roles of ia and morphology in action potential propagation...

Roles of IA and morphology in action potential propagationin CA1 pyramidal cell dendrites

Corey D. Acker & John A. White

Received: 4 July 2006 /Revised: 21 January 2007 /Accepted: 8 March 2007 / Published online: 20 April 2007# Springer Science + Business Media, LLC 2007

Abstract Dendrites of CA1 pyramidal cells of the hippo-campus, along with those of a wide range of other celltypes, support active backpropagation of axonal actionpotentials. Consistent with previous work, recent experi-ments demonstrating that properties of synaptic plasticityare different for distal synapses, suggest an importantfunctional role of bAPs, which are known to be prone tofailure in distal locations. Using conductance-based modelsof CA1 pyramidal cells, we show that underlying “travelingwave attractors” control action potential propagation in theapical dendrites. By computing these attractors, we dissectand quantify the effects of IA channels and dendriticmorphology on bAP amplitudes. We find that non-uniformactivation properties of IA can lead to backpropagationfailure similar to that observed experimentally in thesecells. Amplitude of forward propagation of dendritic spikesalso depends strongly on the activation dynamics of IA. IAchannel properties also influence transients at dendritic

branch points and whether or not propagation failureresults. The branching pattern in the distal apical dendrites,combined with IA channel properties in this region, ensurepropagation failure in the apical tuft for a large range of IAconductance densities. At the same time, these sameproperties ensure failure of forward propagating dendriticspikes initiated in the distal tuft in the absence of someform of cooperativity of synaptic activation.

Keywords Backpropagation . Propagation failure .

Traveling wave attractor . bAP. Dendritic spike

1 Introduction

Backpropagation of axonal action potentials into thedendritic arbor is supported by a wide variety of neuronsand is thought to play an important role in several cellularfunctions, including synaptic integration, and synapticplasticity (Häusser et al. 2000; Stuart and Häusser 2001;Johnston et al. 2003; Gulledge et al. 2005). In spike-timing-dependent plasticity (STDP) protocols, pairings with a shortdelay between the synaptic input and a postsynaptic actionpotential usually lead to long term synaptic potentiation(Magee and Johnston 1997; Markram et al. 1997; Bi andPoo 1998). This potentiation is thought to depend primarilyon depolarization-induced unblocking of NMDA channelsand supralinear NMDA calcium influx. It was recentlydiscovered that STDP when observed at distal synapses, issignificantly different than when observed at proximal syn-apses (Froemke et al. 2005; Sjöstrom and Häusser 2006).

These new finding are consistent with the hypothesisthat backpropagating action potentials (bAPs) play animportant role in the STDP phenomenon providing thenecessary depolarization for potentiation. It is well known

J Comput Neurosci (2007) 23:201–216DOI 10.1007/s10827-007-0028-8

Electronic supplemary material The online version of this article(doi:10.1007/s10827-007-0028-8) contains supplementary material,which is available to authorized users.

Action Editor: Alain Destexhe

C. D. Acker (*)Department of Neuroscience,University of Connecticut Health Center,263 Farmington Avenue, Farmington, CT 06030, USAe-mail: [email protected]

J. A. WhiteDepartment of Biomedical Engineering, Boston University,44 Cummington St., Boston, MA 02215, USA

C. D. Acker : J. A. WhiteDepartment of Biomedical Engineering,Center for BioDynamics, and Center for Memory and Brain,Boston University, Boston, MA, USA

http://dx.doi.orig/10.1007/s10827-007-0028-8

that bAPs propagate decrementally in dendrites of many celltypes including apical dendrites of cortical and hippocampalpyramidal neurons (Stuart and Sakmann 1994; Spruston etal. 1995). Decreased action potential amplitude at distalsynapses leads to decreased NMDA calcium influx whenpaired with synaptic stimulation, which can explain theweakened potentiation observed experimentally (Froemkeet al. 2005). At far distal synapses, pairings that normallyproduce potentiation, have been observed to producedepression instead (Sjöstrom and Häusser 2006). Thisdramatic change in plasticity may be explained by consid-ering action potential failure at these distal regions, whichwould lead to dramatically reduced calcium entry.

The properties of backpropagating action potentials(bAPs), including their amplitudes and widths, likelyimpact the roles that they play. In some CA1 pyramidalcells, bAPs decline initially and then fail (Golding et al.2001). Failures are of particular interest because, as recentlydemonstrated in physiologically similar cortical pyramidalcells (Sjöstrom and Häusser 2006), they may causedramatic changes in cellular function such as a switch fromsynaptic potentiation to depression. Our goal is to betterunderstand the biophysical properties of dendrites thatcontrol the decline and failure of bAPs. It has beendemonstrated that both voltage-gated channel propertiesand morphological properties can contribute to bAP declineand failure (Golding et al. 2001; Vetter et al. 2001). In CA1pyramidal cells, blocking IA blocks the decline in amplitudenormally exhibited (Hoffman et al. 1997), and models showthat altering the distribution of IA can determine whether ornot bAPs reach the end of the dendrite (Migliore et al.1999; Golding et al. 2001). In some circumstances,dendritic branching also contributes to the apparent failureof bAPs in these models (Golding et al. 2001).

In this work, we compare generation and propagation ofAPs in simple cable models with that in a morphologicallycomplex model of apical dendrites of CA1 pyramidal cells.In each case we can compute traveling wave attractors atevery location along the cable or model dendrite. Theseattractors reveal what the propagating action potentialwould look like if it were propagating in a long dendritewith uniform properties. This method allows one toquantitatively and independently study effects due tochannel properties and branching. Since these attractorsdo not depend on propagation direction, the same attractorapplies to both backpropagating action potentials andforward propagating dendritic spikes, and the effects ofchannel properties are investigated in both cases.

Increasing IA conductance density with distance from thesoma along CA1 apical dendrites, along with changingchannel kinetics, leads to declining backpropagating actionpotential amplitude with distance. The rate of decline canbe accurately predicted along with the critical location at

which channel properties no longer support regenerativepropagation and beyond which propagation failure occurs.Before the IA conductance density is sufficient to blockactive propagation, a non-robust action potential is sup-ported that is sensitive to perturbations due to dendriticbranching. The conditions that result in propagation failureat branch points are computed, which for the CA1 modelconsidered ensure propagation failure in the distal tuft for arange of IA distributions, as well as failure of forwardpropagating dendritic spikes initiated in the distal tuft in theabsence of some form of cooperativity of synaptic activation.

