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Saltatory Waves in the Spike-Diffuse-Spike Model of Active Dendritic Spines

S. Coombes1 and P. C. Bressloff2

1Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom2Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA

(Received 9 April 2003; published 9 July 2003)

In this Letter we present the explicit construction of a saltatory traveling pulse of nonconstant profile

in an idealized model of dendritic tissue. Excitable dendritic spine clusters, modeled with integrate-and-fire (IF) units, are connected to a passive dendritic cable at a discrete set of points. The saltatorynature of the wave is directly attributed to the breaking of translation symmetry in the cable. Theconditions for propagation failure are presented as a function of cluster separation and IF threshold.

DOI: 10.1103/ PhysRevLett.91.028102 PACS numbers: 87.19.La, 05.45.Xt

The focus of many mathematical studies in physics hasbeen on waves which propagate with constant speed andconstant profile. However, there is an increasing body ofexperimental data from the natural sciences h ighlightingthe existence of waves which t ravel with nonconstantprofile. For example, when calcium is released from in-

ternal stores into the cytosol of a cardiac myocyte, a waveof increased concentration can travel with a lurchingquality, where activity is seen to jump from store to store[1]. Another example can be drawn from the field ofcomputational neuroscience where neurons that can firevia post inhibitory rebound are known to underlie thegeneration of lurching waves of activity propagatingthrough an inhibitory network [2]. Such lurching wavesare typically referred to as saltatory. In this Letter, wepresent the explicit construction of a saltatory wave in anidealized model of a neuronal dendrite.

In the cerebral cortex, approximately 80% of all ex-citatory synapses are made onto dendritic spines (see

Fig. 1). These are small mushroomlike appendages witha bulbous head a nd a tenuous stem (of length around1 m) and may be found in their hundreds of thousandson the dendritic tree of a single cortical pyramidal cell.The biophysical properties of spines have been linkedwith mechanisms for Hebbian learning [3], the imple-mentation of logical computations [4], coincidence de-tection [5], orientation t uning i n complex cells of visualcortex [6], and the amplification of distal synaptic inputs[7]. The implication of excitable channels in the spinehead membrane for amplification of excitatory synapticinputs was first discussed by Jack et al. [8]. However, it isonly relatively recently that confocal and t wo-photonmicroscopy observations have confirmed the generationof action potentials in the dendrites. Since dendriticspines possess excitable membrane, the spread of currentfrom one spine along the dendrites may bring adjacentspines to threshold for impulse generation, resulting in asaltatory propagating wave in the distal dendriticbranches [9].

The first theoretical study of wave propagation medi-ated by dendritic spines was carried out by Baer and

Rinzel [10]. T hey considered a continuum model of adendritic tree coupled to a distribution of excitable den-dritic spines. The active spine head dynamics is modeledwith Hodgkin-Huxley kinetics while the (distal) den-dritic t issue is modeled with the cable equation. The spinehead is coupled to the cable via a spine stem resistance

that delivers a current proportional to the number ofspines at the contact point. There is no direct couplingbetween neighboring spines; voltage spread along thecable is the only way for spines to interact. Numericalstudies of the Baer-Rinzel model [10] show both smoothand saltatory traveling wave solutions, the former arisingin the case of uniform spine distributions and the latterwhen spines are clustered in groups. The saltatory natureof a propagating wave may be directly attributed to thefact that active spine clusters are physically separated. Inthis Letter, we present an alternative, analytically trac-table treatment of saltatory waves based on the so-calledspike-diffuse-spike (SDS) model of active dendritic

spines [11,12]. The SDS model, which reduces the spinehead dynamics to an all-or-nothing action potentialresponse, was previously used to construct exact solutionsfor smooth waves in the case of a uniform spine density.However, this analysis was limited since it did not capturethe true saltatory nature of a dendritic wave. Here weexplicitly take into account the discrete nature of spineclusters, and explicitly construct the corresponding

FIG. 1. An example of a piece of spine studded dendritictissue (from rat hippocampal region CA1 stratum radiatum)5 m in length. Taken with permission from Synapse Web,Boston University, http://synapses.bu.edu

P H Y S I C A L R E V I E W L E T T E R S week ending11 JULY 2003VOLUME 91, NUMBER 2

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saltator y waves. We also derive dispersion curves for thespeed of the wave as a function of cluster spacing and thespine threshold, and determine the conditions for wavepropagation failure.

Let x represent t he spine density per unit lengthalong a uniform, passive dendritic cable. Denoting thevoltage at positionx on the cable at time t by V Vx;t,the associated cable equation is given by

@V

@t V 2 @

2V

@x2 2rax

VV Vr

; (1)

where and are the membrane time constant and theelectronic space constant of the cable. The parameter r isthe spine stem resistance of an individual spine and ra isthe intracellular resistance per unit length of cable. In the

SDS model, the function VVx;t represents the sequenceof action potentials generated in the spine head at xwhenever the associated subthreshold spine head poten-tial Ux;t, driven by current f rom the shaft, crosses somethreshold h. Given the high resistance of the spine stem,we neglect subthreshold currents into the cable. The

voltage U evolves according to the integrate-and-fire(IF) equation,

CC@U

@t U

rr VU

r; (2)

such that whenever Ucrosses the threshold h it is imme-diately reset to zero. Here CC and rr are the membranecapacitance and resistance of the spine head. Let tjxdenote the jth firing time of the spine head at position xsuch that Ux;tjx h. Then VVx;t Pjt tjxwith t 0 for all t < 0. The shape of the actionpotential is specified by the function t, which can befitted t o the universal shape of an action potential.

In the original formulation of the SDS model, the spinedensity function was taken to be uniform. Although im-pulse propagation failure is known to occur if the spinedensity is below some critical level, the numerical studiesof Baer and Rinzel suggest that propagation may berecovered by redistributing the spines into equally spaceddense clusters. Since interspine distances are of the orderof micrometers and electronic length is typically mea-sured in millimeters, we shall consider spine head voltageat a cluster site to be the local spatial average ofmembrane potential in adjacent spines. Hence, we con-sider a discrete distribution of spines for which x nPm xxm, where xm is the location of the mth spinecluster and n is the number of spines in a cluster. Such adistribution breaks continuous translation symmetry sothat saltatory or lurching waves a re expected rather t hantraveling waves of constant profile. We define a saltatorywave as an ordered sequence of firing times . . . tm1