9 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
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1 Physcs 171/271 - Chapter 9R -Davd Klenfeld - Fall 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 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 self-consstent 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 Posson nputs 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 n asynchronous. Krchoff s Law for One Neuron dv (t) C g L [V (t) E L ] I actve (t) = I appled (t) (9.1) where E L = reversal potental of leak, I actve (t) = all tme-dependent actve currents, and I appled (t) = all tme-dependent appled currents. Thus, smplfyng the notaton slghtly C dv (t) = g L [E L V (t)] I act (t) I app (t) (9.2) Let f = β [I app I c ] (9.3) where f = frng rate, β = gan, and we ncorporate the leak conductance through I c = I o c v c g L (9.4) where the constant v c scales the dependence of frng threshold on the leak conductance. Thus f = β [I app I o c v c g L ] (9.5) 1
2 where f s tme-dependent through the tme dependence of ts arguments. 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. See gan curve reyes eps Network Equatons C dv (t) = g L [E L V (t)] I act (t) I ext (t) I net (t) (9.6) where nputs from the outsde come n through I ext (t) and nputs from other cells n the network come n through I net (t). We consder the form of I ext (t), the nput to the -th neuron, frst,.e., I ext = g n (t) [E n V (t)] (9.7) The conductance weghts the external nput and s descrbed by a frst order equaton dg n τ n g n = G n τ n R n (t) ; t 0 (9.8) where the nput R n (t) s taken to be a Posson process wth mean rate f 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 R n (t) (9.9) 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 f n = G n f n τ n e (t t )/τ n R n (t) (9.10) e (t t )/τ n where the product f 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. 2
3 Look at rato of mean to standard devaton 1 τ n f n tme-constant goes to nfnty and as the frequency goes to nfnty. We now have I ext. Ths goes down as the G n τ n f n [E n V (t)] (9.11) We turn to the current that results from synaptc nputs,.e., I net n g j (t) [E j V (t)] (9.12) where g j (t) s the post-synaptc conductance trggered by pre-synaptc spke τ j dg j g j = τ j G j t j δ (t t j ) ; t 0 (9.13) where the summaton s over the spkes n pre-synaptc neuron j. 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 t j δ (t t j ) (9.14) 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 g j (t) = G j = G j f j = G j f j τ j e (t t )/τ j t j δ (t t j ) e (t t )/τ j (9.15) ]] where the ntegral s just τ j f j = mean number of spkes. The network controbuton to the current to the cell become I net The sum of the external and network currents s thus G j τ j f j [E j V (t)] (9.16) I ext I net = G n τ n f n [E n V (t)] G j τ j f j [E j V (t)] (9.17) 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. 3
4 I ext I net = G n τ n f n (E n E L ) G n τ n f n G j τ j f j (E j E L ) (9.18) G j τ j f j [E L V (t)] The frst term to the rght of the equalty s a constant,.e., ndependent of V (t). We consder ths an an effectve appled current I app G n τ n f n (E n E L ) G j τ j f j (E j E L ) (9.19) 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. The second term appears as a leakage current, n whch the synaptc nput adds to ths leakage. We defne an effectve synaptc conductance, g syn that adds to the leakage conductance g L,.e, g syn G n τ n f n G j τ j f j (9.20) The results of our efforts s that we can wrte Krchoff s law for one neuron, rather than the network, wth effectve parameters, In partcular, (g L g syn ) s the effectve leak current. C dv (t) = (g L g syn ) [E L V (t)] I act (t) I app (9.21) The modfed frng rate s now f = β [I app Ic o v c g L v c g syn ] (9.22) = β G n τ n f n (E n E L ) G j τ j f j (E j E L ) Ic o g L v c G n τ n f n v c G j τ j f j v c = β G j τ j (E j E L v c ) f j G n τ n (E n E L v c )f n (Ic o g Lv c ) We can now dentfy terms n the conductance equaons that correspond to terms n our network equatons. In partcular, Synapses W j G j τ j (E j E L v c ) (9.23) 4
5 s the synaptc effcency, or connecton strength, between neurons n the network. External Drve W n G n τ n (E n E L v c ) (9.24) s the synaptc effcency, or stength, for an external nput to the neuron. Threshold θ I o c g Lv c (9.25) s the threshold denoted θ. Thus the equaton(s) for the frng rate become Rate Equaton f = β W j f j W n f n θ (9.26) We now have derved the rate equatons, under the assumpton that the threshold of the f I 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 v c ; the synaptc reversal potental must be taken relatve to E L v c. Man Result A fnal pont s that we need a dfferental equaton for the rates f, 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 and τ du (t) u = W j f j (t) W n f (t) n θ (9.27) Whle for the dscrete case, we have smply f = β [µ ] (9.28) f (t 1) β W j f j (t) W n f n (t) θ (9.29) In terms of the prevous symmetrc and normalzed notaton, we have S = (2f f max )/2f max, whch ranges between -1 and 1. For the specal case of bnary 5
6 neurons,.e., S = ±1, whch corresponds to β wth a saturatng frng rate, f max, we have S (t 1) W j S j (t) W n S n (t) θ Response of Homogeneous Populaton of Neurons (9.30) We consder a homogeneous populaton, the smplest network, to gan nsght nto behavor of the network therefore f (t 1) = β W W j W and f n f n f j (t) W n f n θ and homogenety across all neurons gves a steady-state (t ) result of (9.31) f = β [ NWf W n f n θ ] (9.32) where the rate s the same for all neurons (f f). Thus f NWf < W n f n θ, f = 0 f NWf > W n f n θ, f 0 For the nonzero case: or f = βnwf β(w n f n θ) (9.33) f = β 1 NWβ (W n f n θ) (9.34) We see that the frng rate, f, monotoncally ncreases as the number of neurons, N, ncreases, untl t reaches a sngularty at N = 1/(T β). 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. 6
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