Here we present an approach based on known biochemical intermediates in the induction of plasticity. In this approach, a mechanistic model is constructed by converting known biological mechanisms to assumptions that are formulated mathematically. These assumptions constitute a model that can be simulated or analyzed under different conditions. The model is constrained by matching the output of the model to experimental results.
As an example we present the calcium-dependent plasticity model (CaDP) of (Shouval et al., 2002
). The CaDP model can explain several observed experimental nonlinearities and can be easily modified by adding components that may account for further experimental observations. Such a model can also be used to simulate various slice plasticity protocols (Shouval et al., 2002
; Cai et al., 2007
) and receptive field plasticity in vivo
(Yeung et al., 2004
; Yu et al., 2008
). Here we focus on STDP-style experiments that are hard to explain by linear superposition models.
The CaDP model is based on three key assumptions.
(1) Calcium elevation in spines determines the sign, magnitude and rate of synaptic plasticity. A moderate elevation in calcium results in LTD whereas a large elevation in calcium levels results in long-term potentiation (LTP) (Figure A, left). We also assume that the rate of plasticity is a monotonically increasing function of calcium, η (Figure A, middle).
Figure 3 The CaDP model can account for various forms of spike timing dependent plasticity. (A) The key functions controlling the CaDP model. Left: The Ω function controls the sign and magnitude of calcium-dependent synaptic plasticity, the gray shading (more ...)
The calcium assumption is based on experimental evidence (Cummings et al., 1996
; Yang et al., 1999
) and has been previously suggested in models of calcium-dependent kinase-phosphatase systems in postsynaptic spines (Lisman, 1989
). Mathematically it is described by the equation:
is the synaptic efficacy of synapse i
is the calcium concentration at synapse i
, and λ is a decay time constant. The functions Ω and η (Shouval et al., 2002
) determine the sign and rate of synaptic plasticity and are depicted in Figure A. Ω is a function of calcium concentration and is defined by two thresholds θd
(Figure A) that control the sign and magnitude of synaptic plasticity.
(2) The source of calcium is influx through NMDA receptors which pass calcium and are gated by both glutamate and voltage. NMDA receptors can therefore report the coincidence of presynaptically released glutamate and postsynaptic depolarization by allowing calcium into a dendritic spine. NMDA receptors are relatively slow-gating receptors, with time constants in the range of 50–200
ms, a scale comparable to time windows for timing-dependent plasticity.
(3) Back-propagating action potentials (BPAP) in the postsynaptic neuron leave a lingering post-action potential current in the dendrite. The BPAP is the source of depolarization. The assumption of a lingering tail is necessary in order to explain a time window for LTD when the postsynaptic spike precedes the presynaptic spike.
The results of this model depend on a variety of parameter assumptions. Although we will focus on accounting for CA3–CA1 plasticity rules, parameters can be adjusted to account for plasticity properties at other synapses.
Two timing windows for LTD
In Figure B we show induction of STDP with the CaDP model. The functions for Ω, η, and the voltage response of the back-propagating action potential are depicted in Figure A, and the NMDA receptor conductance for calcium ions (GNMDA
) is set at an appropriate value. These assumptions produce a three-peaked learning rule (Figure B): post-pre LTD, pre-post LTP, and pre-post LTD at larger values of Δt
. This second LTD window is seen at some synapses (Nishiyama et al., 2000
; Woodin et al., 2003
; Wittenberg and Wang, 2006
) whereas it is absent or less prominent in neocortical synapses examined to date.
If the NMDA conductance is reduced by 30%, single postsynaptic spikes no longer produce LTP at low pairing frequencies (Figure C). Now if a burst of two postsynaptic spikes or more is paired with each presynaptic spike, a three-peaked timing-dependent plasticity curve again results (Figure D). This rule resembles the triphasic rule that is possible at CA3–CA1 synapses (Nishiyama et al., 2000
; Wittenberg and Wang, 2006
). This is illustrated in the CaDP applet available at: http://nba.uth.tmc.edu/homepage/shouval/applets/v1/applet01.htm
. Other proposed mechanistic models also generate a second LTD window (Kitajima and Hara, 2000
; Abarbanel et al., 2002
; Karmarkar et al., 2002
Yet neocortical synapses have multiple mechanisms for LTD including metabotropic glutamate receptor or cannabinoid receptor-dependent signaling (van Rossum et al., 2000
; Sjöström et al., 2003
; Bender et al., 2006
) but lack a prominent second LTD window. Biochemical veto mechanisms have been proposed that can overrule the second LTD window in neocortical synapses (Rubin et al., 2005
) but allow it to be expressed at CA3–CA1 synapses. A difference could also be based on biological heterogeneity, for instance the relative abundance of calcium release in CA1 neurons compared with neocortical pyramidal neurons (Nakamura et al., 2000
). Finally, stochastic properties of synaptic transmission in conjunction with the CaDP model may significantly reduce the magnitude of the second LTD window (Shouval and Kalantzis, 2005
Frequency-dependence of LTP induction by postsynaptic spikes and bursts
In neocortical synapses, LTP results from single postsynaptic spikes at high pairing frequencies, but not at low pairing frequencies (Markram et al., 1997
; Sjöström et al., 2001
). At high enough frequencies LTD is eliminated entirely. This frequency-dependence is qualitatively consistent with results at CA3–CA1 synapses (Wittenberg and Wang, 2006
). Such a transition from bidirectionality to all-LTP falls naturally from the function Ω.
In this simple example we have not included the effects of short-term synaptic dynamics (Tsodyks et al., 1998
). In models, short-term facilitation and depression can alter the frequency-dependence of plasticity (Cai et al., 2007
) and may account for properties of the plasticity induced by multi-spike protocols (Froemke and Dan, 2002
; Wang et al., 2005