Regimes of validity and physiological grounding of the calcium-control-hypothesis (CaDP) framework of synaptic plasticity: parameters, rate-rule averaging, and the predicted pre-before-post LTD window

The calcium control hypothesis — that prolonged moderate postsynaptic calcium elevations induce long-term depression (LTD) while large elevations induce long-term potentiation (LTP) — was given a compact dynamical embodiment by Shouval, Castellani, Blais, Yeung and Cooper (Biological Cybernetics, 2002): two mechanistically distinct models acting at AMPA receptors (the receptors that carry most fast excitatory synaptic transmission) were shown, under explicit approximations, to reduce to the same calcium-dependent plasticity (CaDP) equation, which is converted, under Poisson and time-averaging assumptions, into a multi-synapse BCM-like rate rule whose modification threshold is made metaplastic through slow dynamics of the NMDA-receptor chord conductance (the effective conductance of the NMDA-receptor calcium current). The authors explicitly flagged as unresolved: the physiological determination of the model's parameters and absolute calcium scales; the effect of finite-time fluctuations and non-Poisson spike statistics on the validity of the averaged rate rule; whether the full spike-time-dependent system with metaplasticity develops stable, selective receptive fields; and whether the model's distinctive prediction — an LTD window for presynaptic-before-postsynaptic spike timing — is robust across experimental conditions. Later work, chiefly by the same group, advanced the stability/selectivity component (Yeung et al., PNAS 2004, and network demonstrations) and the stochastic and experimental sides; the physiological parameterization, the validity domain of the rate reduction under realistic spike statistics, whether the full spike-time-dependent system with the proposed metaplasticity itself develops stable, selective receptive fields, and the robustness of pre-before-post LTD — now tied to the question of spike-timing-dependent plasticity (STDP) at physiological calcium levels — remain open and actively pursued.

Background

At excitatory synapses in the mammalian brain, repetitive co-activation of a presynaptic axon and a postsynaptic neuron can persistently strengthen or weaken the synapse (long-term potentiation, LTP, and long-term depression, LTD). Induction of many forms of such bidirectional plasticity requires calcium entry through NMDA receptors — glutamate-gated channels that open only when presynaptic glutamate release coincides with postsynaptic depolarization, and which therefore act as coincidence detectors. Spike-timing-dependent plasticity (STDP) protocols, in which single pre- and postsynaptic action potentials are repeatedly paired at a fixed interval $\Delta t$, typically induce LTP when the presynaptic spike precedes the postsynaptic one ($\Delta t>0$) and LTD when it follows ($\Delta t<0$).

The calcium control hypothesis, proposed by Lisman (1989), posits that the direction of plasticity is set by the amplitude of postsynaptic calcium elevation: moderate, prolonged elevations preferentially activate protein phosphatases (enzymes that remove phosphate groups from target proteins) and produce LTD, while large elevations activate protein kinases (enzymes that attach phosphate groups to target proteins), notably Ca$^{2+}$/calmodulin-dependent protein kinase II (CaMKII, a kinase switched on by binding the calcium-sensing protein calmodulin), and produce LTP. The BCM (Bienenstock–Cooper–Munro) theory is an abstract rate-based learning rule in which a postsynaptically computed modification threshold slides with the neuron's average activity, endowing Hebbian learning with stability, competition, and input selectivity — the emergence of a receptive field, meaning selective responsiveness of a neuron to a particular subset of its inputs (reviewed in Cooper and Bear, 2012); metaplasticity denotes such activity-dependent adjustment of the plasticity machinery itself — 'the plasticity of plasticity'.

Shouval, Castellani, Blais, Yeung and Cooper (2002) asked whether two mechanistically distinct molecular hypotheses for LTP/LTD expression — a phenomenological model of activity-dependent insertion and removal of AMPA receptors (the receptors that carry most fast excitatory postsynaptic current) and a detailed biophysical model of AMPA-receptor phosphorylation by kinases and phosphatases, built on Castellani, Quinlan, Cooper and Shouval (2001) — reduce to the same compact induction rule. Under explicit approximations both map onto a single-synapse calcium-dependent plasticity (CaDP) equation,

$$\frac{dW_i}{dt}=\eta\bigl([Ca]_i\bigr)\Bigl(\Omega\bigl([Ca]_i\bigr)-\lambda W_i\Bigr),$$

where $W_i$ is the synaptic weight of synapse $i$, $[Ca]_i$ is the local postsynaptic calcium concentration in the dendritic spine (the small biochemical compartment forming the postsynaptic side of most excitatory synapses), $\Omega$ is a calcium-control function (no change at baseline calcium, LTD at moderate calcium, LTP at high calcium), $\eta$ is a monotonically increasing calcium-dependent learning rate (equivalently written as $1/\tau([Ca])$), and $\lambda$ sets weight decay. In companion work the same group showed that this equation, driven by calcium entry through a model NMDA receptor together with a back-propagating action potential (the dendritic regenerative event carrying the somatic spike into the dendrite) that possesses a slow depolarizing tail, accounts for pairing-, frequency-, and spike-timing-based induction protocols, and predicts an additional LTD region at pre-before-post timings beyond the LTP window (Shouval, Bear and Cooper, 2002).

