Two computational regimes of a single-compartment neuron separated by a planar boundary in conductance space
Brian Nils Lundstrom, Sungho Hong, Matthew H. Higgs, Adrienne L. Fairhall
11 Two computational regimes of a single-compartment neuron separatedby a planar boundary in conductance space
Brian Nils Lundstrom , Sungho Hong , Matthew H Higgs and Adrienne L Fairhall Department of Physiology and Biophysics, University of Washington, Seattle Veterans Affairs Puget Sound Health Care System, Seattle, Washington Abstract:
Recent in vitro data show that neurons respond to input variance with varying sensitivities. Here, wedemonstrate that Hodgkin-Huxley (HH) neurons can operate in two computational regimes, one that ismore sensitive to input variance (differentiating) and one that is less sensitive (integrating). Aboundary plane in the 3D conductance space separates these two regimes. For a reduced HH model,this plane can be derived analytically from the V nullcline, thus suggesting a means of relatingbiophysical parameters to neural computation by analyzing the neuron’s dynamical system. Introduction
A neuron’s ion channel configuration determines how it processes information. However, therelationship between a particular set of ion channels and the specific computation it performs is stillnot clear, even for basic Hodgkin-Huxley (HH) model neurons (Hodgkin and Huxley, 1952; Aguera yArcas et al., 2003; Yu and Lee, 2003) and simplified versions of the model (Kepler et al., 1992;Gerstner and Kistler, 2002; Murray, 2002). In this paper, we examine information processing in theHH neuron, where this model neuron is taken to be a generic case with typical spike-triggering ionchannels. We focus on the response of the space-clamped HH model to a time-varying synaptic currentinput.We approximate the net synaptic current input at the soma as exponentially-filtered Gaussian whitenoise current (Gerstein and Mandelbrot, 1964; Bryant and Segundo, 1976; Tuckwell, 1988; Mainenand Sejnowski, 1995; Destexhe et al., 2001; Rauch et al., 2003; Rudolph and Destexhe, 2003a;Richardson and Gerstner, 2005), in which the input mean reflects the average number of inputs whilefluctuations, quantified by the variance, generally depend on the degree of neuronal input synchrony(Destexhe and Pare, 1999; Moreno et al., 2002; Richardson, 2004). We use the firing rate-current ( f-I )function as a straightforward means of examining the model’s input/output response properties. Iffiring rate varies most strongly with mean current, we term the neuron an integrator ; if the firing rate issensitive to variance and relatively insensitive to the mean current, the neuron is fluctuation-driven,and we refer to it as a differentiator (Abeles, 1982; Konig et al., 1996; Higgs et al., 2006). Adifferentiating neuron is thus characterized by low or zero firing rate in response to a constant or zerovariance input.Naturally, a spectrum exists between these two classifications.
In addition, the operating mode of agiven neuron often depends on the mean input current. Many neurons function as differentiators atlow, subthreshold mean currents but behave more like integrators at suprathreshold mean currents;other neurons differentiate their input at saturating mean currents. However, the purest neuraldifferentiators, such as specialized coincidence detectors in the auditory brainstem (Oertel, 1983;Reyes et al., 1994; Rothman and Manis, 2003) appear entirely incapable of integration, and never firerepetitively at any level of constant current stimulation.
Modeling studies suggest that neurons can function as either integrators or coincidence detectors(differentiators) based on the nature of incoming input (Gutkin et al., 2003; Rudolph and Destexhe,2003b).
In vitro studies suggest that high conductance inputs facilitate coincidence detection, ordifferentiation (Gonsalves and Paller, 2000; Destexhe et al., 2003; Prescott et al., 2006), and at leastone in vivo study suggests that synchronized inputs more effectively drive cortical spiking (Roy andAlloway, 2001). Recent in vitro studies suggest that neuronal firing rates are affected by the statisticalproperties of approximated synaptic inputs (Chance et al., 2002; Fellous et al., 2003), and that differentpopulations of neurons respond to input variance with differing sensitivities (Higgs et al., 2006;Arsiero et al., 2007).Our aim was to determine the biophysical parameter regime for which the HH neuron retainssensitivity to input variance even at high input means. We conclude that the HH neuron can processinformation in two fundamental ways, as both an integrator and a differentiator, and that, despite thehighly nonlinear nature of the model, a planar boundary in the space of maximal conductancesseparates these two regimes. Further, using a 2D simplification of the HH model, we find that thisplanar boundary can be derived directly from model equations and related dynamical systemproperties, thus demonstrating a simple link between the observed categories of computationalfunction and the biophysical conductance parameters.