2 Materials and methods

2.1 CA1 pyramidal cell models

The computational models of apical dendrites of CA1pyramidal cells used here are based on previously publishedmodels (Migliore et al. 1999). Dendritic morphology ismanipulated extensively and action potential propagation insimplified, surrogate morphologies is compared to that inthe previously published detailed model. In Fig. 1, aperfectly uniform cable model is used to demonstrate therelevance of traveling waves (described below) in thesemodels. Subsequently, a branchless “ball-and-stick” mor-phology is considered consisting of a soma coupled to asingle long dendrite section (Fig. 3(a)). A single compart-ment, cylindrical soma section is used with length anddiameter equal to 11.6 μm so that the surface area matchesthat in the original model. The dendrite is 600 μm in lengthand has a constant diameter equal to 1.8 μm. This is anintermediate diameter, considerably smaller than the largeinitial apical diameter, but considerably larger than the finedistal apical dendritic segments. Effects due to branching inapical dendrites are subsequently studied using the origi-nally published, morphologically detailed CA1 pyramidalcell model (Migliore et al. 1999) with the axon and basaldendrites omitted (Fig. 7(a)).

Voltage-gated ionic currents, their space-dependent distri-butions and dynamics are modeled based closely on previ-ously published models (Migliore et al. 1999). Fast sodium(INa), delayed rectifier potassium (IKDR), and A-type potas-sium (IA) currents, crucial in controlling backpropagation,are included. INa and IKDR are distributed uniformly, i.e.constant conductance density over the length of the dendrite.IA is distributed non-uniformly; the channel density increaseslinearly as a function of distance from soma. Channelactivation dynamics are also space-dependent. The activationcurve for distally situated IA channels is hyperpolarization-shifted by approximately 10mV, mimicking experimentalobservations (Hoffman et al. 1997). Changes in IA channelinactivation observed in response to protein kinase activity

202 J Comput Neurosci (2007) 23:201–216

(Hoffman and Johnston 1998) are not included in this model.While we do not address the implications of modulation ofchannel inactivation, it is important to mention that theseeffects tend to oppose those due to changes in channelactivation, but are significantly weaker. In this model, thetime constant of IA channel activation is approximatelydouble for proximally situated channels compared to distallysituated ones. While this effect has not been published exper-imentally, it is consistent with changes in channel activationrate constants that lead to a hyperpolarization-shiftedactivation curve at distal dendrites (Hoffman et al. 1997).

The consequences of space-dependent IA channel distri-bution and channel activation properties are studiedindependently by separating them as follows. Space-dependent changes in channel dynamics are removed bytaking a weighted average of voltage-dependent equations(Eq. (1)). The resulting averaged dynamics are then applieduniformly to the entire dendrite and channel distributioncan be varied independently.

Favg Vmð Þ ¼ W � Fprox Vmð Þ þ 1�Wð Þ � Fdist Vmð Þ ð1Þ

W, the weight value between 0 and 1, is chosen to be 0.1(predominantly “distal” IA activation dynamics, Fig. 3) andis applied to both the steady state equations and timeconstant equations for IA activation. Alternatively, byholding the distribution of IA fixed, the effects onpropagation due to the variations in IA activation propertiescan be studied (Figs. 4, 5).

2.2 Multicompartmental simulations

Multicompartmental simulations are performed usingMNeuron, custom software written in MATLAB (for moreinformation, contact the author C. Acker). MNeuron takesadvantage of MATLAB’s numerical integrator of sparse,stiff systems of ODEs, ode15s. By default, the relative errortolerance is 10−7 and the absolute error tolerance is 10−6.MATLAB graphical user interfaces are used extensively,while equations for membrane conductances are modular,and can be compiled for improved simulation speed.Intercompartmental coupling strengths for tapering den-drites are calculated as in the literature (Eq. (6.30), Pg. 220,Dayan and Abbott 2001). For the simple cable model, the600 μm length dendrite is divided into 80 equally spaced,7.5 μm length compartments. For the morphologicallydetailed model (Fig. 7), each dendritic segment is dividedinto the minimum number of equal length compartmentssuch that they have a maximum length of 7.5 μm. Thisspatial discretization is adequate for the models presentedhere because only small, quantitative differences appearwhen the maximum compartment size is doubled to 15 μm(data not shown).

2.3 Traveling wave attractors

In uniform dendrites or axons, action potential propagationmay be stable. If so, action potentials propagate as travelingwaves: with fixed waveform and constant velocity. Thissteady propagation corresponds to a time-dependent stablestate; small perturbations lead to changes in waveform thatdecay over time, leaving only a time or position shift withrespect to the unperturbed waveform. We refer to this stablestate as a traveling wave attractor, with attractor for ourpurposes referring to the fact that the state is stable, i.e.attracting. Computing traveling wave attractors is doneusing the same strategy as in classical papers on actionpotential propagation in axons (Hodgkin and Huxley 1952).The nonlinear cable equation (partial differential equation)is transformed to a set of ordinary differential equations(ODEs) by assuming the existence of a traveling wavesolution where all variables depend on space and timeas follows: y(x, t ) = y(x − ct ) (see Section 4.3, pg. 200,Weiss 1996), where c is the wave speed given by:

c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K � d4 � Ra � Cm

s

ð2Þ

d is the cable diameter, Ra is the intracellular resistivity, andCm is the membrane capacitance. The value of K dependson the voltage-gated channel properties. After the transfor-mation, the system of ODEs, the traveling wave equations,is analyzed in XPPAUT (Ermentrout 2002, still unavailablein 1952). The correct value of the parameter K is foundusing a trial-and-error method called shooting in which ahomoclinic orbit corresponding to a stable traveling wave issought. When such an orbit can be found, K is determinedto four significant digits, and is recorded along with theamplitude of the resulting traveling wave waveform, andthe wave speed calculated using Eq. (2). Since the homo-clinic orbit found can correspond to a stable or unstabletraveling wave, simulations of the full cable equations areused to confirm stability. A tutorial for computing travelingwave attractors in XPPAUT, along with the necessary codeand model equations, is available as supplemental material.

For nonuniform dendrite models, one does not expect atraveling wave with unchanging waveform. The expectedchanges in waveform are in fact what we are interested in.However, we proceed by treating each compartment in themodel as if it were an infinitely long cable with uniformproperties. All the properties from each compartment in themodel dendrite (maximal conductances, reversal potentials,etc.) are loaded into XPPAUT along with the dynamicequations. The resulting “local” traveling wave attractorcorresponding to each set of “local” properties is thencomputed as described above. In addition to the local

J Comput Neurosci (2007) 23:201–216 203

properties mentioned above, a constant current is applied sothat the rest state exactly matches the resting state of thelocal dendrite compartment. This adjustment is important toaccount for non-uniform resting potentials that can arise inmodels with non-uniform channel properties. The currentapplied matches the net current flowing into the specificcompartment from neighboring compartments when themodel is at rest (@y@t ¼ 0, for all variables).

Parameters are transferred automatically from MATLABto XPPAUT by a Python script that modifies the XPPAUT.ode file. Two methods are used to confirm that parametersare transferred correctly and that results from the multi-compartmental simulations (MATLAB) are consistent withsimulations of the traveling wave equations (XPPAUT).First, resting potentials are compared between the twoprograms and never differ by more than 3 μV. Second,traveling wave waveforms found in XPPAUT are comparedto those recorded from multicompartmental simulations of auniform dendrite with matching membrane properties/parameters. In Fig. 1, one can see that the resultingwaveforms are very similar, indicating very good corre-spondence between the two different numerical methodsused in this example. In our experience with thesecomparisons, the peak action potential amplitudes foundare not as precise as the resting potentials but have neverdiffered by more than 0.5 mV.

