Dendritic spines are small evaginations of the dendritic surface of several functionally important classes of nerve cells. They have a general knoblike appearance of a bulbous spine head and tenuous stem. They are the principal input/output cells of a given brain region; in parts of the mammalian nervous system, the most common way a synapse is formed between two nerve cells is by the presynaptic component making contact with a dendritic spine. There is now overwhelming experimental evidence for excitable channels and for action potentials in dendrites, and new experimental studies conclude that calcium channels exist in the spine heads and action potentials invade spines.;Also examined is the possibility that the spine stem resistance, the tunable "synaptic weight" parameter identified by Rall and Rinzel, is activity-dependent. We assume that changes in the spine stem conductance ;The focus of the present study is on the threshold and dynamical properties of an excitable spine when the synaptic input is modeled as a steady current source. Two parameter Hopf bifurcation diagrams are constructed showing stability boundaries between oscillations and stable steady states. Investigated is how these boundaries change as a function of cable length, spine stem conductance, and dendritic input current.
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