Hodgkin-Huxley neuron as both integrator and differentiator
Recent studies show that f-I curves from in vitro cortical pyramidal cells, hippocampal CAI pyramidalcells, and neurons from avian auditory brainstem demonstrate changes in sensitivity to input varianceas the mean is increased, and suggest that these changes are related to slow adaptation currents (Higgset al., 2006; Prescott et al., 2006) or slow sodium inactivation (Arsiero et al., 2007). However, manysimple neuron models, including the standard Hodgkin-Huxley (HH) model neuron, the Connor-Stevens model neuron, and the Traub-Miles model neuron, show decreased sensitivity to inputvariance as the mean increases. We have found that for several simple model neurons, lowering themaximal sodium conductance G Na , such as through voltage-dependent slow sodium inactivation, leadsto an increased sensitivity to input variance at high means, as would be the case for differentiatingneurons (Figure 1). In this differentiator operating regime, these neurons do not fire repetitively toconstant, or zero-variance, inputs. Figure 1: Spectrum of integration and differentiation. (a)
Simple differentiators do not respond to zero varianceinputs at steady state.
This non-zero current stimulus undergoes a two-fold increase in mean, [5 10] µ A/cm , followed bya two-fold increase in standard deviation, [2.5 5] µ A/cm , and is presented to two different neuron models. Although notideal, the black trace has integrator characteristics, whereby changes in standard deviation result only in very small firingrate changes. In contrast, firing rates in the gray trace increase as standard deviation increases, but the steady state firingrate is always zero if input variance is zero. Decreased sodium conductance leads to neurons that are betterdifferentiators in (b) the standard Hodgkin-Huxley model neuron and in (c) a neocortical model neuron (p. 124,Gerstner and Kistler, 2002).
Data are plotted for the Hodgkin-Huxley and neocortical models with high (black, solidlines) and low (gray, dotted lines) sodium conductances. Lines represent mean firing rate at steady state in response to inputcurrent; increased input standard deviation generally leads to increased firing rate. Models with low sodium conductance donot fire in response to zero-variance inputs at steady state. Parameters for (b) HH: G Na = [120 82] mS/cm , SD = [0 2 4 6] µ A/cm and (c) cortical: G Na = [50 25] mS/cm , SD = [0 2 3 4] µ A/cm neurons. Standard parameters (Appendix A) wereused unless otherwise noted. Here, we consider only static maximal conductance values rather than any activity-dependent(LeMasson et al., 1993; Giugliano et al., 1999) or voltage-dependent change. The membrane potential( V ) of the space-clamped Hodgkin-Huxley (HH) neuron is described by C dVdt = − G Na m h V − E Na ( ) − G K n V − E K ( ) − G Leak V − E Leak ( ) + I , (1)where C is capacitance, G i are the maximal channel conductances, E i are the reversal potentials, I isthe external, injected current, and [m, h, n] are the channel gating variables, which obey first orderdynamics (Appendix A). If we take this standard HH neuron and systematically lower its maximalsodium conductance, its response to inputs, as evaluated by an f-I curve, changes (Figure 2). When G Na is high, the HH neuron behaves more like an integrator, while when G Na is low, the neuron behavesmore like a differentiator.Conceptually, these differences in the f-I curves (Figure 1, gray and black) result from changes in aneffective voltage threshold for spiking, where spike initiation occurs when enough sodium channelsopen to counter outward currents. The maximal conductance G Na , for example, approximatelyestablishes the spiking threshold. Additionally, the availability of sodium channels depends on thevoltage-dependent inactivation variable h , adjusting the effective threshold value. Differentiation thenoccurs when slow components of the input are below this threshold, and firing only results from inputnoise. In certain regimes, such as when G Na has been lowered sufficiently, this effective threshold isnever surpassed as input mean increases – the voltage-dependent threshold keeps increasing