Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Dendritic Spines: Continuum Theory

  • Steven M. BaerEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_797-1


Spine Density Safety Factor Dendritic Spine Continuum Theory Spine Morphology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The continuum theory for dendritic spines, developed by Baer and Rinzel (1991), is an extension of classical cable theory for which the distribution of spines is treated as a continuum. The theory applies when the interspine distance is much less than the length scale of the dendrite, for example, when the dendrite is populated by a large number of spines. The formulation maintains the basic feature that there is no direct coupling between neighboring spines; voltage spread along dendrites is the only way for spines to interact. With the continuum theory, different spine morphologies, multiple populations of spines, and distributed physiological properties are represented explicitly and compactly by relatively few differential equations. The theory is general so that idealized or realistic kinetic models may be adapted.

Detailed Description

Classical cable theory has provided valuable insights into structure-function relationships and guidance in the design of experiments. Perhaps the best example is Rall and Shepherd’s electrotonic model of the olfactory bulb, which predicted the existence of functional dendrodendritic synapses (Rall and Shepherd 1968). Cable equations have been used successfully to study tapered nerves and extensively branched dendritic trees (Jack et al. 1975; Rall 1977), as well as cables with nonuniform electrical parameters such as specific membrane resistivity (Holmes and Woody 1989).

In the 1970s and 1980s, Rall, Rinzel, Miller, and Segev spearheaded efforts that accelerated the theoretical investigation into the function of dendritic spines (Rall and Rinzel 1971a, b; Miller et al. 1985; Shepherd et al. 1985; Rall and Segev 1987; Segev and Rall 1988). Dendritic spines are small knoblike evaginations of the dendritic surface (see Fig. 1a, b), and a single neuron may have 105spines. These large numbers of spines present a challenge for computational studies, particularly when using compartmental modeling. To address this, Baer and Rinzel (1991) formulated an extension of classical cable theory to model cables with large spine densities. They treated the distribution of spines as a continuum (see Fig. 1). This allowed a single cable with hundreds or thousands of spines to be modeled with just a few differential equations. Baer and Rinzel’s model is
Fig. 1

Continuum of dendritic spines. (a): Cortical pyramidal cells (Modified from Berkley 1897). (b): A schematic magnification from (a) shows the dendrite (cylinder) studded with spines. (c): If there are many spines, the dendritic electrical response can be approximated by considering a continuum of spines (stippled envelope). Although spines are treated mathematically as a continuum, the model is constructed so that spines interact only through spread of dendritic potential (From Baer and Rinzel (1991), Fig. 1)

$$ {\tau}_m\ \frac{\partial {V}_d}{\partial t}=\frac{\partial^2{V}_d}{\partial {X}^2}-{V}_d+{R}_{\infty }{\overline{n}}_i{I}_{ss} $$
$$ {C}_{sh}\frac{\partial {V}_{sh}}{\partial t}=-{I}_{ion}-{I}_{syn}-{I}_{ss}, $$
$$ {I}_{ss}=\frac{V_{sh}-{V}_d}{R_{ss}}. $$

Here, V sh and V d are respectively the membrane potential in the spine head and the dendritic base (or shaft). Equation 1 is the cable equation in terms of dimensionless (electrotonic) length X (computed for the passive cable without spines), τ m is the membrane time constant, and R is the input resistance for a semi-infinite passive cable with circular cross section. The spine density \( \overline{n} \) expresses the number of spines per unit electrotonic length. The spine stem current is denoted by I ss and represents the I ⋅ R voltage drop across the spine stem resistance R ss , as expressed in Eq. 3. The spine head is modeled in Eq. 2 as an isopotential compartment with surface area A sh (μm2) and specific membrane capacitance C m (μF/cm2); individual spines have a capacitance of C sh  = A sh C m (μF). Equation 2 describes the membrane potential in a single spine obtained from a current balance relation for the capacitive, ionic, spine stem, and synaptic currents.

The primary focus of Baer and Rinzel’s study was on threshold properties, e.g., minimum number of synaptically activated spines, or minimal density of spines, for the initiation and spread of activity. They found that propagation was precluded for spine stem resistance (R ss ) either too large or too small (Fig. 2). Moreover, even if R ss was in a suitable range for the local generation of an action potential (resulting from local synaptic excitatory input), the range was shown to be not suitable to initiate a chain reaction of spine firings along the dendrite; success or failure of impulse propagation depends on an even narrower range of R ss values. Also demonstrated was that appropriate clustering of spines can enhance synaptic amplification and spread of activity (Fig. 3).
Fig. 2

Propagation failure for spine stem resistance to large or too small. A dendrite of electrotonic length 3 has 60 uniformly distributed excitable spines. Four spines are synaptically activated near X = 0 using an alpha function stimulation. Electrode recordings in the spine head and dendrite are simulated at X = 0 and X = 2. (a): Voltage transients in the head indicate propagation failure when the spine stem resistance is too low (short-dashed curves) because dendritic loading effectively raises the threshold for a local action potential out of range for the given input. Propagation also fails when the stem resistance is too high (long-dashed curves) because large stem resistance results in a decreased shaft potential; the spreading dendritic depolarization becomes too weak to bring the spine heads in front of the wave to threshold. However, for intermediate values of spine stem resistance (solid curves), a “spike-diffuse-spike” chain reaction occurs and the wave propagates. (b): Peak voltage versus stem resistance in spine head (dashed) and dendritic shaft (solid) (From Baer and Rinzel (1991), Fig. 3, with figure caption modified)

