The Wilson Model of Cortical Neurons Richard B. Wells I. Refineents on the odgkin-uxley Model The years since odgkin s and uxley s pioneering work have produced a nuber of derivative odgkin-uxley-like odels. Recall that odgkin and uxley developed their odel fro data easured in the giant axon of the squid. Therefore the original odel only had voltagedependent channel odels for a + channel and a + channel. There are in addition to these types of channels found in axons a great any other species of voltage-gated channels (VCs), both of the + and + types as well as VCs that conduct Ca + currents. These channels are characterized by different rate constants controlling their channel gating as well as different values of axiu channel conductance. Different researchers have eployed different approaches to odeling the odgkin-uxley channel dynaics. Soe researchers stick very close to the biophysical picture presented in odgkin s and uxley s work. Their expressions for the voltage dependence of the rate constants tend to look very uch like those of odgkin-uxley, and their circuit odels tend to look very Ca C E E E Ca E Figure 1: Basic Wilson odel of cortical neurons. It is a single-copartent odel. Ionotropic synapses (or test current sources) are connected at the terinals on the left. The odel cobines the + voltagegated channel and the leakage channel together in one eleent, odeled as a fictitious persistent + channel. It adds a voltage-dependent Ca + channel, coonly called a T channel, and a Ca + -dependent + channel (the channel) to the basic odgkin-uxley channels. The batteries representing the Nernst potentials are all placed with the + terinal facing the cytoplasic side of the ebrane, and the nuerical values for the various E ters above carry the sign of the Nernst potential. Thus E would be a negative nuber in this circuit odel, while E would be a positive nuber. This is indeed the preferred circuit convention for nuerical odeling because it siplifies the coputer code used. 1
siilar except for the addition of ore conductance & battery segents odeling the additional VCs. Most incorporate the usual leakage-current eleent and treat the various channels separately fro one another. These researchers produce odels that keep as close a connection as possible to identifiable biological entities and which atch ore or less one-for-one with the qualitative picture of the neuron produced through laboratory neurobiology research. Other researchers take a different approach. While their circuit and gating-dynaics odels are designed to produce the sae voltage- and current- wavefors easured by physiologists, their odels are less concerned with retaining a one-for-one atch with the qualitative biological odel than they are with obtaining ore efficient nuerical algoriths for calculating the voltage and current wavefors. The Wilson odel, naed after its odeler, ugh R. Wilson, is one exaple of such a odeling approach. The basic Wilson circuit odel for cortical neurons is shown in figure 1. In appearance this circuit odel is no different fro those typically produced by the researchers adhering to a close correspondence with neurobiological qualitative odels. What is different about the Wilson odel can only be seen when we look at the specific atheatical expressions he uses to odel voltage and current wavefors. II. The Wilson Model Wilson s odel differs fro the odgkin-uxley odel in several ways. First, recall fro the odgkin-uxley odel that the transient + channel responsible for action potential (AP) generation has uch faster dynaics than those of the persistent + channel. The activation tie constant, τ, is saller by approxiately an order of agnitude than the inactivation tie constant, τ h, or the activation tie constant for the + channel, τ n. This eans that the activation of the + channel coes very close to reaching its steady-state value during the AP generation. Wilson takes advantage of this by approxiating the + channel as if its tie constant were zero in the differential equation for activation variable. e then cobines the + channel with the leakage channel (which has a zero-value for its tie constant) and odels both types of channels with a single conductance in series with a battery. e also takes another step, for purposes of reducing coputation tie, and this step departs fro physiological reality in the neuron odel. It ust be viewed as a straight-up nuerical approxiation and not as representative of the biophysics of + channel conductance: e ignores the inactivation gate of the + channel. The arguent ounted in justification of this seeingly radical step is based on another nuerical approxiation. Recall fro the odgkin-