slightly asthe input mean increases until eventually the neuron is unable to spike. This effect, known asdepolarization block, is observed for large inputs in the f-I curves of Figure 2. This conceptualexplanation will be further developed in terms of the neuron’s dynamical system. Figure 2: The HH neuron behaves more like a differentiator as its sodium conductance G Na is lowered. As G Na islowered below ~83 mS/cm , the HH neuron ceases to respond to zero-variance input. Traces on each f-I plot from black tolight gray represent increasing SD, where SD = [0 2 4 6] µ A/cm . The first and fourth f-I plots from the left show the samedata as in Figure 1b. Standard HH values ( G Na , G K , G Leak ) = (120, 36, 0.3) mS/cm were used unless otherwise noted(Appendix A). A planar boundary separates integration from differentiation
We simulated the HH model using a set of 56 different conductance values ( G Na , G K , G Leak ) and fromthese determined f-I curves from each condition. As seen in Figure 2, at high G Na the HH neuronresponds well to noiseless inputs, whereas at low G Na the neuron does not respond to zero-varianceinputs. This transition from noiseless firing to no response is relatively sudden along the range of G Na in which firing occurs. This transition can also be observed when G K or G Leak are increased, or whenthese three conductances are modulated in linear combination according to : G Na − G K − G Leak = , (2)where conductances have units mS/cm . When this sum is less than zero, the HH neuron will not fire toa noiseless current, whereas if the sum is greater than zero, the neuron fires repetitively. Eq. (2)describes the empirically observed plane that passes through the origin in the three-dimensional spaceof HH conductances, which is shown as a line for fixed G Leak in Figure 3a. Here, if the neuron does notfire to a noiseless current, no matter how large the mean, then the neuron is considered to behave morelike a differentiator . In terms of the dynamical system, when the right hand side of Eq. (2) is greaterthan zero, the system undergoes a Hopf bifurcation at a particular mean current (e.g. approx 6.5 µ A/cm in Figure 1b, bottom black trace); however, if the right hand side is less than zero, the onefixed point of the dynamical system is stable for all mean current values (as in Figure 5b and c). Figure 3: The boundary between no response to constant stimuli (differentiation) and firing to constant stimuli(integration) is described by a line in the two-dimensional space of G Na and G K . This boundary is similar for both theHodgkin-Huxley ( a ) and Abbott-Kepler ( b ) neuron models. As G Leak increases, the boundary remains linear with the sameslope but an increasing intercept in G Na . The black X ’s designate the standard parameters for the Hodgkin-Huxley andAbbott-Kepler models. These coefficients are the result of a multiple least squares regression of 56 conductance sets (G Na ,G K , G Leak ) determined via simulation. For a given G Na and G Leak , G K would initially take a relatively lowvalue for which the neuron would fire to some mean input current I . G K was then increasedincrementally and f-I curves computed until the neuron would not fire to any mean input I ; this wasthen one of the boundary sets. As the ratios of the conductances (i.e. N = G Na /G Leak and
K = G K /G Leak )increased, the steady state voltages V at the boundary current value (which we will later define as I* )generally decreased: mean –52.8; std 2.0; range [-48.6 –54.8]. Conductances were chosen according tothe following constraints: N was within the range [50 500], G Na > 50 mS/cm , G Leak = 0.3, 1, or . Thebest fit of these data passes through or very close to the origin (Appendix B). Although for the HH model this is an appropriate criterion, other models can show non-zero firingrates in response to zero-variance stimuli while their f-I curves are differentiator-like as in Figure 2.Further, even neurons that are here classified as integrators, as seen on the left side of Figure 2, displaydifferentiator-like characteristics when mean input currents are low.Having obtained Eq. (2) through simulation, we would like to derive it directly from model equations.However, due to its four differential equations, the dynamical system of the HH model is difficult toanalyze. Therefore, we use the two-dimensional Abbott-Kepler (AK) model, which is derived from theHH model equations and maintains a direct correspondence of parameters (Abbott and Kepler, 1990;Kepler et al., 1992). In the AK model, the four variables of the HH model (
