Proc. Nadl. Acad. Sci. USAVol. 91, pp. 11308-11312, November 1994Neurobiology

Activity-dependent current distributions in model neurons(activity-dependent regulation/neuronal plastcity/calcium-dependent modulation/channel density)

MICAH SIEGEL*, EVE MARDER, AND L. F. ABBOTTtCenter for Complex Systems, Brandeis University, Waltham, MA 02254

Communicated by John J. Hopfield, August 15, 1994

ABSTRACT The electrical activity of a neuron can affectits intrinsic physiological characteristics through a wide rangeof processes. We study a computer-simulated multicompart-ment model neuron In which channel density depends on localCa2+ concentrations. This has three interesting consequencesfor the spatial distribution of conductances and the physiolog-ical behavior ofthe neuron: (i) the model neuron spontaneouslydevelops a realistic, nonuniform distribution of conductancesthat is linked both to the morphology of the neuron and to thepattern of synaptic input that it receives, (i) the response tosynaptic Input reveals a form of intrinsic localized plasticitythat balances the synaptic contribution from dendritic regionsreceiving unequal stimulation, and (i) intrinsic plasticityestablishes a biophysical gain control that restores the neuronto its optimal firing range after synapses are strengthened by"Hebbian" long-term potentiation.

The electrical characteristics of a neuron depend on thedensity and distribution of its ion channels. For a neuron todevelop and maintain a specific repertoire of functionalcharacteristics, ion channels must be continually synthe-sized, transported to appropriate sites, and inserted into thecell membrane. Active channels can be modulated by phos-phorylation (1, 2) and removed from duty through membraneinternalization (3). It is known that cell-cell interactions canplay an important role in controlling the distribution oftransmitter receptors and ion channels on nerve and musclecells (4-6) through a complex series of molecular and cellularmechanisms. Another critical factor that may influence chan-nel densities and distributions is the activity of the neuron(7-11). We have previously developed a Ca2+-dependentregulation scheme (12, 13) to model the effects of activity onmembrane conductances. In single-compartment isopotentialmodel neurons, this regulation has a number of interestingproperties. Model neurons with activity-dependent conduc-tances can spontaneously develop the conductances neededto perform a specific function, they can respond to externalperturbations by adjusting their conductances to maintain adesired level of activity, and, in small neuronal circuits, theycan spontaneously differentiate to develop different proper-ties in response to synaptic input (12, 13). Here we explorethe consequences of neuronal structure on the spatial distri-bution of membrane currents, using a multicompartmentmodel neuron.

MODEL

We consider a multicompartment model neuron (14) with themorphology ofa hippocampal CAl pyramidal cell. The modelneuron has fast Na+ (INa), delayed-rectifier K+ (IK), transientK+ (IA, for Fig. 3 only), voltage-dependent Ca2+ ('ca), andleakage currents (IL). To model the effects of activity-dependent processes affecting channel distribution, we as-

sume that the maximal conductances of INa, IK, and IA aredynamic variables regulated by the local intracellular Ca2+concentration (12, 13). We use Ca2+ as the feedback elementlinking activity to conductances because the average intra-cellular Ca2+ concentration is highly correlated with theelectrical activity of the neuron (12, 15) and because Ca2+ isubiquitous in biochemical pathway regulation and plays a rolein many processes relevant to channel regulation (1, 2, 7-11).Although in our model Ca2+ alone regulates the feedback, inbiological neurons several second messengers may act inconcert to play this role.The maximal conductances in the model approach steady-

state values that are increasing sigmoidal functions ofcalciumconcentration for IK and IA and decreasing sigmoidal func-tions for INa. The time constant controlling the approach tothe steady state is much larger than any other time scalecharacterizing the behavior of the neuron. As a result,significant activity-induced shifts ofconductances occur onlywhen the neuron experiences prolonged periods of modifiedactivity.

