenous and fire independently We can solve if we know the surface potential Assume potential is the same as for an isolated bundle no normal current Assume fiber is long compared to cross section Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Excised Nerve Bundle If we assume that a we know the bundle surface potential 10 and b these same potentials 10 apply to both an isolated and embedded bundle it is possible to write generally 1 I on 3V dV 47rd V le where ISV is the intracellular current and by exploiting Green s second identity 1 1 u oov39 w ag LynS if m Assuming a long circular nerve bundle and uniform surface potential we can approximate this further as 1392 2 1 w 1A 4n fl 8513 F F l Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit 10 Multipoles Higher order expansion of solution to Poisson s equation Monopole dipole quadropole octopole Example two wavefronts in cardiac tissue Bioengineering 6460 W ElectmpWsioogy and Bloelectrlc Sources Bloeecl clly Cardiac Sources Formulation in terms of cells impossible Fibers eg conduction system Volume dipole density Surface dipole density Requires model of propagation Bioengineering 6460 W ElectmpWsioogy and Bloelectrlc Sources Enosecsz Network of Cables Starting from the previous expressions for an infinite cable we can write 1 8 1 lt1 7lt175lt1gtP5 p 47706 V 93739 0 a 1 V 721 ml Which with the usual simplifications we can write in terms of Vm 1 1 Mp 39 V 11quot 47T 0396 x r i 7 i Bioengineering 6460 Electrophysiology and Bioelectric Sources Bioelec tricity Dipole Layer gt We wrote before for the infinite core cable pxyZ 74 6 Now if we have a wavefront across which voltage jumps we can write r 7 VMZAimwi z T 6 mm U In Te Ud U m T6 1 1 1 WV 15 p 47rd Sm p 7 4770 Sm N Bioengineering 6460 Electrophysiology and Bioelectric Sources Biosadmit 12 Bidomain Model Ve 8e Intracellular Vi 8i 2 domains share the same space Membrane separates the domains All properties V g i are macroscopic averages Bioengineering 6460 Electmphysiology and Bioelectric Sources Biosadmit Homogeneous Isotropic Bidomain Starting from the same bidomain equations as before i e iii 7 V mg 31 V 39 G V mm V V K 6 We assume homogeneous and isotropic conditions and can then write from Poisson s equation V 39 Imp v 39 Imp Adding the two gives us V 39 Z V 39 Bioelectric Sources Bioelec tricity Bioengineering 6460 Electmphysiology and 13 Homogenous Isotropic Bidomain Volume Equations We can rewrite the previous equation in terms of Vm as V 39 gcv e V 39 gquVm and with the definitions gige geq and Jeq geq m as V 39 gevcpe Vjeq which is a Poisson s equation NW Bioelectric Sources Bioengineering 6460 Electmphysiology and Bioelec tricity Homogenous Isotropic Bidomain Source model The previous Poisson s equation has the solution 1 a 1 qgt r w JagWW dV 47796 i7 7 which we recognize as an equivalent volume dipolar source equation with 23V Jeq gquVm Actually solving this equation requires additional boundary co ditions geometry etc see Gulrajani Bioelectric Sources Bioengineering 6460 quot E39mmph ii i 14 Uniform Dipole Layer We can integrate the previous expression for volume density over the wave front surface to get the total dipole moment 5V ng gquV which applied to this geometry yields ng Volume 2 gquVm Ad In UDL anisotropy will V V An 96q d A 7 not affect source gquPAn strength Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit Intrinsic Deflection PxyZ From the voltage step across a dipole layer that we have seen previously we can write Vwave Ave 99513 2 wit VI the IntrInSIc deflection 96 W 96 of the wave front N Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Biosadmit J 15 Anisotropic Bidomain We can start again from the basic equations V V39GVltIgtWIi ll 1 V G V e mv15v 5 ll and once again subtract to get V vacbe vGVlt1gt or v G WDC v own with G x G G Bioelectric Sources Bioelec tricity Bioengineering 6460 Electmphysiology and Anisotropic Bidomain Volume Sources Starting from v vacb ev Ggw We recognize this as another Poisson s type equation Ifthe interstitial space in homogeneous ie no rotation we can describe a current source j leq Nt th th oe e e I eq 1 Z prime on the GS and If both spaces are homogeneous denotesloca39ly oriented I I conductivity V 39 G VQDB iv 39 tensorsie leads by the same arguments to 23223539 Jeq GVVm which acts in a homogenous bulk conductivity of GzGiGe To solve forthe extracellular fields we substitute the source term back into one ofthe