V, m, h, and n ) are dividedby time scale into two groups that become V and U , a fast activation and a slow recovery variable,respectively. The 2D AK model equations have the form (Abbott and Kepler, 1990; Kepler et al.,1992; Hong et al., In press): C dVdt = − G Na m ∞ V ( ) h ∞ U ( ) V − E Na ( ) − G K n ∞ U ( ) V − E K ( ) − G Leak V − E Leak ( ) + I , (3) dUdt = G Na V − E Na ( ) m V ( ) h V ( ) − h U ( ) ( ) / τ h V ( ) + G K V − E K ( ) n U ( ) n V ( ) − n U ( ) ( ) / τ n V ( ) G Na V − E Na ( ) m V ( ) ′ h U ( ) + G K V − E K ( ) n U ( ) ′ n U ( ) . (4)Eq. (3) has a similar form to Eq. (1), where the dynamics of the gating variables have been replaced bytheir steady state voltage-dependent values m ∞ , n ∞ and h ∞ . Like the HH model, this model shows aboundary plane between integration and differentiation, which is described by : G Na − G K − G Leak = , (5)where conductances have units mS/cm . The line that is described by Eq. (5) when G Leak = 0.3 mS/cm is shown in Figure 3b. Deriving the planar boundary
The advantage of working with a two-dimensional dynamical system is that we can more easilyunderstand how the empirical boundary equation, Eq. (5), emerges from the dynamical system, Eqs.(3) and (4). We begin from the phase portrait of the AK model. The flow is determined by the V and U nullclines, defined as the set of points for which dV/dt = 0 and dU/dt = 0, respectively. The uniquefixed point of the system is the point of intersection of the two nullclines (Gerstner and Kistler, 2002;Murray, 2002; Izhikevich, 2007). In the AK model, from Eq. (4), the U nullcline is given by U=V . The V nullcline generally takes an N-shape for intrinsically spiking neural models (Izhikevich, 2007), andis given by the solution of = − G Na m ∞ V ( ) h ∞ U ( ) V − E Na ( ) − G K n ∞ U ( ) V − E K ( ) − G Leak V − E Leak ( ) + I , (6)which for simplicity we express as f ( U , V ) = I , (7)where f ( U , V ) ≡ G Na m ∞ V ( ) h ∞ U ( ) V − E Na ( ) + G K n ∞ U ( ) V − E K ( ) + G Leak V − E Leak ( ) .Thus, for the AK model, the shape of the V nullcline in (V,U) space depends nonlinearly on meancurrent I as well as on the parameters of Eq. (7). We will focus on the three maximal conductances (G Na , G K , G Leak ) . This process was the same as for the HH model for 55 conductance sets. Again, as conductance ratios(i.e.
N = G Na /G Leak and
K = G K /G Leak ) increased, steady states voltages V at the boundary current value(which we will later define as I* ) decreased: mean –50.3; std 0.8; range [-48.2 –51.2]. The best fit ofthese data passes through or very close to the origin (Appendix B).For any given set of the four parameters (G Na , G K , G Leak , I) , whether or not the neuron fires in responseto DC current is determined by two things: the shape of the V nullcline and the position of the fixedpoint on this nullcline. If the V nullcline has an N -like shape, as seen in Figure 4a and b, the system isexcitable. However, for some values of I , the V nullcline can flatten and the system becomes unable tocreate a spike; this corresponds to depolarization block and is seen in the top contour of Figure 5a,which is the V nullcline for I=40 µ A/cm and the given conductances. When the system is able tospike, the second factor determining DC response is where the fixed point, the intersection of the V nullcline with the line U = V , lies with respect to the local minimum of the V nullcline or the left“knee” of the curve. When the fixed point is to the right of the knee, the neuron fires repetitively tozero-variance mean current (Figure 4a, 4c), but when the fixed point is to the left of the local minimumit does not (Figure 4b, 4d) . Figure 4: Phase portraits of the Abbott-Kepler neuron model. ( a ) The neuron fires repetitively to a mean current of 50 µ A/cm when sodium conductance is high ( G Na = 120 mS/cm ) such that the conductance values lie above the plane of Eq.