INa, IK, and 'A are expressed in the standard form (16)

I= kmP'hq.(V - Er),

with Er = 50 mV for the Na+ current and E, = -70 mV forthe K+ currents. The maximal conductances g, rather thanconstant parameters as in most neuron models, are heredynamic variables that obey the differential equations (12, 13)

dgF G_FGdt 1 + exp[+([Ca2+] - CT)/A] -

where the variable sign is plus with G = 360 mS/cm2 for INaand minus with G = 180 mS/cm2 for IK. The time constantT controls the rate of activity-dependent modification and islikely to be on the order of many minutes to hours in realneurons. However, to speed up our simulations, we took T =

10 s; we have confirmed that this does not affect our results.The parameters CT = 0.5 ,uM and A = 0.6 ptM determine theshape and position of the sigmoidal curve for the equilibriumvalue of k as a function of intracellular Ca2+ concentration.The Ca2+ current in the model is patterned after that ofthe

Morris-Lecar model (17)

ICa = gCa(1 + tanh[(V - V1)/V2])-(V - ECa),

where V1 = -50 mV, V2 = 10 mV, and Eca = 150 mV. Forthe results shown here, the maximal conductance ofthe Ca2+current was fixed and uniform, kca = 0.03 mS/cm2. Theintracellular Ca2+ concentration satisfies a diffusion equationwith exponential Ca2+ removal and uptake,

*Present address: Computation and Neural Systems Program, Cal-ifornia Institute of Technology, Pasadena, CA 91125.tTo whom reprint requests should be addressed.

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Proc. Natl. Acad. Sci. USA 91 (1994) 11309

a[Ca2+] 2I[Ca2+]t -D-- k[Ca] - Yca,

where y converts from current to Ca2+ concentration, D = 6x 10-6 cm2/s, and k = 1/600 ms (18, 19). We set all of theseparameters constant through all compartments in the modeland, as we mention in the next section, the results describedbelow are qualitatively insensitive to their values through awide range.

RESULTS

S r fzing Condu ce Distributions. Because themaximal conductances in any region of the model neurondepend on the average local intracellular Ca2+ concentration,the distribution of membrane conductances is determined bythe electrical activity and morphology of the cell. Fig. 1shows distributions of Na+ and K+ conductances that arisespontaneously in the model for two patterns of synapticstimulation. Initially, we set the maximal Na+ and K+conductances to zero (Fig. 1A). During the simulation, we

Fio. 1. Distribution ofconductances produced by the model. Thethree columns show the distribution of maximal Na+ conductances(IN.), maximal K+conductances (ihK, and the average intracellularCA2+ concentration ([Ca]). (A) Initially the model neuron had auniform Ca2+ conductance but no Na+or K+conductances. (B) Thesteady-state conductance distribution following prolonged 100-Hzstimulation of synapses with 2-nS peak conductance randomlyplaced on the basal and apical dendrites. (C) The steady-stateconductance distribution following prolonged stimulation of ran-

domly placed 2-nS synapses at 100 Hz on the basal dendrites and 5Hz on the apical dendrites. Note that experimentally induced Ca2+transients would be superposed on the average Ca2+ distributions

given above.

applied random Poisson-distributed synaptic inputs over theapical and basal dendritic trees of the model neuron. As thesimulation progressed, the maximal conductances changed inresponse to the intracellular calcium concentration until asteady state was established. The steady-state conductancedistributions (Fig. 1 B and C) are nonuniform, with 'Naconcentrated at the soma, IK concentrated in the dendriticarbor, and relatively uniform conductances along the axon.The soma and initial segment of the axon can initiate actionpotentials and the axon can conduct action potentials (for adifferent approach to constructing an axon, see ref. 20). Thusan initially passive and electrically homogeneous modelneuron can spontaneously develop active, nonuniform sub-structures driven solely by effects ofmorphology and activityand mediated through intracellular Ca+.