previous expressions and solve with appropriate bo ndary conditions and numerical methods Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Bioelec tricity 16 Anisotropic Equivalent Dipole Layer Thin wavefront approximation Axial symmetry of conductances Bioengineering 6460 Electmphysiology and Bioelec tricity NW Bioelectric Sources Anisotropic Bidomain Equivalent Dipole Layer We can write an approximation based on a source layer as MGZV Vm Gigg with Vp Vd V across distance d which has the solution assuming axial symmetry of 1 r A A A 1 14237 11 z E Vplgimyy gz mzz 97171wa V f dS with X 9t91y 32 9t91ltz Z 2 35 Bioengineering 6460 Electmphysiology and Bloelectrlc Sources Bioelec tricity 17 Oblique Dipole Layer We can rewrite the previous equation as ltIgtexyz E17 S V7315 dS O I 1 And as a modification ofthe potential jump across the wave front A V l 7 o front p 2 2 git get3m Y gii geiC03 7 NW F Bioelectric Sources Bioengineering 6460 Eiectmpngleticggimgj 18 Electrocardiography and Electroencephalography Rob MacLeod BIOEN 6460 W ECG and EEG Bloenglneellng ommfiecrrophysioiogy and Bioelectricity Components of the Electrocardiogram ECG Sources Potential differences within the heart Spatially distributed and time varying Volume conductor Inhomogeneous and anisotropic Unique to each individual Boundary effects ECG measurement Lead systems Bipolar versus unipolar measurements Mapping procedures Analysis Signal analysis Spatial analysis Dipole analysis Simulation and modeling approaches W ECG and EEG Bloenglneellng ommfiecrrophysioiogy and Bioelectricity ECG History and Basics Represents electrical activity not contraction Marey 1867 first electrical measurement from the heart Waller 1887 first human ECG published Einthoven 1895 names waves 1912 invents triangle 1924 wins Nobel Prize Goldberger 1924 adds precordial leads 4pk Milllvolts Malawi 1 o 1 PR Interval QT Interval l7 I OHS complex ECG and EEG Bioengineering 6460 Eieetmpnglegli gr Electrophysiology Overview Pacemaker potential 1 90 Pacemaker cells SA Node AV Node Purkinje Fibers Overdrive suppression Conduction system Varied propagation Ventricular myocytes Electrical coupling Anisotropy The Electrocardiogram ECG 4pk Sinoatrial node Atrium Atrioventricular Bundle of His Purkinje fiber in false tendon Terminal Purkinje h r Ventricular muscle fihnr ft Plateau phase 0 ECG and EEG Time ms 500 us 12 T Vei rep 5 mv L T o 9 S 3322133 Cardiac Activation Sequence and ECG 7 Am D n m PWIVI Swlve H p 5 PO segment ST 599quot R I Q 1 Mi P 39 l a van mt a wave ff T wave 5y i ii a mv P T 4L o g 9 5 R wave The end 39 E N a vi l s P F T i i 7 o as EMU ancl EEU wioengineenng You 7 treczropwsiology and Bloeeclrlclly assumptions Typical assumptions simple geometries ECG and EEG 4W uniform characteristics of tissue Primary versus secondary sources Equivalent Sources Match celltissue structure to current sources Multiple models possible depending on formulation and Bioengineering 6460 r Electropt ysloogy and aloe ecmcny Cardiac Sources Formulation in terms of cells impossible Dipoles multipoles simple but incomplete Volume dipole density hard to describe Surface dipole density good compromise in some problems All require some model of time dependence propagation ECG and EEG Bioengineering 6460Eiectrophysioiogy and Bioseczmizy Heart Dipole Approaches Treat the heart as single dipole Fixed in space but free to rotate and change amplitude Einthoven triangle Vector ECG Vectorcardiogram Lead fields generalization of heart dipole ECG and EEG Bioengineering 6460Eiectrophysioiogy and Bioseczmizy Heart Dipole and the ECG Represent the heart as a single moving dipole ECG measures projection of the dipole vector Why a dipole Is this a good model How can we tell AW ECG an d E E G Bioengineering 6460 Electmphysiology and B ioelec tricity Cardiac Activation Sequence as a Moving Dipole Oriented from active to inactive tissue Changes location and magnitude Gross simplification that is clinical important AW ECG and EEG Bioengineering 6460 Electmphysiology and B ioelec tricity ELL Electrocardiographic Lead Systems Einthoven Limb Leads 18954912 heart vector Einthoven triangle string galvanometer Goldberger 1924 adds augmented and precordial leads the standard ECG Wilson Central Terminal 1944 the indifferent reference Frank Lead System 1956 based on three dimensional Dipole Body Surface Potential Mapping Taccardi 1963 ECG and EEG Bioengineering 6460 Electrophysioogy and