uxley odel that the tie constant for + inactivation, τ h, is roughly of the sae nuerical value as that for the + channel, τ n. Wilson argues that this eans replacing h by (1 n) in the odgkin-uxley expression for approxiates the effects of the inactivation gate with reasonable accuracy. e then ounts an additional arguent, applicable to cortical neurons (neurons in the neocortex of the brain), that justifies an additional approxiation. Many of the neurons found in the neocortex have, in addition to the transient + VC, a nuber of persistent + VCs. For a persistent + channel, the conductance has the for r where g is the axiu channel conductance and r is an experientally-deterined integer. Wilson argues that the ion current through these persistent + channels is uch larger than the current conducted through the transient + conductance, and therefore to a reasonable approxiation the transient channel can be ignored in odeling voltage and current wavefors in the cortical neuron. Note that by taking this step the Wilson odel copletely gives up the ability to odel the transient + current; but since his odel is intended to achieve sufficient siplification of the nuerical calculations to perit its use in odeling sall cortical neural networks, he regards giving up the ability to odel the transient current as an acceptable odeling tradeoff. With these approxiations, the circuit of figure 1 is analyzed using irchhoff s current law to obtain the differential equation dv C ( V E ) ( V E ) ( V E ) ( V E ) + I ( t) = (1) Ca Ca in where I in is an input current (fro synaptic ionotropic channel circuits or fro a test current source), V is the ebrane voltage (the voltage at the top of figure 1), and the other sybols are defined in figure 1 and its caption. There are two new VC channels in the Wilson odel. The first, labeled T, is a channel that conducts Ca + ions. It is widely known in the literature as the T channel. The second is a + channel that has the property that its conductance depends on intracellular Ca + levels (and therefore is tied to the actions of the T channel). This conductance produces a very pronounced hyperpolarization of the ebrane potential and is therefore known as an AP channel ( after-hyperpolarization ). With the batteries drawn in the directions 3
shown in figure 1, the battery voltages, E & etc., are nubers that include the sign of the Nernst potential. Thus, E is a negative nuber, E is a positive nuber, etc. The voltage dependence of the conductances ust also be odeled, and here, too, the Wilson odel departs fro the odgkin-uxley ethod. Recall that the equations for the rate constants voltage dependence in odgkin-uxley involved the calculation of exponentials. This type of calculation is coputationally expensive (i.e. it takes longer to copute e(x) ters). Wilson replaces these exponential functions with second-order polynoial curve fit approxiations, and cobines the calculation of the voltage-dependent rate constants with the calculation of the differential equations for the activation variables. e does this by expressing the channel conductances as Ca = 17.8 +.476 V T R T + 33.8 1 4 V () where V is in V, R, T, and are Wilson s activation variables, and g, g T, and g are conductance paraeters in icro-sieens (µs). Note that is odeled siply as a nonlinear voltage-dependent conductance. The activation variables, which are diensionless, are calculated fro the set of differential equations dr τ R = dt τ T = d τ = ( R R ) ( T T ),, ( 3 T ) R T = 1.4 +.37 V = 4.5 +.116 V + 3. 1 + 8. 1 4 4 V V (3) where the three τ ters are tie constants in sec and V is again in V. With the variable diensions as described above, the ion currents are in nanoaperes (na) and the capacitance C is in nanofarads (nf). This is a whole neuron odel, by which I ean that the quantities above are not expressed in ters of per unit area. At this point there are a couple of things that need to be pointed out. First, although the g and C ters above see very biological in actual fact they are based on curve fits to epirical data and are not necessarily representative of actual biological values. Wilson obtained g values for 4
four different classes of cortical neurons (one excitatory neuron class, one inhibitory neuron class, and two bursting neuron classes). e chose his tie constants in (3) to force agreeent with observed firing rates. Interestingly, for C he used a value of 1 µf/c in his nuerical work. 1 This is peculiar and, I would have to say, erroneous because such a choice is diensionally incopatible with his other quantities. Presuably he didn t notice this because the 1 entered into his coputer equations as a 1. With three adjustable τ ters in the odel it would not be particularly difficult to obtain a curve fit even with the erroneous capacitance value. This is an exaple carelessness in quantitative odeling work, and unfortunately this is not uncoon in the literature. owever, the fact that it is coon does not ake it acceptable. We can note that even a value of C = 1 nf iplies a surface area for the neuron on the order of about.1, which is an absurdly large value since it iplies a neuron with a diaeter on the order of about 15 to µ. Real neurons in the neocortex are only about one tenth this size. Trappenberg in his textbook unconcernedly duplicated Wilson s values in chapter and in the MATLAB scripts. This brings e to the second point. All of the quantities in the Wilson odel have to be regarded as non-physiological quantities. The odel is a straight curve fit aied to reproduce specific neuronal action potentials. The paraeters of the odel cannot be trusted to be exact biological quantities, and cannot be used to explore the physiology of specific cortical neurons. What we can do with the Wilson odel is build a circuit that bio-iics signal processing in