'Na seen in Fig. 1 B and C is large near the soma becausethe large somatic volume acts as a Ca2+ sink, keeping theconcentration there fairly low. 'K is large in the dendritesbecause the background synaptic stimulation and small di-ameters keep the average Ca2+ concentration high. Depend-ing on the geometry of the neuron and the pattern of synapticinput we give, heterogeneous "hot" and "cold" spots some-times develop along the dendritic arbor. The axon is moreuniform, since this region has a relatively constant averageintracellular Ca2+ concentration. gNa is high in the first 10-20jam of the axon, to form a spike-initiation region. This lengthscale is determined by Ca2+ removal and diffusion properties.Although our model of Ca2+ uptake is too simple to beconsidered realistic, the final Ca2+ concentrations in Fig. 1are low in the soma and rise in the proximal dendritic regionsimilar to those measured in stimulated hippocampal neurons(21). Experimental results show a drop in the distal dendriticCa2+ concentration (21) not found in our model. The declin-ing Na+ current density along the proximal dendrites agreesqualitatively with inferred distributions in hippocampal neu-rons (18, 19, 21).

Effects of Local Activity. A comparison of Fig. 1 B and Cillustrates the effect of activity on the distribution of con-ductances. The distribution in Fig. 1B arose when the apicaland basal dendrites were stimulated equally. The resultingcurrent distributions are similar for the two trees. Theconductances shown in Fig. 1C resulted from a prolongedperiod during which the apical arbor received synaptic inputat a lower frequency than the basal arbor. The lower rate ofsynaptic stimulation in the apical dendrites caused a lowerlocal intracellular Ca2+ concentration and hence a larger Na+current and smaller K+ current in that region.

Fig. 2 demonstrates the functional significance of theactivity-dependent distribution of dendritic currents seen inFig. 1. We applied a test stimulation to both apical and basaldendritic arbors of the neurons shown in Fig. 1 B and C. Thisproduced similar, passive excitatory postsynaptic potentials(EPSPs) in both dendrites for the conductance distribution ofFig. 1B (Fig. 2A). By contrast, EPSPs in the apical dendriteof the neuron in Fig. 1C are amplified nonlinearly by activeprocesses leading, in this example, to the echo of a delayedsomatic action potential (Fig. 2B). Local activity-dependentconductances balance contributions coming from dendritictrees, or regions of individual dendrites. Heavily stimulatedregions become leaky and require strong synaptic inputs tosignificantly depolarize the soma. Regions that receivechronically weak stimulation become less leaky-or evenactive-and are much more effective at transferring synapticdepolarization to the soma.Gain Contro. We find that activity-dependent conduc-

tances can also act as gain control elements to set theneuronal response relative to a background synaptic inputlevel. This has important implications when coupled withideas about the role of synaptic plasticity in learning andmemory. In Fig. 3 we randomly divided the input synapses

Neurobiology: Siegel et al.

Proc. Natl. Acad. Sci. USA 91 (1994)

A

'!)' t

B

FIG.. 2. Test stimulation of the neurons in Fig. lB and C. (A) Themodel neuron with conductances distributed as in Fig. lB receiveda series of five synaptic stimuli in the apical and basal dendrites.Responses were similar in the two dendritic regions. (B) The modelneuron with conductance shown in Fig. 1C was stimulated as in A.In this case, the test stimuli show that the basal dendrite respondedpassively, whereas the apical dendritic response was nonlinearlyamplified and gave rise to a somatic action potential (the electrotonicecho of which can be seen). Scale bars: 5 ms and 5 mV.