Bioeectricity Einthoven ECG Bipolarlimb leads Einthoven Triangle Based on heartvector ECG and EEG Bioengineering 6460 Electrophysioogy and Bioeectricity Augmented Leads Provide projections in additional directions Redundant to limb leads ie a no new information L R CWVF L R L avL avF avR CT avR ECG and EEG Bioengineering SASOWEecllophySOOgy and Bioeeclicily V lson Central Terminal Goldberger 1924 and Wilson 1944 Invariant reference Unipolar leads Standard in clinical applications Driven right leg circuit 5quot CENTRAL TERMINAL ECG and EEG Bioengineenng SASOWEecllophySOOgy and Bioeeclicily Precordial Leads C awcula declavwcular Modern clInIcaI standard V1 m 6 Note enhanced precordials on right side of chest and V7 ECG and EEG agengmemg 5450 w Eedrophysoogyand Emsslam Projection Summary sAGITrAL PLANE z FRONTAL PLANE X TRANSVERSE PLANE ECG and EEG agengmemg 5450 w Eedrophysoogyand Emsslam Vectorcardiographic Lead Systems i X SAGITI39AL FRONTAL Y TR ANS VE R S E ECG an d E E G Bioengineering 6460 Electmphysiology and Bioelec tricity Frank Lead System Vectorcardiogram R HEAD ECG an d E E G Bioengineering 6460 Electmphysiology and Bioelec tricity Body Surface Potential Mapping Measurements over entire torso Showed that resulting pattern was not always dipolar More complex source model than dipole required AW Bioelectricity ECG and EEG Bioengineering 6460 Electrophysiology and Body Surface Potential Mapping 118 right arm 119 left arm I t 6 120 e 69 g V 19 27 34 41 48 55 62 89 97 20 28 65 42 49 56 63 82 90 98 21 29 86 43 50 57 64 77 83 91 99 1 8 15 69 73 78 84 92 100 108 114 2 9 16 85 93 101 109 115 a 10 17 24 a2 39 46 53 so 67 71 75 80 86 94 102 110 116 8 Q 4 11 18 25 as 40 47 54 61 68 72 76 81 87 95 103 111 117 D Q I I I 1 2 3 4 5 6 7 8 910 11 12 13 14 15 16 17 18 ECG and EEG Bioengineering 6460 Electrophysiology and W Bioelectricity FeaturePattern Analysis PTCA Mapping LAD RCA LCx AW ECG and EEG Bioelec tricity Bioengineering 6460 Electmphysiology and Lead Vector Burger and van Milaan 1940 s Recall that for a di ole p 2059 Now generalize this idea to VAB IDA DB Lacp c Lypy szz 47m VAB L 3917 L lead vector depends on lead location dipole location and torso geometry and conductivity B amp vM used phantom model oftorso with dipole source to estimate L AW ECG and EEG Bioelec tricity Bioengineering 6460 Electmphysiology and 11 Lead Field Helmholtz s reciprocity VAB for current flowing from P to Q VPQ for the same current flowing from A to B We had before that VAB L quotI7 For any current I injected into the torso via A and B we will get a current density J and from that an electric eld E related by Ohm s Law as j E 039 ECG and EEG Bioengineering 6460 Electmphysiology and Bioelec tricity Lead Field Derivation We can compute the voltage from this current density as P VPQ E dl Q Then assuming E does not change much overthe short distance dl we can approximate VPQ E AZ From the definition of a dipole asp I dlwe can write V i Bow i E5 PQ I I From VAB VPQ by reciprocity we can then define L in terms ofE and I E AW M ECG and EEG Whit Bioengineering 6460 Electmphysiology and Bioelec tricity 12 Lead Field Based Leads McFee and Johnston 1950 s Tried to define leads such that E and l were constant overthe heart volume This way dipole movement would not change L Developed lead system on this basis from torso phantom measurements Performance was improved for homogenous torso but the same for realistic torso ECG and EEG Bioengineering 6460Eleclrophysiology and aloeeczmizy Multipoles Higher order expansion of solution to Poisson s equation gt Monopole dipole gt quadropole octopole Example two wavefronts in cardiac ssue I II ECG and EEG Bioengineering 6460Eleclrophysiology and aloeeczmizy Multipole Based Models R GHTVENTR CLE SEPTUM LEFTVENTWCLE W ECG and EEG mengmeermg ede iEkctmphysmugyand aweecch ECG Example 1 43W ECG and EEG Emengmeermg my 7 Electmphysmugy and Eweecch ECG Example 2 Mnrmll ECG M NH ECG Example 3 H prpv Normal ECG ECG Example 4 PITPlTIlIIIII ECG and EEG Bioengineering 6460 Electmphysiology and Bioelec tricity Basics of the EEG Sources Cortical layer 5 pyramidal cells currents of O78 to 297 pAm Burst of 1000050000 synchronously active pyramidal cells required for detection Equivalent to 1 mm2 of activated cells Modeled as a current dipole ECG and EEG 43911 IA 1 t i quotq w AI 5 IV Bioengineering 6460 Electmphysiology and Bioelec tricity 16

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