(5) but ( b ) does not fire repetitively when the sodium conductance is low ( G Na = 50 mS/cm ). The straight black line is the U nullcline corresponding to dU/dt = 0, where U is a general recovery variable, while the curved black line is the V nullcline, satisfying dV/dt = 0 , where V is the membrane voltage. The gray lines are the trajectories of (V,U) in time, wherethe loops in ( a ) are the limit cycles of spikes. For a mean of 20 µ A/cm and SD of 4 µ A/cm , spike initiation is dominatedby ( c ) the internal dynamics when G Na is high but ( d ) when G Na is low, spikes are initiated by input variance. The fixedpoints in the two cases are ( c ) unstable, open circle, and ( d ) stable, closed circle, respectively. For ( c ) and ( d ), I = 20 µ A/cm . Other model parameters include: G K = 36 mS/cm and G Leak = 0.3 mS/cm . This may not be strictly true. As has been noted for the Fitzhugh-Nagumo model (Izhikevich, 2007),loss of stability by the fixed point may not occur precisely at the local minimum in the V -nullcline, ascan be verified by linearization and local stability analysis. The effect is small in the AK model(Appendix B).When the current is noisy, the position of the fixed point with respect to the minimum determineswhether spike initiation is largely dictated either by the internal dynamics of the system (Figure 4c) orby momentary, large current fluctuations (Figure 4d). These fluctuations cause leftward or rightwardexcursions along the V-axis, which may cause the neuron to cross an upward curving thresholdfunction situated slightly beneath, but approximately parallel to, the V nullcline (Hong et al., In press).If a change in stimulus is gradual, the nullclines alter their geometry, which may prevent thresholdcrossing despite an equally large or larger stimulus perturbation.Each point in the parameter space (G Na , G K , G Leak , I) maps onto one of three possible conditions: nominimum exists in the V nullcline and thus the system is not excitable; or the system is excitable, witha fixed point either to the right or the left of the minimum. For the AK model to implement a perfectdifferentiator, it should not fire spontaneously for any mean current level I . For this to be the case, forvalues of I such that the V nullcline shows a minimum, the nullclines must have the configuration ofFigure 4b or 4d. Thus, as I increases, the V -nullcline minimum will disappear before the fixed pointshifts to the right of the V nullcline minimum. We use this to derive the boundary condition onconductance parameters.Let us consider an example of the behavior of the system. In Figure 5a, we show contours of thesurface Eq. (7), which are a set of V nullclines each for different I . As I increases, the nullclines losetheir N -like shape, and eventually the neuron is no longer excitable and undergoes depolarizationblock. For differentiating neurons, the line U = V always intersects the V nullclines at a place wherethe V nullcline is decreasing, i.e. the derivative of the V nullcline evaluated at the fixed point isnegative: as can be seen in Figure 5, this means that ∂ f / ∂ V > . A boundary conductance ( G Na , G K ,G Leak ) gives, then, a set of I -dependent V nullclines for which this partial derivative has a minimumvalue of zero, as in Figure 5d. Figure 5: V nullclines described by f(U,V) = I . ( a ) The surface of Eq. (7) is plotted as a contour in current I ( µ A/cm ) forthe conductances ( G Na , G K , G Leak ) = (120, 36, 0.3). ( b ) f(U,V) for ( G Na , G K , G Leak ) = (60, 36, 0.3), which is very close to theboundary. ( c ) f(U,V) for ( G Na , G K , G Leak ) = (40, 36, 0.3). Fixed points fall on the line
U=V, where stable and unstable fixedpoints are represented by closed and open circles, respectively. (d)
The partial derivative of f(U,V) with respect to V evaluated for U=V for three sets of conductances.