Fig. 3

Spine clusters. Impulse propagation is contingent on the spatial distribution of spines. Effects of 3 spatial distributions of 24 excitable spines are compared in a cable of electrotonic length 2. Simulated electrode recording positions are X = 0, 1, and 2. Solid curves denote dendritic potentials, and dashed curves denote spine head potentials. Three spines near X = 0 are activated synaptically. (a): With a uniform but low density of spines (\( \overline{n}=12 \) spines per e.l.), an impulse fails to propagate. (b): The same 24 spines are grouped into 8 clusters spaced 0.34 apart. Each cluster has 3 spines and width 0.04 (\( \overline{n}=75 \) within a cluster). Impulse propagation is successful for this case. At X = 1 the simulated electrode recording is between spine clusters; hence, membrane potential is shown for the dendrite only. (c): The 24 spines are grouped into 4 clusters spaced 0.42 apart. Each cluster has 6 spines and width 0.08 (again, \( \overline{n}=75 \) within a cluster). Here the impulse propagates to X = 1 but fails before reaching X = 2. Although the clusters have the same density as in (b), they are too far apart for successful propagation (From Baer and Rinzel (1991), Fig. 4)

The Baer-Rinzel model led to some new and important insights into the electrical interplay between passive and excitable membranes. For example, regions of decreased conduction load (e.g., near sealed ends of nerves) have a higher safety factor and therefore facilitate attenuating waves and further enhance waves that are successfully propagating. In regions of increased conductance load such as common branch points, attenuation of the potential and propagation failure are more likely. Spine densities (passive or excitable) also affect conductive loading and safety factor. A suitable spine density could ensure propagation through regions where the safety factor is low. More generally, propagation in regions of nonuniform loading may be complemented or counteracted by variations in spine density; the dynamics of electrical current flow in cables with spines are dependent on spine distribution and spine morphology.

The continuum model has been applied to a variety of problems, for example, the dynamics of receptor potentials in mechanoreceptors (Bell and Holmes 1992) and, more recently, the effect of noise on spiny dendrites (Coutts and Lord 2013). The continuum theory of dendritic spines has been extended to study spine plasticity (Verzi et al. 2004), and it has spawned other computational approaches, including the spike-diffuse-spike model (Coombes and Bressloff 2003), originally intended as a computationally simple version of the Baer-Rinzel model.


  1. Baer SM, Rinzel J (1991) Propagation of dendritic spikes mediated by excitable spines: a continuum theory. J Neurophysiol 65:874–890PubMedGoogle Scholar
  2. Bell J, Holmes M (1992) Model of the dynamics of receptor potential in a mechanoreceptor. Math Biosci 110:139–174PubMedCrossRefGoogle Scholar
  3. Coombes S, Bressloff PC (2003) Saltatory waves in the spike-diffuse-spike model of active dendritic spines. Phys Rev Lett 91:028102PubMedCrossRefGoogle Scholar
  4. Coutts EJ, Lord GJ (2013) Effects of noise on models of spiny dendrites. J Comput Neurosci 34:245–257PubMedCrossRefGoogle Scholar
  5. Holmes WR, Woody CD (1989) Effects of uniform and non-uniform synaptic ‘activation-distributions’ on the cable properties of modeled cortical pyramidal neurons. Brain Res 505:12–22PubMedCrossRefGoogle Scholar
  6. Jack JJB, Noble D, Tsien RW (1975) Electric current flow in excitable cells. Clarendon, OxfordGoogle Scholar
  7. Miller JP, Rall W, Rinzel J (1985) Synaptic amplification by active membrane in dendritic spines. Brain Res 325:325–330PubMedCrossRefGoogle Scholar
  8. Rall W (1977) Core conductor theory and cable properties of neurons. In: Kandel ER, Brookhardt JM, Mountcastle VM (eds) Handbook of physiology, the nervous system, cellular biology of neurons. American Physiological Society, Bethesda, pp 39–97Google Scholar
  9. Rall W, Rinzel J (1971a) Dendritic spines and synaptic potency explored theoretically. Proc Int Union Physiol Sci XXV Int Congr IX:466Google Scholar
  10. Rall W, Rinzel J (1971b) Dendritic spine function and synaptic attenuation calculations. Prog Abstr Soc Neurosci 1:64Google Scholar
  11. Rall W, Segev I (1987) Functional possibilities for synapses on dendrites and on dendritic spines. In: Edelman GM, Gall WE, Cowan WM (eds) Synaptic function. Wiley, New York, pp 605–636Google Scholar
  12. Rall W, Shepherd GM (1968) Theoretical reconstruction of field potentials and dendrodendritic synaptic interactions in olfactory bulb. J Neurophysiol 31:884–915PubMedGoogle Scholar
  13. Segev I, Rall W (1988) Computational study of an excitable dendritic spine. J Neurophysiol 60:499–523PubMedGoogle Scholar
  14. Shepherd GM, Brayton RK, Miller JP, Segev I, Rinzel J, Rall W (1985) Signal enhancement in distal cortical dendrites by means of interactions between active dendritic spines. Proc Natl Acad Sci U S A 82:2192–2195PubMedCentralPubMedCrossRefGoogle Scholar
  15. Verzi DW, Rheuben MB, Baer SM (2004) Impact of time-dependent changes in spine density and spine shape on the input–output properties of a dendritic branch: a computational study. J Neurophysiol 93:2073–2089PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA