1 Single neuron computation and the separation of processing and signaling

Transcription

1 Physics 178/78 - David Klinfld - Wintr Wk 1 1 Singl nuron computation and th sparation of procssing and signaling This ovrviw srvs to connct simpl notions of statistical physics with nuronal signal intgration, dcision making, and communication. 1.1 Ions and Voltags A good plac to start is with an ovrviw of how snsitiv and robust nrvous systms ar. W considr a fundamntal issu, th scal of th potntials across cll mmbrans rlativ to th thrmal nois floor. W furthr discuss how ths potntials ar usd to form pulss, or action potntials, that allow on cll to communicat with anothr. W shall s that th Boltzman nrgy srvs to dfin a band of voltags that can b usd to intgrat and procss nrvous activity, and that th sam nrgy acts as a barrir to isolat intgration of inputs from output and communication. 1. Th Nrnst Potntial Bags comprisd of lipid walls provid a mans to sparat high concntrations from low concntrations of ions. Why is this usful? If w rcall a littl bit of thrmodynamics, w raliz that a diffrnc in ion concntration lads to a potntial, and thus work can b don. Thus, on could concivably build a dvic that is capabl of communicating to all agnts along th surfac of a bag, or som topological quivalnt, that a disturbanc has occurrd at a point. Lt s run through this. Considr a world that consists of two compartmnts, labld in and out. Th compartmnts ar sparatd by a thin wall. To mak this situation somwhat ral, w will fill both compartmnts with watr and considr th diffusion of ions in th watr. W tak ths ions to b Na + and Cl, th ions that rsult from dissolving ordinary tabl salt. On th insid of th box, th concntration of ions is dnotd [Na + ] in and [Cl ] in and on th outsid thy ar dnotd [Na + ] out and [Cl ] out, whr concntration is in units of molculs/cm 3 or, in chmistry, in units of mols pr litr or Molar(M). To gt a fl for th scal of mols/litr, lts put it into trms rlvant for th siz of a cll, i.., ions pr cubic micromtr. In a biological cll, th ion concntration is about 0.15 M, so w hav about 10 8 ions/µm 3 in a cll. Suppos w st th box up so that, initially, [Na + ] in = [Cl ] in and [Na + ] out = [Cl ] out. This insurs that individual compartmnts ar lctrically nutral. Furthr, w impos [Na + ] out > [Na + ] out, so that th ion concntrations ar initially gratr on th outsid than th insid. Sinc thr is a wall btwn th two sids, thr is clarly no intraction btwn th two compartmnts. 1

2 Suppos w now rmov th wall in ntirty. What happns? Th Na + ions will flow down thir concntration diffrnc, so that ultimatly [Na + ] is th sam on both sids of th mmbran. Similarly, th Cl ions will flow down thir gradint. Thus aftr som tim th concntration of ions is th sam on both sids, so that no nrgy is xtractd from th concntration diffrnc. A sid rmark is that thr is a transint potntial diffrnc if th mobility of Na + and Cl ar diffrnt. What is th trick ncssary to gt th concntration diffrnc across th two sids of th wall to turn into a sustaind lctrical potntial? Th ida is to mak a hol that allows only on kind of ion to pass. This is calld an ion slctiv por. To b concrt, w opn up a hol that allows [Na + ] ions, but not [Cl ] ions, to pass. This is a Na + slctiv por, or Na + slctiv channl. Rcall that [Na + ] is highr outsid than insid. Th procss works as follows: Initially, th [Na + ] movs down its concntration gradint, drivn by diffusion. As Na + ions mov across th wall, th solutions in th two compartmnts ar no longr lctrically nutral. Positiv charg (from th Na + ) lavs th outsid and builds up on th insid. This lads to an lctric fild across th wall. Th lctric filds points from th insid to th outsid. dirction that opposs motion of additional Na + ions. This fild is th In tim, th lctric fild causd by th initial movmnt of ions points from th insid to th outsid. This fild is th dirction that opposs motion of additional Na + ions and will prvnt any mor Na + ions from moving. As this point th systm is in quilibrium. Th rsult is that th concntration diffrnc in Na + ions across th wall has bn usd to form a diffrnc in lctrical potntial across th cll. Th rlation btwn concntration diffrncs and lctrical potntial is givn by th Nrnst quation, which quats th lctrical potntial, V, with th chmical potntial, µ, causd by th concntration diffrnc, i.., µ = ( ) F N T,V = k B T lnz N = k BT ln ξn N! N = k BT lnn + constant (1.1) for N >> 1 so that w can approximat N! by Strling s formula. Thus V = k BT ln[na+ ] out [Na + ] in (1.) W s immdiatly that V is on th ordr of k BT 5mV. Bfor w worry about dtails, lt s think about how big this is on th scal of th lctrical nois xpctd for this procss.