The top ( G Na = 120 mS/cm ) and bottom ( G Na = 40 mS/cm ) tracesrepresent approximately integrating and approximately differentiating neurons, respectively. The middle ( G Na = 60 mS/cm ) trace lies approximately at the boundary between the two regimes. For this AK model, (G K , G Leak ) were (36, 0.3)mS/cm . For the fixed point to occur to the left of the minimum, we require that dU/dV < 0 for all fixed points( U ( I ), V ( I )) . Thus, the condition is given by dUdV U , V = − ∂ f / ∂ V ∂ f / ∂ U U , V = . (8)One of the two fixed-point conditions is that all fixed points, independent of I , must satisfy U = V .Thus, fixed points for all I fall somewhere along the line U = V . Since we want to characterize thesystem’s behavior for all I , we will thus evaluate the condition at U = V . Further, we will only considerthe numerator of Eq. (8) (see also Appendix B), giving the condition ∂ f U , V ( ) ∂ V U = V = . (9)0Figure 5d shows some examples of the function ∂ f / ∂ V , evaluated for several values of theconductances. Our boundary criteria require not only that the derivative is zero, but that zero is theminimal value for any I . Thus, at the point ( V (I*),U (I*) ), where the derivative takes the value zero,the function ∂ f / ∂ V must be at a minimum. Hence, we require that for some value of V , V* = V (I*) ,where V* is a function of I and the conductance parameters, Eq. (9) and the following holdsimultaneously: ddV ∂ f U , V ( ) ∂ V U = V ⎛⎝⎜⎜ ⎞⎠⎟⎟ = . (10)Because both boundary equations Eqs. (9) and Eq. (10) are homogenous in G Na , G K and G Leak , we candescribe the boundary plane with two variables. For example, we can rewrite Eq. (9) as: A ( V *) G N a + B ( V *) G K + G Leak = A ( V *) G Na G Leak + B ( V *) G K G Leak = − A ( V *) N + B ( V *) K = − , (11)where N and K are conductance ratios. From Eq. (10), we obtain another constraining equation andmust then solve the following system: A ( V *) B ( V *) A ( V *) B ( V *) ⎛⎝⎜ ⎞⎠⎟ NK ⎛⎝⎜ ⎞⎠⎟ = − ⎛⎝⎜ ⎞⎠⎟ , (12)where A and B are coefficients of Eq. (9) as written explicitly in Eq. (11), A V ( ) = m ∞ V ( ) ⎡⎣ ⎤⎦′ h ∞ V ( ) V − E Na ( ) + m ∞ V ( ) h ∞ V ( ) , B V ( ) = n ∞ V ( ) , and A and B are coefficients of Eq. (10), A V ( ) = m ∞ V ( ) ⎡⎣ ⎤⎦′′ h ∞ V ( ) V − E Na ( ) + m ∞ V ( ) ⎡⎣ ⎤⎦′ h ∞ V ( ) ⎡⎣ ⎤⎦′ V − E Na ( ) + m ∞ V ( ) ⎡⎣ ⎤⎦′ h ∞ V ( ) + m ∞ V ( ) h ∞ V ( ) ⎡⎣ ⎤⎦′ , B V ( ) = n ∞ V ( ) ⎡⎣ ⎤⎦′ . The general solution is: N = − B ( V *) A ( V *) B ( V *) − A ( V *) B ( V *) K = A ( V *) A ( V *) B ( V *) − A ( V *) B ( V *) . (13)The forms of N and K are highly nonlinear due to the dependence on the unknown value V* (Figure6a), where V* > -51 mV for positive conductance ratios. Rather than solve for V* for every parameterset, we observe that the relevant range of V* is highly constrained by the need to generate conductancevalues in a physiological range. The range used, V* = [-50.5 -48] , gives N = [50 500], which was aconstraint for the simulation. In this very narrow range of V* , N and K are linearly related to oneanother; these are plotted against one another in Figure 6b. When we fit the line of N vs. K , we obtain1coefficients of ~1.55 and 16.5, which are very close to the coefficients of G K and G Leak in Eq. (5),respectively. Exact agreement between simulated and calculated results is probably prevented byimprecision in gathering the simulated boundary points.The coefficients of G K and G Leak in Eq. (5) can be determined analytically from the coefficients A and B of Eq. (11). Specifically, for a given value V* , the coefficient for G K is – B ( V* ) /A ( V* ) and thecoefficient for G Leak is - ( V* ), as plotted in Figure 6c and 6d. Given Eq. (13), – B /A and - arethe slope and y -intercept of the linear relationship N vs. K . Regression, as in Figure 6b, takes intoaccount many values of V* and does not require knowing V* a priori , while using A and B gives ananalytic result for a given V* . Figure 6: Conductance ratios
N = G Na / G Leak and
K = G K / G Leak plotted (a) as functions of V and (b) against eachother. The coefficients of the linear regression fit are close to the simulated coefficients of Eq. (5). The voltage rangeplotted in ( b ) is from -50.5 to –48 mV, where lower voltages correspond to higher ratios. Coefficients (c) –B / A and (d) –1 / A of Eq. (9) can be plotted as a function of V* , where these values of V* give ratios of N = [50 500].