3 Results

3.1 Traveling waves in a uniform cable model of a CA1pyramidal cell dendrite

Action potential propagation in neuronal dendrites canexhibit variations in waveform, such as changes in ampli-tude, including distinct drop-offs in amplitude referred to aspropagation failures (Golding et al. 2001). These waveformchanges result from the non-uniform properties of dendritesincluding non-uniformly distributed voltage-gated ionchannels (Migliore et al. 1999; Golding et al. 2001), andmorphological features such as branching (Goldstein andRall 1974; Manor et al. 1991). While our goal is to betterunderstand the complex propagation in these nonuniformstructures, it is informative to first study propagation insimplified models with these non-uniformities removed.

In a uniform cable model of a CA1 pyramidal cell apicaldendrite, action potentials quickly approach steady-statepropagation after a brief initial transient (Fig. 1). While thismodel has a finite length, it is useful to remember that ininfinitely long, uniform cables, steady state propagationcorresponds to a traveling wave with a fixed waveform andconstant propagation velocity. In reality, one finds that theuniform cable model need not be very long to approximate

an infinite cable. Instead, the propagating action potentialquickly approaches the traveling wave expected in theinfinite version (computed as described in Section 2).

The observed propagation behavior can be described as asystem approaching its stable state, which in this case is atraveling wave. Since this state is stable, i.e. attracting, werefer to it as the traveling wave attractor. The “attractive-ness” of this stable state would refer to the rate at which itis approached (for limit cycles, this rate can be computedusing Floquet multipliers, Acker et al. 2003). In this model,the action potential’s amplitude is near its steady-state valueafter the peak of the wave has advanced approximately150 μm. Since the model is much longer than 150 μm, theattractiveness of the stable state is sufficiently strong such thatthis model may be considered an infinitely long uniformcable. The concept of attractiveness will be important when

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Fig. 1 In a uniform dendrite model, a backpropagating actionpotential quickly becomes a traveling wave after a brief initialtransient. (a) A current pulse (1.0 nA, 0.5 ms) is applied to firstdendritic compartment to elicit the action potential. Time snapshots at0.5, 0.8, and 1.1 ms after onset of pulse are shown. Grayscale shadesof lines match inset legend text. Dashed line is peak amplitude oftraveling wave attractor of this model (see panel (b)). Modelparameters as in Fig. 3(c), light gray, except dendrite length is1,200 μm (only first 400 μm shown), and soma is excluded. Steady-state membrane potential is −65 mV. (b) Traveling wave attractor forthis model computed as described in Section 2 (dashed line, wavespeed=0.274 m/s) along with three time snapshots of bAP from panel(a) (solid lines), plotted with their peaks aligned


later we study propagation behavior in non-uniform den-drites. It should be mentioned that this model is bistable; restis also a stable state, which is why a pulse of current isnecessary to trigger the propagating action potential behavior.

3.2 Action potential sensitivity to IA conductanceand action potential failures

It has been demonstrated previously that blocking IAchannels pharmacologically in CA1 pyramidal neuronsleads to increased amplitudes of backpropagating actionpotentials (Hoffman et al. 1997). While this effect isintuitive, it is important to quantitatively study thisrelationship between channel conductance density andaction potential propagation. This can be done by comput-ing traveling wave stable states while varying the channelproperty of interest. This analysis reveals how smooth

changes in channel properties can cause abrupt changes inpropagation behavior, such as propagation failure similar tothat seen experimentally, when there is a change in thesystem’s stable states.

By computing traveling wave attractors, we compute thesensitivity of action potential amplitude to IA channelconductance density (Fig. 2). The same model is used asin Fig. 1 except that the model’s maximal IA conductance,gKA, is varied. As expected, increasing gKA leads todecreasing peak amplitudes of the propagating actionpotential. At the published value of gKA (48 mS/cm2, themodel’s set point in this analysis), the slope of this curve,the instantaneous sensitivity to gKA, is −0.3 mV/mS/cm2. Ina non-uniform dendrite model with varying IA conductancedensity, changes in action potential amplitude can beexpected to reflect this sensitivity. When gKA increasespast 190 mS/cm2 in this model, there is a sudden change inthe system’s steady state: the traveling wave attractordisappears and the propagating action potential is no longera stable state. A sudden change in the number or nature of asystem’s stable states is known as a bifurcation. In thiscase, the (stable) traveling attractor disappears beforereaching zero amplitude, which indicates annihilation bymerging with an (unstable) traveling wave repeller. Thisbifurcation leaves the rest state as the only remaining stablestate without the possibility of sustained action potentialpropagation.

3.3 IA distribution controls backpropagation amplitudein a branchless model and can lead topropagation failure

Experimentally it has been shown that in some neurons,backpropagating action potentials can fail at a certainlocation along the dendrite (Golding et al. 2001). Sincethe distributions and properties of voltage-gated ionchannels are thought to vary fairly smoothly along thelength of the dendrites, it may seem unlikely that channelproperties play a role in these failures. For the samereasons, branch points may seem much better candidatessince they exist at distinct locations, and one of theselocations might correspond to the location at whichpropagation fails. However, using reconstructions of neu-rons, a clear “responsible” branch point was not found(Golding et al. 2001). Here, we use a branchless model andshow how non-uniform channel properties can lead tobackpropagation behavior, including backpropagationfailure, that mimics that seen in experiments.

In a “ball-and-stick” model including a soma and asingle section of constant diameter apical dendrite, back-propagation is observed for three distributions of IA (Fig. 3,panel (b) shows IA distributions). One of the distributions isthe uniform distribution considered previously (Fig. 1)

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gKA=190 (Bifurcation Point)

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Fig. 2 Smooth increases in maximal IA conductance can lead toabrupt failure of action potential propagation. (a) Peak amplitudes ofthe model’s steady-state action potential waveforms (traveling waveattractors) are plotted as a function of maximal IA conductance, gKA(mS/cm2). No traveling wave attractor exists when gKA is greater than190 (failure, or bifurcation point), i.e. the model does not support apropagating action potential. (b) Traveling wave attractors for valuesof gKA, for which attractors exist (solid lines). Dashed line is thesteady-state membrane potential for gKA=191 mS/cm2


while the other two show increasing densities with distancefrom soma. The peak amplitude of the propagating actionpotential, triggered by a brief somatic current pulse, iscalculated every 0.05 ms and plotted as a function ofdistance from the soma (Fig. 3(c), solid lines). Twoimportant qualitative features of backpropagation amplitudeare apparent that match experimentally observed back-propagation amplitude in these cells. First, the amplitude ofthe backpropagating action potential tends to decrease as afunction of distance from the cell soma (Johnston et al.1999; Golding et al. 2001). Second, backpropagation canapparently fail, i.e. the AP amplitude can drop off quicklywith distance beyond a specific dendritic location (Goldinget al. 2001). This change in slope (Fig. 3(c), black line)

becomes obvious when the negative second derivative ofthese peak voltage data are computed (Fig. 3(d)).