3 1.3 Thrmal Fluctuations in Voltag Bcaus thr is a por, thr is a conductanc G across th cll. This lads to a fluctuation in th potntial, known as th Johnson nois, of siz δv = 4kB T f G = kb T C = ( kb T ) ( L ɛ m ) 4πa = 1 a ( kb T ) ( ) 1 c m π (1.3) whr w usd f = 0 df (πf(c/g)) = G 4C. (1.4) Furthr, C = ɛ m (ara/thicknss), so that for a thin dilctric sphr of thicknss L and radius a, C = ɛ m 4πa L. For most all clls, th ratio ɛm L is c m ɛ m L F 1.0x10 14 µm. (1.5) For a nuron of radius a = 0µm, th nois lvl is found to b δv 0µV. Th important rsult that for clls th nois lvl is much lss than th thrmal voltag k B T/, whr k B T 5mV (1.6) Only at th smallst structur, th synaptic vsicl with radius a 10 µm = 10nm, will th nois lvl approach th thrmal voltag, so that for a 10 µm w hav δv 10mV. Actually, th cas of vsicls is intrsting, in that w can turn th argumnt around and look at th fluctuation in th numbr of ions across th cll. In synaptic vsicls, th mmbran potntial V is st by a hydrogn ion, or ph gradint. Thn V = k BT ] out ln[h+ = k BT [H + ] in (ph in ph out ). (1.7) Typically, ph in 5 and ph out 7.5. Th varianc in th transmmbran voltag in trms of ion concntration is δv = V δ[h + ] [H + in = k BT δ[h + ] in (1.8) ] in [H + ] in W quat nois lvl this with th xprssion for Johnson nois to gt δ[h + ] in [H + ] in = 4 ( k B T C = k B T ) 1 c m πa (1.9) An intrsting numbr is th valu of th radius a for which th fluctuations in ion concntration ar of ordr unity, i.., δ[h+ ] in [H + ] in 1. W call this a crit, whr 3

4 a crit = ( ) 1 30nm (1.10) π k B T c m This is on th small sid of th obsrvd radius of vsicls, or synaptosoms, which is not too bad as an stimat of th smallst cll. A slightly diffrnt way to look at th nois issu is to rcall that Q = N = C V, whr N is th numbr of ions rsponsibl for th nt charg imbalanc across th mmbran. If w substitut this xprssion for C into th formula for Johnson nois, th Signal-to-Nois ratio for th cll is Signal Nois V δ ( V ) = 1 N V k B T/ (1.11) Th dpndnc on th squar root of N is intuitiv. Not that th abov xprssions nd to b modifid if th conductanc is not st by a static por, but a por that fluctuats opn and closd. In th lattr cas, th fluctuations cut off at som high tmporal frquncy. But th major story holds. In th largr pictur, th cll mmbran may hav slctivly prmabl channls to mor than on ion. Thus th final potntial may rprsnt a stady stat rathr than an quilibrium valu. In fact, natur uss two ions, K + and Na +, to st two potntials, on low and on high. Th prsnc of a voltag-dpndnt (nonlinar) conductanc, on of th major findings of th cntury, allows th cll to switch btwn ths two lvls. Lt s rviw th gist of this - th dtails will b th topic of anothr lctur. In th simplst, or Hodgkin Huxly cll, thr ar thr major conductancs, thos du to K +, Na +, and Cl. Th quilibrium potntial for ach of ths ions is V K + = k BT ln [K+ ] out [K + ] in 75mV V Na + = k BT ln [Na+ ] out [Na + ] in V Cl = k BT ln [Cl ] out [Cl ] in +50mV 55mV Th stady stat potntial for a systm with ths thr ions and associatd mmbran prmabilitis P Na +, P K +, and P Cl, is V SS = k BT lnp K+[K + ] out + P Na+[Na + ] out + P Cl [Cl ] in P K +[K + ] in + P Na +[Na + ] in + P Cl [Cl ] out 50mV (1.1) This rlation is drivd by considring th diffusion quation for th total flux of ions through th cll mmbran and is found is standard txtbooks an th Nrst- Plank quation. Th important point is that th stady stat potntial is dominatd by th ion that has th maximal conductanc. Th rsting lvl is dominatd by th K + conductanc. 4

