# Dendritic Spines

**DOI:**https://doi.org/10.1007/978-1-4614-7320-6_132-1

## Keywords

Dendritic Spine Synaptic Input Membrane Area Spine Head Calcium Conductance## Definition

Many types of neurons receive synaptic inputs on dendritic spines which are short protrusions of membrane composed of a bulbous “head” connected to the dendrite by a thin “stem” or “neck” (Yuste 2010). Although spines are typically small in size, they occur in high densities along dendrites and may compose 40–60 % of the total dendritic surface area. There are two straightforward methods for including spines in computational models of neurons without representing every spine explicitly.

## Detailed Description

Spines are widely regarded to serve a biochemical function by providing an isolated compartment where highly localized calcium signals and subsequent reactions can occur. An electrical function was initially proposed by Rall who showed that the steady-state attenuation of a signal from the spine head to the dendrite could be expressed as V_{SH}/V_{BI} = 1 + R_{SS}/R_{BI}, where V_{SH} and V_{BI} are the voltages at the spine head and dendrite, respectively; R_{SS} is the spine stem resistance; and R_{BI} is the input resistance at the dendrite. Rall suggested that changes in spine stem diameter might be important for plasticity because of its effect on R_{SS}, but studies searching for such changes produced mixed results. Others have suggested that if voltage-dependent conductances are present on spines, then spines might amplify inputs and could otherwise increase the computational possibilities of the cell. Many of the more interesting possibilities suggested by theoretical studies require a large R_{SS}, but whether or not R_{SS} is sufficiently large has been controversial.

Because the proportion of membrane area occupied by spines can be large, single-neuron models that ignore spines are bound to produce erroneous results. There are two straightforward methods for including spines in models without coding each and every spine, as long as these implicitly included spines do not receive synaptic input. First, if dendritic and spine membranes are assumed to have the same specific membrane resistance and capacitance, R_{m} and C_{m}, spines can be implicitly included in a model by increasing C_{m} and reducing R_{m} according to the proportion of total membrane area contributed by spines. We call this the R_{m}–C_{m} method. For example, if spines increase the membrane area of a dendrite by 33 %, then spines can be implicitly modeled by multiplying the dendrite C_{m} by 1.33 and dividing dendritic R_{m} by 1.33. Alternatively, one can keep R_{m} and C_{m} constant but increase length and diameter in such a way as to keep intracellular resistance constant (i.e., keep length/diameter^{2} constant). We call this the l-d method. For example, if spines increase membrane area by 33 %, let F = 1.33 and multiply length by F^{2/3} and diameter by F^{1/3}.

These two methods for including spines in models implicitly provide identical results with passive models, but they make very different assumptions when voltage-dependent conductances are present. The first method, where R_{m} and C_{m} are changed, assumes that spines have passive membrane, while the second method, where length and diameter are changed, assumes that the spines have the same densities of voltage-dependent conductances as the dendrite. The cell should have a lower input resistance with the second method but should also respond more nonlinearly with voltage perturbations. Needless to say, results may be very different.

Nevertheless, it is straightforward to modify either method to include spines possessing different densities of voltage-dependent conductances than present on the dendrite. With the first method, one should change dendritic R_{m} and C_{m} as described above but also modify the maximum conductance density (g-bar) for each conductance by \( {\overline{\mathrm{g}}}_{\mathrm{d}-\mathrm{new}}=\left({\overline{\mathrm{g}}}_{\mathrm{d}-\mathrm{old}}{\mathrm{A}}_{\mathrm{d}}+{\overline{\mathrm{g}}}_{\mathrm{s}\mathrm{pine}}{\mathrm{A}}_{\mathrm{s}}\right)/{\mathrm{A}}_{\mathrm{d}} \) where A_{d} is the membrane area of the dendritic segment and A_{s} is the membrane area of all spines on that segment. With the second method, one should modify the maximum conductance density for every conductance on each dendrite by \( {\overline{\mathrm{g}}}_{\mathrm{d}-\mathrm{new}}=\left({\overline{\mathrm{g}}}_{\mathrm{d}-\mathrm{old}}{\mathrm{A}}_{\mathrm{d}}+{\overline{\mathrm{g}}}_{\mathrm{s}\mathrm{pine}}{\mathrm{A}}_{\mathrm{s}}\right)/\left({\mathrm{A}}_{\mathrm{d}}+{\mathrm{A}}_{\mathrm{s}}\right) \) and then change length and diameter of the dendrite as described above. These expressions assume spine head and neck have the same channel densities. If this is not true, one could calculate an equivalent \( {\overline{\mathrm{g}}}_{\mathrm{spine}} \) to use.

These methods are easily incorporated into single-neuron simulators such as NEURON or GENESIS and work well for voltage-dependent conductances with fixed reversal potentials. However, these methods are subject to error when calcium and calcium-dependent conductances are present and particularly so if these conductances have different densities in spines and dendrites. Calcium accumulation and removal can be different in spines and dendrites, and different intracellular calcium concentrations will affect the calcium Nernst potential and activation of calcium-dependent conductances differently. The R_{m}–C_{m} method is preferred if dendrites have calcium conductances and spines do not. The l-d method is preferred if calcium conductances are on both dendrites and spines, but activation of calcium-dependent potassium conductances in spines remains problematic. It is possible to compensate for these types of errors by adjusting calcium pool parameters or by considering spine calcium and calcium-dependent conductances as separate conductances with their own parameters when conductances are merged into dendrites. These errors are not likely to be significant in short-time simulations but can be seen to accumulate over longer time periods.

## Reference

- Yuste R (2010) Dendritic spines. MIT Press, Cambridge, MAGoogle Scholar