1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations

Transcription

1 Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys a set of conductancebased equatons for the transmembrane potental that leads to subthreshold and spke actvty. Our goal s to derve an expresson for the frng rate of the neuron n terms of the synaptc nput to the cell. The equatons are solved n a self-consstent manner n the sense that the output of each neuron contrbutes to the synaptc nput of every other neuron. We are motvated to ths proof as a means of connectng sngle-cell equatons, whch are complcated, to smplfed network equaton, whch are smple n that each cell s specfed solely by ts frng rate. Our goal s to understand the weghts W j n terms of cellular propertes. We follow the dervaton lad down by Sompolnsky, whch holds for averagng over nputs that arrve wth a Posson (exponental) dstrbuton for a large networks. A smlar dervaton, whch holds for averagng over a large number of spkes by a sngle synaptc nput, was gven by Ermentrout. In each case the crtcal ssue s that the network dynamcs are asynchronous. Krchoff s Law for One Neuron dv (t) C g L [V (t) E L ] I actve (t) = I appled (t) (1.1) where E L = reversal potental of leak, I actve (t) = all tme-dependent actve currents, such as the Na, K, and Ca 2 actve currents, and I appled (t) = all tme-dependent appled currents. The appled current contans terms from external nputs as well as lateral, or network nteractons. Thus, C dv (t) = g L [E L V (t)] I actve (t) I appled (t) (1.2) where g L s the leakage conductance of the membrane and E L s the leakage reversal potental, nomnally the restng potental when all synaptc nputs are turned off. Let Λ = f [ I appled I offset] (1.3) where = f ( I appled I offset) = frng rate, f s the functonal form of the frng nput-output curve. Ths relaton gans possble tme dependence through the tme 1

2 dependence of t s arguments, and for sem-lnear relatons such as Λ = f [I] has unts of nverse Coulombs. We ncorporate the leak conductance through I offset = I offset c E c g L (1.4) where the constant E c scales the dependence of frng threshold on the leak conductance. Thus Λ = f [ ] I appled Ic offset E c g L. (1.5) Ths formalsm assumes that changes n conductance shfts the threshold level of the f I curve, but does not effect the slope of the curve. Ths s approxmately true, at least based on the experments of Reyes and Abbott (Neuron). It corresponds to a shftng threshold for the nput-output curve rather than a change n the gan or slope of the curve. Network Equatons C dv (t) = g L [E L V (t)] I actve (t) I external (t) I network (t) (1.6) where we have st the appled nputs nto two terms: nputs from the external world come n through I external (t) and nputs from other cells n the network (lateral or feedback connectons) come n through I network (t). The g L [E L V (t)] term wll term to be pvotal as we wll augment the leakage conductance wth synaptc conductances. We consder the form of I external (t), the nput to the -th neuron, frst,.e., I external = g n (t) [E n V (t)] (1.7) The conductance weghts the external nput and s descrbed by a frst order equaton dg n τ n g n = G n τ n λ n (t) ; t 0 (1.8) where the nput λ n (t) s taken to be a Posson process wth mean rate Λ n. The formal soluton for the state (steady state of nhomogeneous response) s gven by g n (t) = G n e (t t )/τ n λ n (t) (1.9) 2

3 so that the average over a large number of nputs, ether separate presynaptc nputs or many post-synaptc potental from one nput by a very slow synapse, s g n (t) = G n = G n Λ n e (t t )/τ n λ n (t) (1.10) G n Λ n = G n Λ n τ n e (t t )/τ n e (t t )/τ n where the product Λ n τ n s just the number of post-synaptc nputs (or presynaptc spkes from all neghbors, snce we do not nclude synaptc depresson and other tme-dependent synaptc effects) that occur n the tme-constant of the postsynaptc cell. 1 Look at rato of mean to standard devaton,.e., τ. Ths fracton decreases n Λn as ether the tme-constant goes to nfnty or the frequency goes to nfnty. We now have I external G n τ n Λ n [E n V (t)] (1.11) We turn to the current that results from synaptc nputs,.e., I netwrk n g j (t) [E j V (t)] (1.12) where g j (t) s the post-synaptc conductance trggered by pre-synaptc spke τ j all dg j g spkes j = τ j G j δ (t t j ) ; t 0 (1.13) t j where the summaton s over the spkes n pre-synaptc neuron j and we recall that the unts of δ(t) are 1/s. We replace the spatal summaton n synaptc nput by the ensemble average, as n the case of the external nput. Ths holds for Posson frng rates among the neurons n the network. Thus g j (t) = G j e (t t )/τ j all spkes t j δ (t t j ) (1.14) so that the average over a large number of nputs (ether separate presynaptc nputs or many post-synaptc potental from one nput by a very slow synapse) s 3