5 1.4 Fundamntals of th Action Potntial Th ssntial ingrdint that allows signaling is that th conductanc for Na + is voltag dpndnt. This is rminiscnt of th drain-to-sourc conductanc in a FET, which is a nonlinar function of th voltag to th gat. A nic way to think of this, which w will discuss in dtail, is that th currntvoltag rlationship has an ssntial nonlinarity. Thus for small disturbancs of th mmbran potntial, th cll rturns to th rsting potntial. Howvr, for currnt injctions byond som critical valu, th potntial will jump to a nw quilibrium point. A simplifid modl maks us of a voltag dpndnt chang in th conductanc for on of two ions. To b concrt, w tak a cll with a solly Ohmic potassium currnt, G K +, and a voltag dpndnt sodium conductanc, G Na +(V ), that has a valu of zro blow a thrshold potntial, V th, and that is constant abov V th with valu G Na +(V ). Thus w hav a currnt-voltag rlation givn by I(V ) = { GK +V G K +V K + if V < V th (G K +G Na +(V )) V (G K +V K + + G Na +(V )V Na +) if V > V th whr, in this approximation, V Na + and V K + ar th Na + and K + Nrnst potntial for sodium and potassium, rspctfully. This rlation is discontinuous at V th and Ohmic blow and abov this potntial. W considr a puls of currnt that causs th cll to chang from th lowr to th uppr curv. This rprsnts th front of th action potntial. Th shift in potntial from V = V K + to V = G K + V K + +G Na + (V ) G K + +G Na + (V ) occurs in roughly 10 4 s. On th longr tim scal of 10 3 s, rlaxation procsss associatd with th Na + currnt and th activation of an additional voltag dpndnt K + currnt caus th front to dcay, so w ar lft with a puls. Th critical lsson is that nurons us two voltag lvls, and at last on voltag dpndnt conductanc, to shift btwn th two lvls. W now know how nurons signal, and that th signaling, at last in principl, oprats wll abov th physical limits to mmbran nois. W put off th issu of variability in th transmission of spiks btwn clls - suffic it to say that at som synapss,.g., thos formd as calycs, th failur rat is vry low indd. 1.5 Sparation of Dndritic Intgration and Communication Our final point concrns how a nuron prforms logic, which is to say how it sparats th intgration of synaptic inputs from th dcision making that lads to th production of an action potntial. W rquir a band of voltags ovr which th cll can intgrat, that is summat, synaptic inputs. Th rang of this band must clarly b largr than th scal of thrmal nois and also larg compard to th activation of th Na + -basd action potntial. Givn th xprimntal fact that th Na + channl turns on through th action of 4 chargs, th rang of synaptic intgration 5

6 is xpctd to xcd V > k B T/4 6 mv. This rang corrsponds to th diffrnc btwn th K + rvrsal potntial (th lowst voltag for inhibitory inputs) and th activation of th Na + -basd action potntial, a rang of about 1-1/-tims k B T/ 35 mv. Othrs may argu that th lvl is th rvrsal potntial for Cl, th dominant inhibitory input in mammals. In this cas th rang is about 1-tims k B T/ 5 mv. Eithr way, w s that th scal for intgration is always of ordr k B T/ and is always larg compard to th nois lvl across th mmbran. How larg ar th post-synaptic potntials that imping on th cll? This distribution has bn masurd by a numbr of invstigators in pair-lctrod masurmnts, whr currnt is injctd into th prsynaptic cll to induc an action potntial and masurd in th post-synaptic. cll. Th typical valus ar around 0.5 mv or lss, or a vry small fraction of k B T/4. But a small prcntag, mayb 5 %, com in at a fw millivolts. Thus coactivation of a small numbr of inputs can, in principl, driv a nuron to spik. Th issu is an opn rsarch qustion; s handout on Mrsic-Flogl data. Th abov argumnt suggst that th cll has hadroom for intgration without firing an action potntial for insignificant changs in input. W now considr how a cll can isolat th synaps from intgration. Th ida is that th action potntial must b larg nough activat a procss whos turn-on occurs far from th rang of synaptic activation. This implis that th activation of synaptic transition, which occurs ovr a rang voltag, must b sparatd from th rang of voltag V > k B T/4 6 mv that govrns th activation of th inward Na + currnt. Synaptic activation dpnds on a chmical cascad that is initiatd by th activation of a high-thrshold voltag gatd (N-typ) Ca + currnt. This currnt paks at intracllular potntials of about + 5 mv, significantly lss than V Na +. Thus thr is hadroom of ordr of k B T/ 50 mv that sparats th turn-on of th action potntial from th turn-on of synaptic transmission, so that dndritic intgration pr s cannot lad to synaptic rlas, or communication, until th thrshold for spiking is crossd. Furthr, th shap of th action potntial will impct th total flux through th high-thrshold voltag gatd Ca + currnt to influnc synaptic rlas. 6