Neuroscientists are currently hypothesizing on how voltage-dependent channels, in dendrites with spines, may be spatially distributed or how their numbers may divide between spine heads and the dendritic base. A new cable theory is formulated to investigate electrical interactions between many excitable and/or passive dendritic spines. The theory involves a continuum formulation in which the spine density, the membrane potential in spine heads, and the spine stem current vary continuously in space and time. The spines, however, interact only indirectly by voltage spread along the dendritic shaft. Active membrane in the spine heads is modeled with Hodgkin-Huxley (HH) kinetics. Synaptic currents are generated by transient conductance increases. For most simulations the membrane of spine stems and dendritic shaft is assumed passive. Action-potential generation and propagation occur as localized excitatory synaptic input into spine heads causes a few excitable spines to fire, which then initiates a chain reaction of spine firings along a branch. This sustained wavelike response is possible for a certain range of spine densities and electrical parameters. Propagation is precluded for spine stem resistance (R(ss)) either too large or too small. Moreover, even if R(ss) lies in a suitable range for the local generation of an action potential (resulting from local synaptic excitatory input), this range may not be 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. The success or failure of local excitation to spread as a chain reaction depends on the spatial distribution of spines. Impulse propagation is unlikely if the excitable spines are spaced too far apart. However, propagation may be recovered by redistributing the same number of equally spaced spines into clusters. The spread of excitation in a distal dendritic arbor is also influenced by the branching geometry. Input to one branch can initiate a chain reaction that accelerates into the sister branch but rapidly attenuates as it enters the parent branch. In branched dendrites with many excitable and passive spines, regions of decreased conductance load (e.g., near sealed ends) can facilitate attenuating waves and enhance waves that are successfully propagating. Regions of increased conductance load (e.g., near common branch points) promote attenuation and tend to block propagation. Nonuniform loading and/or nonuniform spine densities can lead to complex propagation characteristics. Some analytic results of classical cable theory are generalized for the case of a passive spiny dendritic cable. New expressions for electrotonic length and input resistance reflect the cumulative effect of having many passive spines, and they depend explicitly on spine density n̄ and individual spine electrical parameters. As n̄ increases, the electrotonic length increases proportionally to √n̄, whereas the input resistance decreases reciprocally with √n̄, i.e., higher spine density means more membrane available for current loss. If R(ss) is increased the electrotonic length decreases, and the input resistance increases. However, in the expected parameter range, where the spine head resistance R(sh) substantially exceeds R(ss), the effect of changing R(ss) is negligible. The initiation of a chain reaction by synaptic input in spines depends on whether the voltage-dependent channels are located in the spine heads or in the dendrite; direct activation of more synapses is needed when the channels are distributed along the dendritic shaft than when they are isolated in the heads. On the other hand, an established propagating wave is little influenced by the location of the channels. Also, the minimal number of synapses required for initiation and the wave's peak dendritic potential are much more sensitive to changes in R(ss) for the case of spine head channels than for dendritic channels. With the continuum theory, different morphologies and distributed physiological properties may be represented explicitly and compactly by just a few differential equations. The equations may be integrated with standard finite difference methods adapted to treat branching geometries. The theory is general so that idealized or complex kinetic models may also be adapted. These features mean that models are easily communicated, that computations can be readily reproduced by other investigators, and that spines can and should be taken into account where appropriate.
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