RESUMEN

Dendritic spines are small, mushroom-like protrusions from the arbor of a neuron in the central nervous system. Interdependent changes in the morphology, biochemistry, and activity of spines have been associated with learning and memory. Moreover, post-mortem cortices from patients with Alzheimer's or Parkinson's disease exhibit biochemical and physical alterations within their dendritic arbors and a reduction in the number of dendritic spines. For over a decade, experimentalists have observed perforations in postsynaptic densities on dendritic spines after induction of long-term potentiation, a sustained enhancement of response to a brief electrical or chemical stimulus, associated with learning and memory. In more recent work, some suggest that activity-dependent intraspine calcium may regulate the surface area of the spine head, and reorganization of postsynaptic densities on the surface. In this paper, we develop a model of a dendritic spine with the ability to partition its transmission and receptor zones, as well as the entire spine head. Simulations are initially performed with fixed parameters for morphology to study electrical properties and identify parameters that increase efficacy of the synaptic connection. Equations are then introduced to incorporate calcium as a second messenger in regulating continuous changes in morphology. In the model, activity affects compartmental calcium, which regulates spine head morphology. Conversely, spine head morphology affects the level of local activity, whether the spines are modeled with passive membrane properties, or excitable membrane using Hodgkin-Huxley kinetics. Results indicate that merely separating the postsynaptic receptors on the surface of the spine may add to the diversity of circuitry, but does not change the efficacy of the synapse. However, when the surface area of the spine is a dynamic variable, efficacy of the synapse may vary continuously over time.

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RESUMEN

Populations of dendritic spines can change in number and shape quite rapidly as a result of synaptic activity. Here, we explore the consequences of such changes on the input-output properties of a dendritic branch. We consider two models: one for activity-dependent spine densities and the other for calcium-mediated spine-stem restructuring. In the activity-dependent density model we find that for repetitive synaptic input to passive spines, changes in spine density remain local to the input site. For excitable spines, the spine density increases both inside and outside the input region. When the spine stem resistances are relatively high, the transition to higher dendritic output is abrupt; when low, the rate of increase is gradual and resembles long-term potentiation. In the second model, spine density is held constant, but the stem dimensions are allowed to change as a result of stimulation-induced calcium influxes. The model is formulated so that a moderate amount of synaptic activation results in spine stem elongation, whereas high levels of activation result in stem shortening. Under these conditions, passive spines receiving modest stimulation progressively increase their spine stem resistance and head potentials, but little change occurs in the dendritic output. For excitable spines, modest stimulation frequencies cause a lengthening of both stimulated and neighboring spines and the stimulus eventually propagates. High-frequency stimulation that causes spines to shorten in the stimulated region decreases the amplitude of the dendritic output slightly or drastically, depending on initial spine densities and stem resistances.

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RESUMEN

The spread of electrical activity in a dendritic tree is shaped, in part, by its morphology. Conversely, experimental evidence is growing that electrical and chemical activity can slowly shape the morphology of the dendrite. In this theoretical study, the dendritic spines are dynamic elements, with biophysical properties that change in response to patterns of electrical activity. Recent experiments and diagrammatic models suggest that activity-dependent processes can regulate structural modifications in dendritic spines as well as their distribution along the dendrite. This study considers how local changes in spine structure (minutes to hours) can influence patterns of electrical activity along the dendrite; and how electrical activity due to synaptic events and excitable membrane dynamics can, over time, influence the morphology of the dendrite. The model presents a slow subsystem for structural synaptic plasticity associated with long-term potentiation. A perturbation problem evolves naturally when the spine stem shortens, since the ratio of spine stem resistance to input resistance is small. Hence, the difference between the spine head and dendritic potentials become negligible. This paper presents an asymptotic expansion of head potential in terms of dendritic potential. This leads to a reduced model for post-synaptic restructuring that captures the dynamics of the full model in a briefer computation period when the spines are well connected to the dendrite.

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RESUMEN

We develop a one-dimensional physical model of the crawling movement of simple cells: The sperm of a nematode, Ascaris suum. The model is based on the assumptions that polymerization and bundling of the cytoskeletal filaments generate the force for extension at the front, and that energy stored in the gel formed from the filament bundles is subsequently used to produce the contraction that pulls the rear of the cell forward. The model combines the mechanics of protrusion and contraction with chemical control, and shows how their coupling generates stable rapid migration, so that the cell length and velocity regulate to constant values.