3.4 Local traveling wave attractors predictbackpropagation behavior and role of IA distribution

For each of the three dendrite models with differingdistributions of IA (Fig. 3(c)), traveling wave attractors canbe calculated for each compartment in the multicompart-mental model. As described in Section 2, each compartmentis assumed to be an infinite cable with uniform propertiesand steady-state traveling waves are computed for eachcompartment. When the peak amplitudes of these travelingwave attractors (Fig. 3(c), dashed lines) are compared to the

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Fig. 3 Backpropagation amplitude in a ball-and-stick CA1 pyramidalcell model: the effects of non-uniformly distributed IA conductance.Voltage-gated ion channel distributions and dynamics are as publishedpreviously for CA1 pyramidal cells (Migliore et al. 1999). (a) Cartoon ofmodel neuron (not to scale). Soma is coupled to the apical dendrite,modeled as a 600 μm long branchless cable with a constant diameterof 1.8 μm (see Section 2). (b) Three distributions of IA are compared.GKA, the maximal conductance density (mS/cm2), is plotted as a functionof distance from soma for the three cases. GKA increases in density with a

slope of 0, 0.2, or 0.5 (mS/cm2/μm, light gray, dark gray, and black lines,respectively). (c) Backpropagating action potentials (bAPs) are triggeredat the model’s soma using a brief current pulse. The peak amplitude ofeach bAP is calculated every 50 μs and plotted as a function of distancefrom the model’s soma (solid lines). Gray scale shades correspond to theIA distributions in panel (b). Dashed–dotted lines indicate restingmembrane potentials. (d) Negative of second spatial derivative of peakamplitude data from panel (c). Amplitude data are first resampled to auniform grid and results are filtered using a Hamming window


peak amplitudes of simulated backpropagating actionpotentials (Fig. 3(c), solid lines), one can see that they aretypically very similar. The differences near the soma aredue to the same transient decrease in action potentialamplitude seen previously in the hypothetical uniform cabledendrite model (Fig. 1). These transients disappear withinthe first 200 μm, and subsequently, the bAP amplitudestrack the attractors very closely, indicating that theattractors are controlling the propagation behavior.

In the cases of nonuniform IA distribution it is clear thatthe traveling wave attractor is sensitive to the shiftingchannel conductances. In the models with increasing IAconductance density, the attractor amplitudes decreasenearly linearly with distance from the soma and the slopeof this decline depends on the slope of the increase in IAconductance. In the uniform model (light gray), the attractoramplitude remains constant for the length of the dendrite. Inthe other two cases with rising IA conductance, travelingwave attractor amplitudes fall with distance with slopes of−0.04 and −0.12 mV/μm for the intermediate and steepestcases (dark gray and black dashed lines, respectively). Afterthe initial transients, backpropagating action potentialsfollow these underlying attractors very closely, falling inamplitude with similar slopes. This close correspondenceindicates that the backpropagating action potential isclosely tracking a slowly changing steady state, given bythe traveling wave attractors. Thus, the change in mem-brane properties can be considered “adiabatic.” It isimportant to keep in mind that for more abrupt, non-adiabatic changes in IA channel density than thoseconsidered in Fig. 3, it may not be possible for thepropagating action potential to track the changing underly-ing steady state, and that discrepancies between simulationsand traveling wave attractors may arise. Abrupt featuressuch as boundaries, step-wise changes in channel kinetics,and branch points can never be considered to be adiabaticchanges and instead lead to transients that can be measuredby comparing simulations to traveling wave attractors orsimulations that do not include the studied non-adiabaticchange in parameters.

Transients in action potential amplitude can be observedat the dendrite tip depending on IA channel distribution(Fig. 3(c)). In the two cases in which the backpropagatingaction potential still has significant amplitude near the endof the dendrite, there is a transient increase in actionpotential amplitude. These rises in amplitude are due to theinteractions of the propagating waves with the sealedboundary at the end of the dendrite, shown previously tocause a transient increase in amplitude (Goldstein and Rall1974). Notice that the traveling wave attractor does notdeviate near the boundaries of the model dendrite. Eventhese terminal compartments are assumed to be infinite

cables in order to compute the travel wave attractors. Thisuseful property of traveling wave attractors allows one toquickly quantify the effects of the sealed dendrite tip. Sincethe attractors predict propagation behavior very accuratelyin the absence of any abrupt change in morphology such asa sealed tip, any discrepancy between the simulated actionpotential and the attractor is attributable to the change inmorphology. Later, this strategy will be used to dissect theeffects of dendritic branching from the effects of voltage-gated ion channels (Fig. 7).

In the case of steepest IA conductance density (Fig. 3,black lines), backpropagation seems to fail near 300 μm asindicated by the large drop in the rate of decrease of peakamplitude with distance (Fig. 3(d)). In this case thetraveling wave attractor disappears near 300 μm in exactlythe same way that the attractor disappears when gKA isincreased sufficiently in uniform models (Fig. 2). Thissudden disappearance of the underlying traveling wavecauses the dramatic change in backpropagation in theabsence of any changes in dendritic morphology and asmentioned, the location of attractor failure is a very goodpredictor of the location of action potential failure. Thisexplains how non-uniform, but smooth channel distribu-tions alone, in the absence of any critical branch point canlead to abrupt decline in action potential amplitude asobserved experimentally (Golding et al. 2001).

3.5 Distance-dependent changes in IA channel dynamicshave dramatic effects on backpropagation amplitude

In Fig. 3 we showed, as other groups have shown (Miglioreet al. 1999; Golding et al. 2001), that the distribution of IAhas a significant impact on bAPs in model CA1 pyramidalcells. Increasing IA density decreases bAP amplitude andcan cause backpropagation failure. These effects on back-propagation can be accurately predicted by the underlyingtraveling wave attractors. Next, we show how changes in IAchannel kinetics, independent of distribution, can dramati-cally affect backpropagation amplitude.

Dynamic properties of IA change considerably whenmeasured near the soma compared to when measured farfrom the soma in CA1 pyramidal cells (Hoffman et al. 1997).These two “types” of IA that have been observed onproximal and distal apical dendrites differ significantly intwo ways. First, IA recorded more than 100 μm from thesoma activates at lower voltages (by approximately 10 mV)compared with that recorded proximal to the soma (Hoffmanet al. 1997; Migliore et al. 1999). Second, IA in the distaldendrite activates approximately twice as rapidly as thatrecorded proximally (Hoffman et al. 1997; Migliore et al.1999; see Section 2). Qualitatively, both of these differ-ences allow distal IA channels to react more rapidly to


upswings in potential during action potential onset. Wetherefore refer to the IA dynamics observed distally as“fast” and proximally as “slow.”