into two groups or patterns, pi and p2. Initially, high-frequency stimulation of one group of synapses combinedwith low-frequency siuaon of the other group fired theneuron, but low-frequency background stimulation of bothgroups failed to do so (Fig. 3A). We then potentiated the pisynapses to simulate "Hebbian learning" of this pattern ofstimulation. We denote the potentiated synapses by PI. Afterpotentiation, high-frequency stimulation of the P1 synapsesproduced high-frequency firing. However, because of thestrength of the potentiated P1 synapses, low-frequency stun-ulation of P1 also fired the cell when accompanied either byhigh- or low-frequency stimulation Of P2 (Fig. 3B). Thisillustrates a common problem associated with modification ofsynaptic strengths: potentiated synapses may become strongenough to dominate other synaptic inputs and saturate theresponse of the neuron. Saturation is a problem endemic toany physical system with limited dynamic range. A neuronwith many potentiated synapses may respond to virtually anysynaptic input by firing near its mxalrate, thereby losingthe ability to code information through a graded firing rate.Conversely, a neuron with strongly depressed synapses maystop firing altogether and drop out of network activity.

Fig. 3C illustrates that activity-regulated conductances canameliorate this problem. We exposed the model neuron tolow-frequency, background stimulation of both P1 and P2synapses for a prolonged period. The initially elevated elec-trical response to these background inputs following poten-tiation of the P1 synapses slowly down-regulated the Na+conductances and up-regulated the K+ conductancesthroughout the cell. After this gain control regulation, neitherthe background stimulus nor the "unlearned" pattern P2

produced firing; only the "learned" pattern P1 fired theneuron (Fig. 3C). The activity-dependent currents eliminatedthe excessive background-induced activity allowing the neu-ron to respond selectively to the learned input pattern.

Robustness. We have tested these results over a wide rangeof parameter values and in three different cells, and we findthat the physiological behavior and the conductance distri-bution is stable through diverse extrinsic and intrinsic con-ditions. The model neuron converges to its steady-stateconductance distributions independent of the initial conduc-tance distributions we choose for the model (e.g., Fig. 1 B andC do not depend on Fig. 1A; we took the initial conductancedistribution in Fig. 1A for simplicity). The final conductancedistribution and physiological behavior of the cell are essen-tially unaffected by variations over orders ofmagitude in thevarious electrical parameters characterizing the passivemembrane.We also ran numerical simulations in which kca, the

maximal conductance of Ica, varies as a Ca2+-dependentdynamic variable like other currents in the model. Because inour model 1ca is a small fraction of the total electrical currentcarried into the cell, the steady-state conductance distribu-tion is unaffected by this change. We find that the localdistribution grows smoother when ICa becomes the dominantelectrical current. With a spatially uniform exponential up-take mechanism for Ca2+, the two parameters governing thediffusion and uptake of Ca2+ set the effective length scaleover which conductances vary in the model cell; the overallconductance distribution is also qualitatively independent ofthese parameters. A more complicated buffer distribution orkinetic model ofCa2+ buffering and diffusion would affect thedistribution of free Ca2+ and therefore would change thedistribution ofconductances in our model cell. Nevertheless,it is interesting that the simplistic first-order model used hereproduces anything that resembles a verisimilar channel dis-tribution.

DISCUSSIONOur modeling studies indicate that the local Ca2+ concentra-tion can regulate conductances effectively to produce threeinteresting effects. (i) Realistic, nonuniform current distri-butions develop spontaneously on the basis of local intracel-lular Ca2+ concentrations that depend on both electricalactivity and cell morphology. This suggests that the time-averaged Ca2+ concentration in a region of a neuron mayshape the types and strengths of conductances in that area.(ii) Different spatial patterns of stimulation produce differentdistributions of conductances. As a result, the depolarizationdelivered to the soma from a given dendritic region dependson the strength of a synaptic input relative to the chronicbackground level in that region rather than solely on theabsolute strength of the input. (iii) Activity-dependent reg-ulation acts as a gain control mechanism- to compensate forthe effects oflong-term synaptic potentiation and depression.The model discussed here suggests a plausibe biophysicalmechanism for the neuron to make this gain adjustmentautomatically. It could be augmented by including slow,Ca2+-dependent regulatory effects on synaptic strengths aswell.We do not mean to imply that neuronal structure and

activity in themselves determine the distribution and numberof channels in biological neurons and networks. The effectsof trophic factors, growth factors, and hormones will modifysecond-messenger pathways and therefore will be integratedwith the direct effects of activity. Thus the molecular envi-ronment may sculpt or transform the influence ofactivity andstructure on channel distribution. It is also important toremember that neuronal structure is itself influenced by localCa2+ concentration and activity (23-26). Therefore one ex-

