AMPA receptors mediate the majority of fast excitatory synaptic transmission in the CNS, and evidence suggests that AMPA receptor trafficking regulates synaptic strength, a phenomenon implicated in learning and memory. There are two major mechanisms of AMPA receptor trafficking: exocytic/endocytic exchange of surface receptors with intracellular receptor pools, and the lateral diffusion or hopping of surface receptors between the postsynaptic density and the surrounding extrasynaptic membrane. In this paper, we present a biophysical model of these trafficking mechanisms under basal conditions and during the expression of long-term potentiation (LTP) and depression (LTD). We show how our model reproduces a wide range of physiological data, and use this to make predictions regarding possible targets of second-messenger pathways activated during the induction phase of LTP/LTD.

We present a model of protein receptor trafficking within the membrane of a cylindrical dendrite containing small protrusions called spines. Spines are the locus of most excitatory synapses in the central nervous system and act as localized traps for receptors diffusing within the dendritic membrane. We treat the transverse intersection of a spine and dendrite as a spatially extended, partially absorbing boundary and use singular perturbation theory to analyze the steadystate distribution of receptors. We compare the singular perturbation solutions with numerical solutions of the full model and with solutions of a reduced one-dimensional model and find good agreement between them all. We also derive a system of Fokker-Planck equations from our model and use it to exactly solve a mean first passage time (MFPT) problem for a single receptor traveling a fixed axial distance along the dendrite. This is then used to calculate an effective diffusion coefficient for receptors when spines are uniformly distributed along the length of the cable and to show how a nonuniform distribution of spines gives rise to anomalous subdiffusion. Introduction.Neurons are amongst the largest and most complex cells in biology. Their intricate geometry presents many challenges for cell function, in particular with regard to the efficient delivery of newly synthesized proteins from the cell body or soma to distant locations on the axon or dendrites. The axon contains ion channels for action potential propagation and presynaptic active zones for neurotransmitter release, whereas each dendrite contains postsynaptic domains (or densities) where receptors that bind neurotransmitter tend to cluster. At most excitatory synapses in the brain, the postsynaptic density is located within a dendritic spine, which is a small, submicrometer membranous extrusion that protrudes from a dendrite. Typically spines have a bulbous head which is connected to the parent dendrite through a thin spine neck. Given that hundreds or thousands of synapses and their associated spines are distributed along the entire length of a dendrite, it follows that neurons must traffic receptors and other postsynaptic proteins over long distances (several 100μm) from the soma. This can occur by two distinct mechanisms: either by lateral diffusion in the plasma membrane [8,26,1,7] or by motor-driven intracellular transport along microtubules followed by local insertion into the surface membrane (exocytosis) [17,20,29]. It is likely that both forms of transport occur in dendrites, depending on the type of receptor and the developmental stage of the organism.

We present a model for the diffusive trafficking of protein receptors along the surface of a neuron's dendrite. Distributed along the dendrite are spatially localized trapping regions that represent submicrometer mushroom-like protrusions known as dendritic spines. Within these trapping regions receptors can be internalized via endocytosis and either reinserted into the surface via exocytosis or degraded. We calculate the steady-state distribution of receptors along the dendrite assuming a constant flux of receptors inserted at one end, adjacent to the soma where receptors are synthesized, and use this to investigate the effectiveness of membrane diffusion as a transport mechanism. We also calculate the mean first passage time of a receptor to travel a certain distance along the cable and use this to derive an effective surface diffusivity.

AMPA receptor trafficking in dendritic spines is emerging as a major postsynaptic mechanism for the expression of plasticity at glutamatergic synapses. AMPA receptors within a spine are in a continuous state of flux, being exchanged with local intracellular pools via exo/endocytosis and with the surrounding dendrite via lateral membrane diffusion. This suggests that one cannot treat a single spine in isolation. Here we present a model of AMPA receptor trafficking between multiple dendritic spines distributed along the surface of a dendrite. Receptors undergo lateral diffusion within the dendritic membrane, with each spine acting as a spatially localized trap where receptors can bind to scaffolding proteins or be internalized through endocytosis. Exocytosis of receptors occurs either at the soma or at sites local to dendritic spines via constitutive recycling from intracellular pools. We derive a reaction-diffusion equation for receptor trafficking that takes into account these various processes. Solutions of this equation allow us to calculate the distribution of synaptic receptor numbers across the population of spines, and hence determine how lateral diffusion contributes to the strength of a synapse. A number of specific results follow from our modeling and analysis. (1) Lateral membrane diffusion alone is insufficient as a mechanism for delivering AMPA receptors from the soma to distal dendrites. (2) A source of surface receptors at the soma tends to generate an exponential-like distribution of receptors along the dendrite, which has implications for synaptic democracy. (3) Diffusion mediates a heterosynaptic interaction between spines so that local changes in the constitutive recycling of AMPA receptors induce nonlocal changes in synaptic strength. On the other hand, structural changes in a spine following long term potentiation or depression have a purely local effect on synaptic strength. (4) A global change in the rates of AMPA receptor exo/endocytosis is unlikely to be the sole mechanism for homeostatic synaptic scaling. (5) The dynamics of AMPA receptor trafficking occurs on multiple timescales and varies according to spatial location along the dendrite. Understanding such dynamics is important when interpreting data from inactivation experiments that are used to infer the rate of relaxation to steady-state.

Ca2+-calmodulin-dependent protein kinase II (CaMKII) is a key regulator of glutamatergic synapses and plays an essential role in many forms of synaptic plasticity. It has recently been observed that stimulating dendrites locally with a single glutamate/glycine puff induces a local translocation of CaMKII into spines that subsequently spreads in a wave-like manner towards the distal dendritic arbor. Here we present a mathematical model of the diffusion, activation and translocation of dendritic CaMKII. We show how the nonlinear dynamics of CaMKII diffusion-activation generates a propagating translocation wave, provided that the rate of activation is sufficiently fast. We also derive an explicit formula for the wave speed as a function of physiological parameters such as the diffusivity of CaMKII and the density of spines. Our model provides a quantitative framework for understanding the spread of CaMKII translocation and its possible role in heterosynaptic plasticity.