3.6 “Fast” IA limits backpropagation amplitudeand promotes backpropagation failure

In order to study the effects on propagation of the twodifferent types of IA dynamics observed in CA1 pyramidalcell, we study two dendrite models, one with “fast” and onewith “slow” IA (equations from Migliore et al. 1999). Forthese simulations, the distribution of IA is the same in bothcases and matches the black lines in Fig. 3. For the “slow”form of IA, the traveling wave attractor predicts that theaction potential amplitude should decrease by 5.3 mV every100 μm (Fig. 4, black dashed line, slope = 53 mV/mm). Inthe corresponding multi-compartmental simulation (solidblack), we see that this is an accurate prediction away fromthe boundaries. Backpropagation behavior changes signif-icantly when the “fast” form of IA is incorporated in themodel dendrite. In this case, the action potential isexpected to have smaller amplitude initially, to decreasemore rapidly with distance (12 mV every 100 μm), and tofail to backpropagate at a distance of 245 μm (gray dashedline). Again, multi-compartmental simulations confirmthese predictions; the action potential follows the attractorclosely and declines significantly more rapidly after250 μm (solid gray).

3.7 A transition in IA dynamics can leadto backpropagation failure

In Fig. 4 we showed how the faster reacting, distal form ofIA leads to reduced backpropagation amplitude in modelscompared to the slower, proximal form. An abrupttransition from the proximal to the distal form at a givenlocation can therefore be expected to cause poor propaga-tion, or possibly no active propagation, distal to thetransition location. We test this hypothesis in the modelfor two locations of the transition from “slow” to fastkinetics (200 and 400 μm, Fig. 5). The traveling waveattractors are almost identical up until 200 μm with only aslight difference due to small differences in restingmembrane potential along the dendrite in the two models.With the transition to fast IA dynamics occurring at 200 μm(gray lines), a step decrease in traveling wave attractoramplitude occurs, followed by its near immediate disap-pearance. Since this change in channel kinetics is abruptand not adiabatic as for shifting conductance densities, onedoes not expect the propagating action potential to followthe underlying steady state, i.e. traveling wave attractor,closely. The transient effect of this non-adiabatic change inchannel kinetics is directly observable by comparing solidlines and dashed lines in Fig. 5: the action potential’samplitude drops off sharply just before 200 μm. The smallledge seen before the second drop off is due to largechanges in waveform shape (not shown) near the discon-tinuity. With the transition to distal IA dynamics occurringat 400 μm, the traveling wave attractor (dashed black line)disappears and the simulated action potential (solid blackline) fails near 400 μm.

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Fig. 4 Effects of activation properties of IA on backpropagatingaction potential amplitude. Two identical dendrite models arecompared except that in one case IA is of the “slow reacting” type(found on proximal dendrites, black lines), while in the other case IA isof the “fast reacting” type (found on distal dendrites, gray lines).Amplitude of the action potential is plotted for the two cases (solidlines) along with amplitude of the traveling wave attractors (dashedlines). When the model includes the fast reacting version of IA, thetraveling wave attractor disappears at 245 μm. Dashed–dotted linesshow resting membrane potentials for the two cases

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Fig. 5 Transitions in activation properties of IA strongly influencebackpropagation amplitude and backpropagation failure. Same exper-iment as in Fig. 4 except activation dynamics of IA abruptly switchfrom the “slow,” proximal type to the “fast,” distal type at either200 μm (gray lines), or 400 μm (black lines)


In Fig. 5, the amplitudes of the simulated backpropagat-ing action potentials drop proximal to the point at which thetraveling wave attractor drops off or disappears. Thisapparently non-causal behavior occurs when the leadingedge of the propagating action potential (distal to the peak)propagates in the portion of the dendritic membrane thatcannot support the traveling wave. For this reason, thewaveform’s peak amplitude can depend on properties distalto the instantaneous peak location. Although step changesin IA channel kinetics cannot be considered adiabatic,induced transients are mainly restricted to the regionimmediately surrounding the location of the change inkinetics.

3.8 Propagation of dendritic spikes is affectedby changes in IA dynamics

Dendritic spikes can be generated in distal dendrites ofpyramidal cells in region CA1 and elsewhere (Häusser et al.2000; Golding et al. 2002; Gasparini et al. 2004). Largeconductance and especially highly spatio-temporallycorrelated excitatory inputs landing on small diameterdendrites can cause large membrane depolarizations. Re-cruitment of dendritic Na+ or Ca2+ voltage-gated conduc-tances can lead to dendritic spikes (Häusser et al. 2000).How these spikes propagate towards the cell soma dependson distributions of voltage-gated channels and channelkinetics, as well as morphology and branch pointsencountered. In Fig. 6 we show how changes in IAdynamics, which, as we have demonstrated, strongly affectaction potential backpropagation, also strongly affectforward propagation of dendritic spikes.

In the same dendrite model used in Figs. 4 and 5, a brief,large amplitude current pulse (4 nA, 0.5 ms) is applied tothe most distal compartment of the dendrite to trigger adendritic spike. In Fig. 6(a), the amplitude of the forwardpropagating spike is shown in the same two cases as inFig. 4. The “slow” (found on proximal dendrites, blacklines) and “fast” (found on distal dendrites, light gray lines)types of IA are incorporated with identical distributions.Dashed lines show the traveling wave attractor amplitudesfor the two cases, which are identical to those in Fig. 4.There is a large difference in the amplitude of forwardpropagation between the two cases. When the slow type ofIA is present on the entire dendrite, the dendritic spike iscarried all the way to the soma, where it elicits an actionpotential. As it propagates towards the soma, its amplitudesteadily increases as predicted by the traveling waveattractor. In contrast, when the fast type of IA is presentalong the entire dendrite, the dendritic spike fails topropagate and its amplitude quickly decays to zero. Thiscan be explained in terms of the traveling wave attractors.The distal dendrite, where the spike is generated, does not

support a propagating action potential as indicated by theabsence of the traveling wave attractor. This situation isdifferent than the case of failing backpropagating actionpotential, which propagates for a certain distance beforefailing. Here the dendritic spike is a transient effect of thedendritic stimulation and never propagates as a travelingwave. By the time the spike reaches the portion of thedendrite that exhibits a traveling wave attractor andtherefore supports a propagating action potential, itsamplitude is severely diminished and is therefore unableto elicit a propagating action potential in the proximaldendrite.

