# Dendritic Spines: Continuum Theory

**DOI:**https://doi.org/10.1007/978-1-4614-7320-6_797-1

## Keywords

Spine Density Safety Factor Dendritic Spine Continuum Theory Spine Morphology## Synonyms

## Definition

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).

^{5}spines. 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

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 }(*μ*m^{2}) and specific membrane capacitance *C* _{ m }(*μF*/cm^{2}); 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.

*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).

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.

## References

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