8 Derivation of Network Rate Equations from Single- Cell Conductance Equations

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1 Physcs 178/278 - Davd Klenfeld - Wnter Dervaton of Network Rate Equatons from Sngle- Cell Conductance Equatons We consder a network of many neurons, each of whch obeys a set of conductancebased, Hodgkn-Huxley 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 to connect sngle-cell equatons, whch are complcated, to smplfed network equatons, whch are smple n that each cell s specfed solely by ts frng rate. Our goal s to understand the weghts W j that defne the synapses between pars of cells n terms of cellular propertes. We follow the dervatons lad down by Sompolnsky and colleagues and by Ermentrout and colleagues, whch holds for averagng. The crtcal ssue s that the network dynamcs and that the rate vares slowly over tme,.e., slow on the tme-scale of the tme-constant of the neuron. n asynchronous. 8.1 Krchhoff s law for one neuron C dv (t) + g L [V (t) E L ] = I actve (V, t) + I appled (t) (8.8) where E L = reversal potental of leak current, I actve (V, t) = all voltage- and tmedependent actve currents, and I appled (t) = all tme-dependent appled currents. Let r(t) = f{i app (t) I c } (8.9) where r = frng rate, f{ } s the nonlnear gan functon and I c s a threshold current. A typcal but not exclusve choce s f{µ} = f We ncorporate the leak conductance through max 1 + tanh(µ). (8.10) 2 I c = I o c + v c g L (8.11) where the constant v c scales the dependence of frng threshold on the leak conductance. Thus r(t) = f{i app (t) I o c v c g L }. (8.12) 1

2 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. FIGURE - Gan curve of Reyes 8.2 Network equatons We now consder a network of neurons n whch the appled currents I app (t) have two contrbutons, ones from the outsde come n through I ext (t) and nputs from other cells n the network come n through I net (t). C dv (t) = g L [E L V (t)] + I act (V, t) + I ext (t) + I net (t). (8.13) Averagng over external nputs We consder the form of I ext (t), the external nput to the -th neuron, frst,.e., I ext (t) = g n (t) [E n V (t)] (8.14) where E n n the reversal potental for external synaptc nput. The conductance weghts the external nput and s descrbed by a frst order equaton for t 0,.e., dg n all nputs (t) τ n + g n (t) = G n τ n δ (t t n ) (8.15) t n where the maxmum conductance G n and the tme-constant τ n s taken as the same for each synaptc nput, a smplfcaton that permts the summaton to be taken as over all spkes from all external nputs to the -th postsynaptc neuron. The formal soluton for the state (steady state of nhomogeneous response) s gven by g n (t) = G n e (t t )/τ n all nputs t n δ (t t n ). (8.16) The external nput s taken to be an nhomogeneous Posson process wth a mean rate r n (t) that evolves on a tme scale that s much longer than τ n. Thus g n (t) = G n e (t t )/τ n = G n r n (t) e (t t )/τ n = G n r n (t) τ n all nputs t n δ (t t n ) (8.17) 2

3 where the product r n (t)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 perod of one tme-constant of the post-synaptc cell. We now have I ext (t) G n τ n r n (t) [E n V (t)]. (8.18) Averagng over synaptc nputs We next turn to the current that results from synaptc nputs,.e., I net (t) n g j (t) [E j V (t)] (8.19) where g j (t) s the post-synaptc conductance trggered by pre-synaptc spke τ j dg j (t) + g j (t) = τ j G j t j δ (t t j ) (8.20) where the summaton s over the spkes n pre-synaptc neuron j. We replaced 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 ) (8.21) 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 e (t t )/τ j t j δ (t t j ) = G j r j (t) e (t t )/τ j = G j r j (t) τ j (8.22) where r j (t) s the slowly varyng rate of spkng of neuron j and τ j r j (t) corresponds to the mean number of spkes n a tme perod of τ j. The network contrbuton to the current to the cell become I net (t) G j τ j r j (t) [E j V (t)]. (8.23) 3

4 8.2.3 Recaptulaton of network equatons The sum of the external and network currents s thus I ext (t) + I net (t) = G n τ n r n (t) [E n V (t)] + G j τ j r j (t) [E j V (t)] (8.24) where N s the number of neurons n the network. Ths expresson has constant terms and voltage dependent terms. Let s expand ths expresson and 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. I ext (t) + I net (t) = G n τ n r n (t) (E n E L ) + + G n τ n r n (t) + G j τ j r j (t) (E j E L )(8.25) G j τ j r j (t) [E L V (t)]. The frst term to the rght of the equalty s a constant,.e., ndependent of V (t). We consder ths as an effectve appled current I app (t) G n τ n r n (t) (E n E L ) + G j τ j r j (t) (E j E L ). (8.26) 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 (t) that adds to the leakage conductance g L,.e., g syn (t) G n τ n r n (t) + G j τ j r j (t). (8.27) 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 syn (t) s the effectve leak current and C dv (t) = [g L + g syn The modfed frng rate s now (t)] [E L V (t)] I act (V, t) + I app (t). (8.28) 4

5 r (t) = f{i app (t) Ic o v c g L v c g syn (t)} (8.29) = f{g n τ n r n (t)(e n E L ) + = f{ G n τ n r n (t)v c G j τ j r j (t)(e j E L ) Ic o g L v c G j τ j r j (t)v c } G j τ j (E j E L v c ) r j (t) + G n τ n (E n E L v c )r n (t) (I o c + g L v c )} 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 v c ) (8.30) 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 ) (8.31) s the synaptc effcency, or strength, for an external nput to the neuron. Threshold θ I o c + g L v c (8.32) s the threshold denoted θ. Thus the equaton(s) for the frng rate become Rate Equaton r (t) = f{ W j r j (t) + W n r n (t) θ} (8.33) 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. 5

6 8.3 Man result A fnal pont s that we need a dfferental equaton for the rates r, 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 (t) as the nput to the cell, we have and τ du (t) Whle for the dscrete case, we have smply + u (t) = W j r j (t) + W n r (t) n θ (8.34) r (t + 1) f{ r (t) = f{µ (t)} (8.35) W j r j (t) + W n r n (t) θ} (8.36) where, n terms of the prevous symmetrc and normalzed notaton, we have S (t) = (2r (t) r max )/2r max, whch ranges between -1 and +1. For the specal case of bnary neurons,.e., S = ±1, whch corresponds to β wth a saturatng frng rate, f max, we have S (t + 1) sgn W j S j (t) + W n S n (t) θ. (8.37) 6

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