In Figs. 6(b) and (c), the time domain responses to thedendritic current pulse as measured at the soma andstimulation site, respectively, are shown. In addition to thetwo cases shown in panel a, the two examples from Fig. 5are also shown in which transitions from proximal to distalIA dynamics take place at 200 and 400 μm usingintermediate shades of gray. At the distal tip (600 μm),there is a small but clear difference in amplitude betweenthe case that successfully forward propagates and the

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Fig. 6 Forward propagation of dendritic spikes is influenced byactivation properties of IA. (a) Current pulses (4 nA, 0.5 ms) areapplied to the most distal dendritic compartment (600 μm) in order toelicit dendritic spikes that travel right to left. Peak amplitudes of theforward propagating spike are shown (solid lines) for the same twocases considered in Fig. 4 (black for “slow” IA and light gray for“fast” IA). Open symbols are peak amplitudes of time waveforms atsoma and distal tip. Dashed lines are traveling wave attractoramplitudes (identical to Fig. 4). (b) Somatic responses for the twocases shown in panel a (light gray and black lines), along with twoadditional cases in which IA dynamics switch from slow to fast at 200and 400 μm (intermediate gray scale lines). The somatic actionpotential elicited in the case of “slow,” i.e. proximal IA activation isclipped. (c) Dendritic spikes recorded at stimulation site. Same formatas panel (b)


others, which all overlap very closely. However, thisdifference is not sufficient to explain the observed results,since use of a larger current pulse cannot rescue forwardpropagation in the cases shown that fail (data not shown).The voltage responses measured at the soma (distance=0 μm) are similar (approximately 1.5 mV amplitudes) in allcases except when the slow (proximal) form of IA is presentalong the entire dendrite (black lines), which leads to a fullamplitude somatic action potential (voltage trace clipped).

When the density of IA rises unceasingly along thelength of the apical dendrites as in Fig. 6, the scenario thatforward propagation of distally generated dendritic spikes isonly possible when IA is of the slow form seems likely.While this form of IA was observed experimentally on theproximal regions of CA1 apical dendrites and not on thedistal regions (Hoffman et al. 1997), the kinetics of IA wasshown to be phosphorylation dependent (Hoffman andJohnston 1998). It is therefore possible that under differentcircumstances with different protein kinase activity in thedistal dendrites, IA might be in the slow form in the distaldendrites thereby permitting the possibility of forwardpropagation. Since there is some evidence that the IAconductance density in the distal tuft might drop (Rhodes etal. 2004), it is also necessary to keep in mind the possibilitythat forward propagation of distally generated dendriticspikes might be possible to some extent if IA if theconductance is sufficiently reduced. Recall that withreduced channel densities, the fast form of IA permitspropagation (Fig. 4, proximal region).

Finally, it is important to mention that even with channelproperties conducive to forward propagation, the extent offorward propagation may be limited by dendritic branching.Branching is especially unfavorable for forward propagat-ing dendritic spikes that are initiated on fine branches andattempt to invade the dendritic arbor compared to back-propagating action potentials. More on this will bediscussed after the effects of branching on propagation arepresented in detail in the context of backpropagating actionpotentials. Without conditions (both channel properties andbranching) conducive to forward propagation of solitary,distally generated dendritic spikes, successful propagationto the soma may require some form of “cooperativity.”Synaptic inputs may trigger several dendritic spikes onseparate branches simultaneously, or synaptic activationmay lead to sufficient dendritic depolarization to permitsuccessful forward propagation (Jarsky et al. 2005).

3.9 Contributions of dendritic morphologyto backpropagating action potential amplitude

Dendritic morphology, including variations in diameter ofdendritic processes, and patterns of branching, shapeselectrical signaling properties of neuronal dendrites (Rall

1959; Goldstein and Rall 1974; Mainen and Sejnowski1996; Vetter et al. 2001). Modeling studies have shown thatamplitude of backpropagation can be dramatically differentin models that feature different morphologies but identicaldistribution and properties of ion channels (Vetter et al.2001). We now reincorporate the originally published,apical dendrite morphology data from a reconstruction ofa CA1 pyramidal cell (Migliore et al. 1999). By holdingchannel properties the same as in the simple cable model,one can compare backpropagation amplitude in the twomodels. This comparison, with the help of the localtraveling wave information, allows one to tease out effectsdue specifically to dendritic morphology in non-uniformdendrite models.

3.10 Using traveling wave attractors to computeaction potential transients in dendrite modelswith non-uniform channel properties

Cable models with uniform channel properties have aconstant steady-state behavior given by the traveling waveattractor. Propagation velocity depends on diameter asgiven by Eq. (2), while the waveform of the propagatingaction potential represented by the attractor, plotted as afunction of time, is independent of diameter. Branches orabrupt changes in diameter can lead to failure (see Fig. 9below), or cause transient changes in the waveform (e.g., inamplitude) and permanent propagation delays or advances.In models with uniform channel properties, these amplitudetransients and changes in timing can be quantitativelyunderstood as a function of the geometric ratio (GR, Eq. (3)below, Goldstein and Rall 1974) calculated at each branchpoint or diameter change location (Goldstein and Rall 1974;Manor et al. 1991). The GR is defined as (Rall 1959):

GR ¼

X

j

d3=2j

d3=2p

: ð3Þ

Diameters of the j daughter branches, dj, form thenumerator with the diameter of the parent branch, dp, inthe denominator. Branch points with GR > 1 cause transientdecreases in amplitude, those with GR<1 cause transientincreases in amplitude, while with GR=1, the amplitude isunaffected (Goldstein and Rall 1974).

In dendritic models with non-uniform membrane prop-erties, quantitatively predicting effects of branching is moredifficult, because the steady-state propagation behavior isno longer constant along the dendrite. In cases for whichchanges in channel properties can be considered adiabatic(e.g., Fig. 3), calculated responses in the absence ofbranching follow the traveling wave attractor closely. In


these cases, it is possible to calculate the transientamplitude effects of branching by calculating the differencebetween amplitudes from simulations and amplitudes oftraveling wave attractors computed along the dendrites.

3.11 Effects of branching on backpropagation amplitude

A diagram of the morphologically detailed CA1 pyramidalcell model is shown in Fig. 7(a) (Migliore et al. 1999). Inthis diagram, angles between sections were chosen ran-domly, with the exception that the longest series of sections,with the distal tip lying 830 μm from the cell’s soma, isplotted on the vertical axis. Backpropagation along thisseries of sections, which will be referred to as the “principalbranch,” is studied here (all branches are studied in Fig. 8).Locations of changes in diameter along with branch pointlocations are indicated by open circles on the modeldiagram. Finally, Fig. 7(e) shows the geometric ratio values(GR, Eq. (3)) for each branch.

Somatic action potentials are triggered using briefcurrent pulses as before, and resulting backpropagatingaction potential amplitude as a function of distance fromsoma along the principal branch for three cases is shown inFig. 7(b), (c), and (d), solid lines. In these three cases, the

transition from “proximal” to “distal” IA dynamics occurs atthree different locations (distance=150, 300, and 450 μm inFig. 7(b), (c), and (d), respectively). The amplitude of thelocal traveling wave attractors is shown using dashed linesin each of the three cases.