sall neural networks without the pretense that the odel itself tells us anything about the quantitative physiology of the neocortex. Wilson points this out in his paper. Because Wilson s purpose in developing this odel was to perit siulation of sall neural networks, he also needed to develop a way for action potentials fro one neuron to stiulate the opening of synaptic channels in another. To do this he added two ore differential equations. Each neuron is given a firing variable that is an abstract index of the aount of neurotransitter it is currently releasing fro its presynaptic terinals. Let each neuron be identified with an integer label. For neuron j its firing variable is described by the equation df j [ = 1 ( f j + U V j Ω ]) τ (4) syn where τ syn is a synaptic tie constant, Ω is a threshold (in V), and the function U is called the eaviside step function, defined as 1.R. Wilson (1999), Siplified dynaics of huan and aalian neocortical neurons, J. Theor. Biol., 375-388. ed after Oliver eaviside (185-195), a proinent 19 th century physicist. 5
U ( x) 1, =, x > x. (5) If neuron j akes a synaptic connection to postsynaptic neuron i, neuron i is linked to the firing variable of neuron j by a synapse activation variable S i,j. The equation describing this variable is dsi, j 1 = ( Si, j + f j τ syn ). (6) The synapse activation variable deterines the conductance of the synaptic channel of neuron i through a synaptic conductance / synaptic battery circuit such as we used earlier to odel synapses in the odgkin-uxley circuit. The synaptic current in neuron i is given by I syn i () t g S ( E V ) =, (7) syn i j syn where g syn is the peak synaptic conductance, E syn is the synaptic battery potential, and V is the ebrane voltage of neuron i. III. Scaling. A casual exaination of equations () and equations (3) shows that the nuerical constants used in Wilson s equations span a range of about 4 orders of agnitude. This can often be a serious issue affecting the accuracy of a coputational odel. Every arithetic operation a coputer perfors introduces sall round-off errors, and over tie these errors can build up to the point where they lead to serious inaccuracies. This is especially the case when there is a very large spread in the nubers, as is the case in Wilson s odel. Fortunately, there is usually a very siple way to avoid this proble, and this trick is called scaling. As an exaple of how scaling works, let s re-write the expression for R in equation (3): 4 V V 1.4 +.37 V + 3. 1 V = 1.4 + 3.7 + 3. R =. 1 1 6
The coefficients are now all of the sae order of agnitude but V is replaced in the expression by the scaled value v = V 1. This is a siple trick that yields identical results except for the fact that the coputer will now calculate v instead of V. But since we know the relationship between the two, we can easily get the ebrane potential fro the scaled variable after the calculations have copleted. In order for scaling to work we ust apply it consistently over all the equations. Thus (1) (3) becoe 1 dv E E E E C v v Ca v v = Ca + Iin () t 1 1 1 1 (1s) Ca R 1 = 17.8 + 47.6 v + 33.8 v T T 1 1 (s) dr τ R = dt τ T = d τ = ( R R ) ( T T ),, ( 3 T ) R T = 1.4 + 3.7 v + 3. v = 4.5 + 11.6 v + 8. v. (3s) We siply ultiply the conductances and the capacitor by 1, rescale the Nernst potentials by dividing the by 1, and adjust the constants in () and (3) to ake the consistent with the scaled ebrane voltage v. This is what Wilson does in his coputer progra, and Trappenberg did the sae scaling trick in his MATLAB script. In the coputer siulation progra the scaling factor can be directly incorporated into the paraeters of the odel, and this, too, is what Wilson and Trappenberg have done. The battery potentials in the odel are E = 5 V.5, E = -95 V -.95, E Ca = 1 V 1., and E = -95 V -.95. Note that E = E because these are both + channels. Wilson odeled four classes of cortical neurons, and the axiu conductance values for each type are as follows. For all four types, g =.6 µs 6 (scaled units), τ T = 14 sec, and τ = 45 sec. For the excitatory regular spiking ( RS ) neuron, 7
g T =.1 µs.1 scaled units, g =.5 µs 5. scaled units, τ R = 4. sec (4. in coputer units). For the inhibitory fast spiking (FS) neuron, g T =.5 µs.5 scaled units, g = (no AP channel), τ R = 1.5 sec (1.5 in coputer units). For the continuous bursting (CB) neuron, g T =.5 µs.5 scaled units, g =.95 µs 9.5 scaled units, τ R = 4. sec (4. in coputer units). Finally, for the intrinsic bursting or IB neuron, Wilson reported two subtypes characterized as shown in the following table. TABLE I: IB Neuron Paraeters first subtype second subtype g T =.8 µs.8 g T =.1 µs 1. g =.4 µs 4. g =.34 µs 3.4 τ R = 4. sec τ R = 4. sec Despite the abiguity in Wilson s value for the circuit capacitance, we can deterine its value fro an exaination of Trappenberg s MATLAB script. ere we see that C enters into the calculations as unity, which iplies 1C = 1 or C =.1 nf, corresponding to a neuron diaeter of about 18 µ (a biologically realistic value). Finally, it is iportant to note that the sae scaling factor ust be applied to Ω, g syn and E syn when synaptic inputs are incorporated into the odel. Representative scaled values are given in Wilson s paper. 8