11310 Neurobiology: Siegel et al.

Proc. Natl. Acad. Sci. USA 91 (1994) 11311

APi P2

Pi

Pi

P2

P2

B

P1 P2

P1

P1

P2

P2

ILHI

C

jJJJS ej

20 Hz - 80 Hz

FIG. 3. Regulation of conductances following synaptic potentiation. Synapses were randomly divided into two groups, P1 and p2. The threepanels in A, B, and C show the response of the model neuron to stimulation of P1 at 80 Hz and p2 at 20 Hz (Top), p1 at 20 Hz and p2 at 80 Hz(Middle), and both pi and p2 at 20 Hz (Bottom). (A) Initial responses before potentiation of p1 synapses. (B) Responses ofthe same model neuronimmediately following potentiation of the P1 synapses, labeled P1 to denote potentiation. The increased strength of these synapses causedhigh-frequency firing (50-90 Hz) for all three stimulation paradigms. (C) After a prolonged period of background stimulation (20-Hz stimulationof both P1 and p2), represented by the broken arrows, activity-dependent changes in the conductances of the model neuron resulted in theresponses shown. The neuron fired when the potentiated Pi synapses were stimulated at 80 Hz, but not in response to high-frequency stimulationof p2 or low-frequency stimulation of both patterns. For this figure, we added an IA (27) to the model to allow for more variation in the firingrate. Activity-dependent regulation of the maximal conductance of IA was similar to that of IK. Synaptic conductances were g. = 4 nS forunpotentiated synapses and g. = 8 nS for potentiated synapses. Scale bars: 100 ms and 40 mV.

pects that multiple levels of feedback are called into play todetermine how channels are localized in biological neurons.The model proposed in this paper could, in principle, be

tested directly. Because the number and distribution ofNa+-channels can be directly determined with immuno-electron microscopy (22), it should be possible to determinehow channel density is affected by local stimulation andcorrelated with local Ca2+ concentration, either in slice or incultured preparations. Our model predicts that Na+ channeldensity should be reduced in regions that show persistenthigh levels of intracellular Ca2+. There are likely to beneurons in which our simple of model of intracellular Ca2+ ishighly inaccurate. Nevertheless, in these neurons the pre-diction that mean Ca2+ concentration correlates with channeldensities may be valid.The intrinsic form of activity-dependent plasticity we

model here acts more slowly than Hebbian synaptic plastic-ity, and it has a stabilizing rather than a destabilizing influ-ence on activity. Activity-induced conductance changes mayaffect regions rangng from around 20 pm to the size of theentire neuron. This reflects the fact that conductances areregulated by both local and global processes. Local effectscontrolled by local second-messenger concentrations mayregulate processes such as vesicle fusion, membrane inter-nalization, and channel phosphorylation to enhance weakerand depress stronger areas of synaptic input, as in Figs. 1Cand 2. Global modifications involving gene expression andprotein synthesis may serve to keep the neuron operatingover an appropriate range of firing rates, as in Fig. 3. Actingtogether, intrinsic plasticity (both local and global mecha-nisms) and synaptic plasticity produce a system that can storecorrelations locally, while still retaining sensitivity to all

inputs and maintaining a full dynamic range offiring frequen-cies.

We thank Russ Fricke for providing morphological data for theCA1 pyramidal cell, Zachary F. Mainen for help with our graphicalinterface, and Michael Hines for writing and generously supportingthe public-domain program NEURON. This research was supported byNational Institute of Mental Health Grant MH46742 and NationalScience Foundation Grant DMS-9208206.

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