For a given V* , these coefficients can be used to find the coefficients of the boundary Eq. (5). The values of V* used in ( c ) and ( d ) arethe same as those used in ( b ) to plot N and K , demonstrating that exponentially spaced V* give rise to uniformly spacedvalues in ( K,N) . Hodgkin-Huxley boundary equation
The coefficients of the HH boundary equation, Eq. (2), differ by a constant from those of the AKboundary, Eq. (5). The HH model is less excitable than the AK model, since less outward conductanceis needed to suppress firing. In the approximations made in the 2D reduction of HH to AK, the largestsource of error probably stems from the approximation of infinitely fast dynamics for the sodiumactivation variable, such that m ≈ m ∞ . Around the typical voltage threshold, m has a time constant of2~0.3-0.5 msec. This will cause the true m to lag behind the approximated m ≈ m ∞ and to take a lowervalue at any point in time leading up to the threshold, causing the increased excitability of the AKmodel. Comparing m dynamics during spiking near the boundary with those of m ∞ (not shown), wefind that multiplying m ∞ by a constant factor of 3/4 provides a better approximation to m around thethreshold. When this factor is included in the AK model, the boundary equation coefficients extractedfrom simulations are found to be 2.09 and 22.14, in good agreement with those determined by the HHequations in Eq. (2). With this correction, the V- nullcline of Eq. (9) can be used to analytically find thecoefficients as above : 2.07 and 21.3. Slow sodium inactivation
While we have considered only steady state conductances, this approach may help to conceptualize thebehavior of a system subject to certain types of adaptation, such as slow sodium inactivation (Arsieroet al., 2007). That the addition of slow sodium inactivation effectively lowers maximal sodiumconductance for steady state can be seen from the model equations. Slow sodium inactivation may beintroduced (Fleidervish et al., 1996; Miles et al., 2005) through an additional slow sodium inactivationvariable s : I Na = G Na m hs V − E Na ( ) , (14)where s has the same form as the inactivation variable h but with a much slower time constant. Theaddition of s lowers the effective sodium pool for spiking. At steady state, when s is approximatelyconstant due to its slow kinetics relative to spiking, this addition is equivalent to lowering G Na .Through the sigmoidal voltage dependence of s , changes in mean current leads to effective changes of G Na ; as mean current increases, s decreases, decreasing the effective sodium pool. In other words, incontrast to spike-dependent adaptation currents, the voltage dependence of sodium inactivationprovides feedback regardless of spiking activity. Thus, slow sodium inactivation, which in some senseis similar to voltage-dependent adaptation currents (Benda and Herz, 2003), is one mechanism bywhich the functional role of a neuron could change from integrating to differentiating without changesin channel density. Discussion
Models of a variety of neurons share the same form as the Hodgkin-Huxley (HH) neuron (Ermentrout,1998; Shriki et al., 2003), and evidence suggests that single-compartment models can capture keyproperties of in vivo and in vitro neurons (Destexhe et al., 2001). Here, we show specifically how aratio of inward and outward conductances affect the HH neuron’s sensitivity to input variance. Thebalance of outward to inward currents affects neuronal excitability. The well-known coincidencedetection of auditory neurons (Reyes et al., 1996; Trussell, 1997) is enhanced by both a low-thresholdpotassium channel, which effectively decreases the membrane time constant (Reyes et al., 1994;Rathouz and Trussell, 1998; Rothman and Manis, 2003), and low availability of sodium channels(Svirskis et al., 2004). In dendrites, membrane excitability is modulated by changes in sodium or This is, of course, not a true V -nullcline of the HH model since we are not specifying a dependentvariable to plot it against. But, as in the AK case, we assume that that the slow gating variables n and h are not functions of V and can then be ignored when partial differentiating with respect to V . This isanalogous to assuming some sort of mapping from the 3D HH (i.e. excluding m ) to a 2D model, andagain we see that coefficients of Eq. (9), with the 3/4 correction factor multiplying m ∞ , are related tothe coefficients of Eq. (2)3potassium currents (Colbert et al., 1997; Johnston et al., 1999). Low sodium conductance, which leadsto spike frequency adaptation, may be the result of intrinsic low channel density (Melnick et al., 2004)or slow sodium inactivation (Fleidervish et al., 1996; Miles et al., 2005), which can be modulated bysecond messenger systems (Cantrell and Catterall, 2001).Neurons demonstrate a spectrum of integration and differentiation, and slow adaptation currents maycorrelate with shifts along this spectrum, at least for more complicated neurons such as neocorticalpyramidal neurons (Higgs et al., 2006). However, diverging f-I curves, as in Figure 2 or Arsiero et al. (2007), would more likely be related to voltage-dependent rather than spike-dependent adaptation. Previous work has demonstrated that increased shunting, a likely consequence of strong excitatory andinhibitory synaptic input, can allow M -current adaptation to effectively switch a neuron’s operatingregime from integration to coincidence detection, or differentiation (Prescott et al., 2006). This may bea specific case of our general result, where the M -current essentially increases G K and shuntingincreases G Leak . In some cases, the synaptic conductance alone may be large enough to switch a neuronfrom integration to differentiation. For example, Destexhe and Paré (1999) found that the inputresistance of pyramidal cells in the parietal cortex decreases approximately five-fold during intensenetwork activity. In the standard HH model, our results indicate that this change would place the cellnear the boundary between integrator and differentiator function. Because synaptic conductances arehighly variable on both short and long time scales, a postsynaptic neuron may fluctuate dynamicallyacross the functional boundary.