4 g j (t) = G j = G j Λ j e (t t )/τ j G j Λ j = G j Λ j τ j e (t t )/τ j e (t t )/τ j all spkes t j δ (t t j ) (1.15) where the ntegral s just τ j Λ j = mean number of spkes. The network contrbuton to the current to the cell become I network The sum of the external and network currents s thus G j τ j Λ j [E j V (t)] (1.16) I appled = I external I network (1.17) G j τ j Λ j [E j V (t)] = G n τ n Λ n [E n V (t)] where N s the number of neurons n the network. Ths expresson has constant terms and voltage dependent terms. Let s put all the voltage terms wth respect to E L, so that the nput gans a term that appears n form smlar to that of g L. That s. we add and subtract terms to form I appled = G n τ n Λ n (E n E L ) G n τ n Λ n G j τ j Λ j (E j E L ) (1.18) G j τ j Λ j [E L V (t)] The frst two terms to the rght of the equalty are constant,.e., ndependent of V (t). We consder ths an an effectve appled current, denoted I appledeffectve I appled effectve G n τ n Λ n (E n E L ) G j τ j Λ j (E j E L ) (1.19) The mportant thng s that the external and network terms have a voltage dependence that s proportonal to the dfference between the synaptc reversal potental and the leakage reversal potental,.e., the restng potental. 4

5 The second term appears as a leakage current, n whch the synaptc nput adds to ths leakage. We defne an effectve synaptc conductance, g synapse shunt that adds to the leakage conductance g L,.e, g synapse shunt G n τ n Λ n G j τ j Λ j (1.20) The results of our efforts s that we can wrte Krchhoff s law for one neuron, rather than the network, wth effectve parameters.in partcular, (g L g synapse shunt ) s the effectve leak current. C dv (t) = (g L g synapse shunt ) [E L V (t)] I actve (t) I appled (1.21) The nteractons between neurons enter ths equaton though through the synaptc nputs and ther effect on g synapse shunt. We fnd ths term from the modfed frng rate as Λ = f [ I appled = f I offset c G n τ n Λ n (E n E L ) E c g L E c g synapse shunt ] = f G j τ j (E j E L E c ) Λ j G j τ j Λ j (E j E L ) I offset c G n τ n (E n E L V c ) Λ n (1.22) G j τ j Λ j E c ( ) Ic offset g L E c g L E c G n τ n Λ n E c It s worth notng that the equatons were derved under the assumpton that spkng s a Posson process. Clearly a sngle cell model can be determnstc. Ths one must add thermal nose or parameter varaton to randomze the actual tme of spkng about the desred norm. We can now dentfy terms n the conductance equatons that correspond to terms n our network equatons. In partcular, Synapses W j G j τ j (E j E L E c ) (1.23) s the synaptc effcency, or connecton strength, between neurons n the network. External Drve W n G n τ n (E n E L E c ) (1.24) s the synaptc effcency, or strength, for an external nput to the neuron. 5

6 Threshold θ I offset c g L E c (1.25) s the threshold denoted θ. Thus the equaton(s) for the frng rate of the cell becomes Rate Equaton Λ = f W j λ j W n Λ n θ (1.26) We now have derved the rate equatons, under the assumpton that the threshold of the frng rate versus current curve s shfted by the changes n synaptc conductance and that the cell receves multple nputs (PSPs) durng each ntegraton perod (nomnally τ j ). It s not surprsng that the synaptc effcency depend on the conductances,.e., W j G j. It s surprsng that the synaptc effcency depends on the combnaton E j E L E c ; the synaptc reversal potental must be taken relatve to E L E c. Man Result A fnal pont s that we need a dfferental equaton for the rates λ, or for the dscrete case a dfference equaton, so that the rates evolve over tme. Ths s equvalent to lettng the nput to the neuron evolve wth the tme-constant of the neuron. Thus for the contnuous case, usng our prevous notaton µ as the nput to the cell, we have τ du (t) and, for example, take f to be sem-lnear: u = W j λ j (t) W n λ (t) n θ (1.27) Λ = f [µ ] (1.28) Whle for the dscrete case, we have smply Λ (t 1) f W j Λ j (t) W n Λ n (t) θ (1.29) In terms of the prevous symmetrc and normalzed notaton, we have S = (2λ λ max )/2λ max, whch ranges between -1 and 1. For the specal case of bnary neurons,.e., S = ±1, whch corresponds to λ wth a saturatng frng rate, λ max, we have S (t 1) W j S j (t) W n S n (t) θ (1.30) 6

7 Response of Homogeneous Populaton of Neurons We consder a homogeneous populaton, the smplest network, to gan nsght nto behavor of the network therefore Λ (t 1) = f W W j W and Λ n Λ n Λ j (t) W n Λ n θ and homogenety across all neurons gves a steady-state (t ) result of (1.31) Λ = f [ NW Λ W n Λ n θ ] (1.32) where the rate s the same for all neurons (Λ Λ). Thus f NW Λ < W n Λ n θ then Λ = 0 f NW Λ > W n Λ n θ then Λ 0 For the nonzero case: Λ = fnw Λ f(w n Λ n θ) (1.33) where we us the captal Λ snce the output s pecewse lnear,or Λ = W n Λ n θ 1 fnw (1.34) We see that the average frng rate, Λ, monotoncally ncreases as the number of neurons, N, ncreases, untl t reaches a sngularty at N = 1/(W f). In realty, saturaton occurs frst, whch can be ncluded by addng, for example, a term proportonal to -[I I c ] 2 to the gan (f I) functon, whch stll allows us to get an analytcal result. 7