The 2002 paper then converts the single-synapse rule into a multi-synapse rate-based rule. Assuming that NMDA receptors are the dominant calcium source (calcium current proportional to the NMDA-receptor chord conductance $G$ — the effective conductance of the NMDA-receptor current, averaged over the voltage trajectory — the presynaptic release rate $x_i$, and a voltage factor $B(V)$), that spine calcium follows presynaptic activity with decay time constant $\tau_{Ca}$ at quasi-steady state (calcium equilibrating much faster than the synaptic weight changes), that presynaptic spikes are Poisson, that firing rates vary slowly, and that fluctuations are small relative to their means, the averaged calcium at each synapse is proportional to $\tau_{Ca}\,G\,x_i\,y$ with $y$ the postsynaptic rate; with a quadratic ansatz for $\Omega$ the averaged rule takes a BCM-like form with modification threshold

$$\theta_M=\frac{\theta_0}{G\,\tau_{Ca}},$$

where $\theta_0$ is a constant of the ansatz. Making the conductance $G$ itself a slowly varying, activity-dependent variable then supplies a concrete NMDA-receptor-based mechanism of metaplasticity — a sliding modification threshold with a molecular substrate.

The authors state that the numerical parameter values, including the fitted $\tau(Ca)$, are illustrative rather than physiologically definitive, since many kinetic constants and absolute calcium scales are unknown; that the impact of finite-time fluctuations and non-Poisson spike statistics on the validity of the averaged rule has not been analyzed; that whether the full spike-time-dependent system with the proposed metaplasticity develops stable, selective receptive fields comparable to BCM-like models was under examination but not demonstrated; that the predicted second LTD region for pre-before-post timing had been observed in only limited experimental settings, so its robustness across experimental conditions was unconfirmed; and that whether the full set of equations (induction rule, NMDA-conductance metaplastic dynamics, and network interactions) yields stable, biologically plausible receptive-field development remained to be shown. Together these issues delimit the regimes in which the framework and its derived rate rules can be confidently applied.

Problem Statement

Determine the physiological validity and the regimes of applicability of the calcium-control-hypothesis framework — the calcium-dependent plasticity (CaDP) model derived in Shouval, Castellani, Blais, Yeung and Cooper (2002) — by resolving the four components the source leaves open. The framework rests on the single-synapse induction equation

$$\frac{dW_i}{dt}=\eta\bigl([Ca]_i\bigr)\Bigl(\Omega\bigl([Ca]_i\bigr)-\lambda W_i\Bigr),$$

where $W_i$ is the synaptic weight of synapse $i$; $[Ca]_i$ is the local postsynaptic spine calcium concentration; $\Omega$ is the calcium-control function, negative (driving long-term depression, LTD) at moderate calcium and positive (driving long-term potentiation, LTP) at high calcium; $\eta$ is a monotonically increasing calcium-dependent learning rate, equivalently written $1/\tau([Ca])$; and $\lambda$ sets weight decay. Calcium is sourced chiefly through NMDA receptors (glutamate-gated channels that require coincident presynaptic release and postsynaptic depolarization), whose effective chord conductance $G$ is itself slowly modulated by prior activity, providing metaplasticity — activity-dependent adjustment of the plasticity machinery.

1. Parameters: determine the calcium-control functions $\Omega$ and $\eta$ (equivalently $\tau(Ca)$), the calcium thresholds $\theta_D$ and $\theta_P$ separating the no-change, LTD, and LTP regimes of $\Omega$, and the absolute intracellular calcium scales of the model from physiological measurements, replacing the illustrative fitted values used in the source.

2. Averaging validity: establish the domain of validity of the averaged multi-synapse rate rule. The source's derivation from the full spike-time-dependent CaDP system assumed Poisson presynaptic statistics, quasi-steady-state spine calcium (calcium equilibrating much faster than weights change), slowly varying firing rates, and fluctuations small relative to their means; it yields averaged calcium proportional to $\tau_{Ca}\,G\,x_i\,y$, with $\tau_{Ca}$ the spine-calcium decay time constant, $G$ the NMDA-receptor chord conductance, $x_i$ the presynaptic release rate at synapse $i$, and $y$ the postsynaptic firing rate, and a BCM-like (Bienenstock–Cooper–Munro-like) rule with modification threshold $\theta_M=\theta_0/(G\tau_{Ca})$, where $\theta_0$ is a constant of the quadratic ansatz for $\Omega$. Quantify when this reduction fails under each of these violations of its assumptions — finite-time fluctuations; non-Poisson presynaptic spike statistics (renewal — spike intervals drawn independently from a common non-exponential interval distribution — correlated, or bursty); and rapidly varying rates — and characterize the averaged dynamics (including $\theta_M$) in those regimes, stating in each case whether the averaged rule and its modification threshold $\theta_M$ remain valid or break down.

3. Full-system receptive fields: demonstrate — or refute — that the full spike-time-dependent CaDP system with the proposed $G$-based metaplasticity develops stable, selective receptive fields (selective responsiveness of a neuron to a subset of its inputs) comparable to BCM-like models, both in single neurons with many synapses and in networks.

4. Experimental prediction: establish experimentally whether the predicted LTD window for pre-before-postsynaptic spike timing (the second LTD region at positive $\Delta t$, where $\Delta t$ denotes the interval from the presynaptic spike to the postsynaptic spike in a paired-spike protocol) is a robust feature of NMDA-dependent plasticity across experimental conditions (e.g. preparations, extracellular calcium levels, pairing frequencies and burst structure) or an artifact of particular conditions.

The four components are distinct open issues flagged for the same framework in the source record; together they bear on delimiting the regimes in which the reductions and derived rate rules of the calcium-control framework can be confidently applied, and on assessing its distinctive experimental predictions. They are scientifically linked — the same calcium-control functions and NMDA-calcium model determine the rate rule, the metaplastic threshold, and the spike-timing predictions — and progress on one component does not close the others.

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