Editing Synaptic plasticity (section)

==Theoretical mechanisms==
A bidirectional model, describing both LTP and LTD, of synaptic plasticity has proved necessary for a number of different learning mechanisms in [[computational neuroscience]], [[neural networks (biology)|neural networks]], and [[biophysics]]. Three major hypotheses for the molecular nature of this plasticity have been well-studied, and none are required to be the exclusive mechanism:
# Change in the probability of glutamate release.
# Insertion or removal of post-synaptic AMPA receptors.
# [[Phosphorylation]] and de-phosphorylation inducing a change in AMPA receptor conductance.

Of these, the latter two hypotheses have been recently mathematically examined to have identical calcium-dependent dynamics which provides strong theoretical evidence for a calcium-based model of plasticity, which in a linear model where the total number of receptors are conserved looks like

:<math>\frac{d W_i(t)}{d t}=\frac{1}{\tau([Ca^{2+}]_i)}\left(\Omega([Ca^{2+}]_i)-W_i\right),</math>

where

* <math>W_i</math> is the [[synaptic weight]] of the <math>i</math>th input axon,
* <math>[Ca^{2+}]</math> is the concentration of calcium,
* <math>\tau</math> is a time constant dependent on the insertion and removal rates of neurotransmitter receptors, which is dependent on <math>[Ca^{2+}]</math>, and
* <math>\Omega=\beta A_m^{\rm fp}</math> is also a function of the concentration of calcium that depends linearly on the number of receptors on the membrane of the neuron at some fixed point.

Both <math>\Omega</math> and <math>\tau</math> are found experimentally and agree on results from both hypotheses. The model makes important simplifications that make it unsuited for actual experimental predictions, but provides a significant basis for the hypothesis of a calcium-based synaptic plasticity dependence.<ref>{{cite journal | vauthors = Shouval HZ, Castellani GC, Blais BS, Yeung LC, Cooper LN | title = Converging evidence for a simplified biophysical model of synaptic plasticity | journal = Biological Cybernetics | volume = 87 | issue = 5–6 | pages = 383–91 | date = December 2002 | pmid = 12461628 | doi = 10.1007/s00422-002-0362-x | s2cid = 7753630 | url = http://physics.brown.edu/physics/researchpages/Ibns/Lab%20Publications%20(PDF)/converging.pdf | author5-link = Leon Cooper }}</ref>