In Fig. 7(b–d), backpropagation amplitude is initiallygreater than that of the local traveling wave attractor(leftmost shaded area). This discrepancy is due mainly tothe initial transient caused by the large amplitude somaticaction potential. Subsequently, after the action potentialpasses 65 μm, we observe that its amplitude is less than thatpredicted by the local traveling wave attractors. In Fig. 7(b),the action potential soon interacts with the location of thetransition from slow to fast IA dynamics. In Figs. 7(c–d),this transition occurs more distally, allowing us to see thatthe action potential amplitude consistently falls below thatof the traveling wave attractor by approximately 4 mV for adistance of roughly 200 μm. Recall that in a branchlessmodel with the same channel properties and distributions(Fig. 5), there is very little amplitude discrepancy. There-fore, we can attribute this 4 mV difference in amplitude tothe branches and changes in dendrite diameter encounteredalong this section of dendrite. Notice that in this region,between 50 and 350 μm, we see a cluster of GR values

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Fig. 7 Backpropagation ampli-tude in a morphologicallydetailed CA1 pyramidal cellmodel. (a) Diagram illustratingdendrite morphology using pub-lished soma and CA1 apicaldendrite morphology data.Angles between dendrite sec-tions are chosen from a proba-bility distribution. The longestseries of branches is shown as avertical path. Open circles indi-cate locations of changes indendrite diameter and branchpoints. (b), (c), (d). Backpropa-gation amplitude (solid), ampli-tude of underlying travelingwave attractor (dashed), andresting potentials (dashed–dot-ted) are shown for three cases.Dynamics of IA change from“proximal” to “distal” dynamicsat locations 150, 300, and450 μm in panels (b), (c), and(d) respectively. (e) GR (geo-metric ratio) values for allbranches and changes in diame-ter along longest, “principal”branch. Shaded regions indicateclusters of branches with GR<1


greater than one (Fig. 7(e)). It has been shown previouslythat an action potential’s amplitude transiently decreasesbefore passing a branch point with a GR value greater thanone (Goldstein and Rall 1974). In this section of this model,these effects add up to approximately 4 mV.

When propagation passes 320 μm, the discrepancybetween the action potential amplitude and that of the

traveling wave attractor switches from −4 mV to approxi-mately +5 mV (Fig. 7(d)), backpropagation has failed by thispoint in panels (b) and (c). Notice that in this region (shaded)between 350 μm to 500 μm, there is a cluster of GR valuesless than one. Branches with GR values less than onetransiently increase action potential amplitude (Goldsteinand Rall 1974). In this region of this model, the effects of

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mGKA = 0.5

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Simulations

TWA

Failure locations (b)(a)

Fig. 8 Action potential propagation failures, without failure of theunderlying traveling wave attractor, occur depending on IA channelproperties and distribution, and are due to the influence of dendriticbranches. (a) Peak amplitudes in all branches of the CA1 pyramidalcell model (see Fig. 7) with three different IA channel distributions. IAconductance, gKA, rises linearly with distance (see Fig. 3) with slopemGKA from an initial conductance of 48 mS/cm2. Thick black lines arefrom apical trunk (Fig. 7). Location of the disappearance (failure) of

traveling wave attractor (TWA) along principal branch is indicated byvertical dotted line (no failure in bottom panel). Simulation peakvoltage data is pooled, smoothed, and the minimum of the secondderivative (see Fig. 3(d)) is used to determine overall location ofaction potential failure (vertical dashed line). Bottom panel shows GR(geometric ratio) values for all branches. (b) Failure locations of TWA(dotted) are plotted along with failures computed from simulation data(dashed) as a function of slope of IA conductance distribution, mGKA


these branches on bAP amplitude accumulate to approxi-mately 5 mV.

3.12 Distal dendritic branches can cause backpropagationfailures when intrinsic properties would otherwisesupport active propagation

The distal portion of the model CA1 dendrites thatcorresponds to the apical tuft of these cells is comprisedof many bifurcations of processes already with smalldiameters. These bifurcations lead to values of geometricratio (GR) significantly greater than one. As discussedpreviously, such branches lead to transient decreases inbackpropagating action potential amplitude. If strongenough, these decreases may lead to propagation failuredepending on intrinsic membrane excitability properties(Goldstein and Rall 1974). We find that failures do occur inthe distal region of the CA1 model due to high-GRbranches in regions with a substantial amount of IA. Inthese cases, membrane excitability is sufficient to sustainpropagation, as indicated by the traveling wave attractor;however, failure is observed in spite of this.

In Fig. 8, the distribution of IA is varied but the transitionfrom proximal to distal IA dynamics (varied in Fig. 7) isfixed at 150 μm. Peak membrane potential values for everymodel compartment are plotted as a function of distancefrom the soma. Failure locations predicted by location oftraveling wave disappearance are compared to failurelocation found from model simulations (using secondderivative data, see Fig. 3(d)) for each of the three IAchannel distributions. Decreasing the slope of the rising IAconductance density increases the amplitudes of back-propagating action potentials and smoothly delays the pointat which the intrinsic dynamics no longer sustain activepropagation. However, in simulations, strong effects due tobranching between 400 and 600 μm are seen and causefailures in many dendritic branches in this region for asignificant range of IA distributions. When the failurelocations predicted by the traveling wave attractor arecompared to those found in simulations (Fig. 8(b)), onesees a clear divergence of the two curves plotted as afunction of the slope in IA conductance density. Inparticular, failures in simulations precede the disappearanceof the attractor. Two non-adiabatic changes are possiblesources of divergence of simulations and attractors:boundary effects and branches. Because our sealed-endboundaries lead to increases in membrane potential relativeto predictions from attractors, as can be seen from the casesin Fig. 8 for which bAPs do not fail, the effects ofboundaries would be to reduce the likelihood of failure. Incontrast, high-GR branches promote failure of bAPs. Thus,we conclude that the unpredicted failures seen in Fig. 8(b)are caused by dendritic branching.

3.13 Propagation failure at branch points depends on IAchannel properties

Ion channel densities and properties affect action potentialamplitude directly. This is evident in branchless models andeffects can be predicted by computing the underlyingtraveling wave attractors (Figs. 1–6). Additionally, actionpotential transients at branch points with specific parent anddaughter branch diameters depend on channel densities andproperties (Manor et al. 1991). For example, the samebranch point can cause anywhere from negligible effects onaction potential amplitude to propagation failure, dependingon membrane excitability properties. Spurred by resultsfrom Fig. 8, we explored how failure at branch pointsdepends on GR and GKA, the conductance underlying IA(Fig. 9). As expected, larger values of GR make propaga-tion more difficult; thus, only small values of GKA allowaction potentials. The “distal,” faster version of IA makesthe bAP more prone to failure. For GKA approaching themaximal values that permit propagation, the morphologicalperturbation sufficient to cause propagation failure becomesvanishingly small. For this reason GR values approach 1near these critical values of GKA (Fig. 9 open symbols).