Although we divide neural behavior into integration and differentiation as classified by the variancedependence of f-I curves, there are other possible ways to determine the computation that a singleneuron performs on its current inputs. One example is white noise analysis, which we have exploredextensively in applications to the Hodgkin-Huxley model (Aguera y Arcas et al., 2003), reducedmodels (Aguera y Arcas and Fairhall, 2003; Hong et al., In press), and neurons of avian brainstem(Slee et al., 2005). While we have found, suggestively, that the computation as described by alinear/nonlinear model changes as a function of G Na /G K (R. Mease, unpublished data), here we do notattempt to connect these results.In conclusion, we find that a plane separates two computational regimes in the space of maximal ionicconductances in the Hodgkin-Huxley and Abbott-Kepler models. Hyperplanes have been noted toseparate models with different firing characteristics (Goldman et al., 2001; Taylor et al., 2006); in thiscase we show how the plane can be derived from characteristics of the V nullcline. Since V nullclinesfor reduced models of neuronal dynamics can in principle be obtained experimentally from I(V) plots(Izhikevich, 2007), this method may provide a means of relating biophysical properties withcomputational ones. Specifically, one may be able to infer certain biophysical parameters and theirrelated effect on the neuron’s computation from an experimentally obtained set of
I(V) plots. Finally,although we focus here on maximal conductances, as might be regulated by homeostatic mechanisms,many forms of adaptation, such as slow sodium inactivation, change conductances over time, so thateach adaptation state has a different maximal conductance. Time-dependent movement of the systemthrough different computational states may be an effective way to describe the functional role ofadaptation.4
Acknowledgments
We thank W. J. Spain, Rebecca Mease, Michele Giugliano, and Larry Sorenson for helpful discussionsand Michele Giugliano for comments on the manuscript. This work was supported by a Burroughs-Wellcome Careers at the Scientific Interface grant and a Sloan Research Fellowship to ALF; BNL wassupported by the Medical Scientist Training Program, a fellowship from the National Institute ofGeneral Medical Sciences (T32 07266) and an ARCS fellowship; MHH was supported by a VA MeritReview to WJS.5
Appendix A: Biophysical Modeling
The single-compartmental conductance-based Hodgkin-Huxley (HH) model neuron (Hodgkin andHuxley, 1952) was used with standard parameters (Koch, 1999; Dayan and Abbott, 2001; Gerstner andKistler, 2002) except as noted. In addition to Eq. (1), the following equations comprise the HH model: dndt = α n (1 − n ) − β n n , dmdt = α m (1 − m ) − β m m , dhdt = α h (1 − h ) − β h h , α n V ( ) = V + ( ) − e − V + ( ) β n V ( ) = e − V + ( ) /80 , α m V ( ) = V + ( ) − e − V + ( ) β m V ( ) = e − V + ( ) /18 , α h V ( ) = e − V + ( ) /20 β h V ( ) = + e − V + ( ) , where steady state gating values, such as n ∞ , are equal to α / ( α + β ). Standard parameters for HH are: G Na = 120 , G K = 36 , and G Leak = 0.3 mS/cm ; E Na = 50 , E K = -77 , and E Leak = -54.4 mV; and
C = 1 µ F/cm . For the cortical neuron model of Figure 1, standard parameters are: G Na = 50 , G Kfast = 225 , G Kslow = 0.225, and G Leak = 0.25 mS/cm ; E Na = 74 , E Kfast = E
Kslow = -90 , and E Leak = -70 mV; and
C = 1 µ F/cm .Equations for the Abbott-Kepler model are derived from the HH model (Abbott and Kepler, 1990;Kepler et al., 1992; Hong et al., In press). Equations were solved numerically using fourth-orderRunge-Kutta integration with a fixed time step of 0.05 msec, or 0.025 msec for the AK model. Injectedcurrent was simulated by a series of normally-distributed random numbers that were smoothed by anexponential filter ( τ = 1 msec). Spike times were identified as the upward crossing of the voltage traceat -20 mV (resting potential = -65 mV) separated by more than 2 msec. Appendix B: Examining f(U,V)