The propagation failure data in Fig. 9 qualitativelyexplain why morphology-induced propagation failures areobserved in the distal region of the present CA1 model

1 2 3 4 5 6 7 8 9 10 11

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GK

A (

mS

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2 )

Failure for distal and proximal IA

Failure for distal IA

Fig. 9 Propagation failure at branch points depends on IA conduc-tance density and channel activation properties. Lines indicate criticalgeometric ratio (GR) and IA conductance density (GKA) pairs that leadto propagation failure in uniform cable models with a single stepchange in diameter. Shaded areas indicate regions of propagationfailure for proximal and distal IA dynamics. A constant, uniformcurrent is adjusted along with GKA to maintain pre-stimulus steady-state membrane potential at −65 mV. Maximal IA conductancedensities for stable propagation are plotted at GR=0 for proximaland distal IA kinetics (square and circle, respectively)


(Fig. 8). Failures at branch points occur when the fast formof IA is present on the distal dendrites and the channel’sconductance density increases with distance at a rate greaterthan mGKA=0.18 mS/cm2/μm (Fig. 8(b)). Failures clusteraround 450 μm where GR values at branch points aretypically 2 (Fig. 8(a) bottom). At 450 μm, IA conductancedensities are greater than 130 mS/cm2 for the range ofdistributions that lead to failure. These results correspondwell with the data for solitary branch points, which causepropagation failure for IA conductance densities greaterthan 150 mS/cm2 when GR=2 (Fig. 9).

4 Discussion

In this study we take a closer look at how the transientpotassium current IA affects backpropagation of actionpotentials and forward propagation of dendritic spikes. Wefind that the activation dynamics of this current can have anespecially large impact on propagation. Distally locatedchannels tend to activate at lower potentials and activatemore quickly. These distal channel activation properties canlead to backpropagation failure, similar to that seenexperimentally. Also, these properties can make it impos-sible for distally generated dendritic spikes to reach thesoma. Because the activation properties of these channelsare modulated by common intracellular second messengers(Hoffman and Johnston 1998), it is possible that the cell isable to tune or alter the propagation behavior of back-propagating action potentials and dendritic spikes for somepurpose. Interestingly, activation properties of sodiumchannels are also subject to modulation by the same secondmessengers and therefore, these channels can also beexpected to play a role in “backpropagation modulation”(Gasparini and Magee 2002; Frick et al. 2004).

We discovered that dendrites can have underlyingtraveling wave attractors that govern backpropagatingaction potentials and forward propagating dendritic spikes.We describe how to compute these attractors for any model,morphologically simplified or complex. For cases in whichconductance densities change adiabatically, these attractorsprovide nearly perfect descriptions of simulated behavior.For cases in which parameters change abruptly, thetraveling wave attractor typically remains qualitativelyuseful. Depending on the properties of the dendrite, theunderlying attractor can change. Changes in attractoramplitude can predict the actual backpropagation behaviorfrom simulations well, depending on the dendrite’s mor-phology. Also, the underlying traveling wave attractor candisappear, and this disappearance has important consequen-ces for dendritic action potentials. In the absence of a stabletraveling wave attractor it is difficult or impossible to

trigger a propagating dendritic spike. Synaptic inputs mustbe large enough such that the membrane potential crossesthreshold at any neighboring region that does support atraveling wave. Also, propagating action potentials thatencounter the loss of the traveling wave attractor are likelyto fail. The failures we see in models resemble thoseobserved experimentally (Golding et al. 2001). The failurelocation is typically predicted by the point of attractordisappearance. In cases of branch points with largegeometric ratios and large values of GKA, failure can bemore prominent than predicted by the attractor (Figs. 8, 9).

Changes in dendritic morphology, including branchingand dendritic terminations, cause changes in the propagat-ing action potentials (Goldstein and Rall 1974). Associatedchanges in action potential amplitude are not reflected bytraveling wave attractors since dendritic diameter is onlyrelevant for the attractor’s wave speed and not its timedomain waveform (Hodgkin and Huxley 1952; Goldsteinand Rall 1974; Weiss 1996). The resulting discrepanciesbetween the action potential’s amplitude and that of theunderlying attractor appear a certain distance before theaction potential arrives at the change in dendritic morphol-ogy. This distance depends on the spatial extent of theaction potential which is approximately equal to (exact fortraveling waves in uniform cables) the product of the timedomain extent (ex. half width) and the wave speed. Wavespeed is proportional to the square root of dendriticdiameter in exactly the same way as is the dendrite’selectrotonic length (Goldstein and Rall 1974). Therefore,dendrites with large diameter are expected to have largewave speed and spatially extended action potentials. Theseaction potentials can be expected to react well in advanceof, and possibly more strongly to, any dendritic branchpoint or termination they encounter. Given these complex-ities, it seems that the best way to understand the effects ofdendritic morphology in a particular model may be to do aswe do in this study: compare propagation in the morpho-logically complex model of interest with propagation in asurrogate model with branch points or terminationsremoved.

With the help of traveling wave attractors and morpho-logically simplified models, we are able to distinguisheffects on action potential propagation due to branching andchanging dendritic diameter from those due to voltage-gated ion channels in a morphologically complex dendritemodel. In the model CA1 pyramidal neuron dendrite,effects due to morphology are clearly visible and dependon whether the dendrite is becoming effectively wider ornarrower. Typically, these effects are minor; however, distaldendritic branches in the model we studied have particu-larly large values of GR and can cause action potentialfailures in the distal dendrites for a range of IA channelconductances in this region. These effects can be under-


stood in terms of the interplay between morphology andmembrane properties (Fig. 9).

The analysis in this paper focused on the effects of time-invariant parameters, including the spatial distribution ofGKA and branch points of a given geometric ratio (GR).Traveling-wave-attractor-based analysis could be extendedto more complex cases including time-varying inputs,assuming that such inputs could be approximated as adiabatic(e.g., broadly distributed in space and stationary in time).

Excitatory synapses in the distal dendritic region likelyfollow different learning rules than those in more proximalregions if backpropagation failure is indeed prevalent. Recentexperiments in neocortical pyramidal cells demonstrate thatresponses to spike-timing-dependent plasticity protocols arecrucially sensitive on backpropagation (Sjöström andHäusser 2006). We speculate that similar mechanisms areoperating in CA1 pyramidal cells, along with more clearlyestablished dependence of CA1 plasticity on local dendriticspikes (Golding et al. 2002). The behavior of bAPs anddendritic spikes in distal regions depends strongly on bothchannel density and the phosphorylation-determined kinet-ics of IA (Hoffman and Johnston 1998). For this reason, it isimportant to make difficult experimental measurements ofchannel densities and kinetics (or phosphorylation states) inorder to understand the rules of plasticity of proximal anddistal synapses more deeply.

Acknowledgements The authors would like to thank: GeorgiMedvedev (Drexel University) and Eugene Wayne (Boston University)for early, stimulating input; Bard Ermentrout (University of Pittsburgh)for technical help with XPPAUT; and Jonathan Bettencourt, Kyle Lillis,and Theoden Netoff (NDL, Boston University) for feedback and helpediting the final manuscript.

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roles of ia and morphology in action potential propagation...

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