The calcium-dependent plasticity (CaDP) model of Yeung, Shouval, Blais and Cooper (PNAS, 2004) shows that a fast calcium-driven Hebbian learning rule, combined with a slow homeostatic regulation of postsynaptic calcium, jointly produces stable synaptic weights and input selectivity in a simulated spiking neuron. In that model, calcium enters only through N-methyl-D-aspartate receptors (NMDARs) and stabilization is implemented as regulation of NMDAR permeability, even though the authors acknowledged that homeostatic plasticity is still observed when NMDARs are blocked and that calcium can enter through several parallel pathways. The open problem is the multi-source instantiation of this framework: extend the model so that postsynaptic calcium derives from parallel entry pathways (NMDARs, voltage-gated calcium channels, and intracellular stores) with homeostatic regulation acting on one or more of them, and determine (i) whether synaptic stability and correlation-based input selectivity, including the symmetry breaking between equivalent input channels, are preserved, lost, or qualitatively altered; (ii) which calcium entry pathway dominates the homeostatic signal under which activity conditions, consistent with the partially conflicting experimental record; and (iii) how the model's predicted dendritic calcium dynamics compare with direct measurements during homeostatic plasticity. Substantial adjacent experimental and modeling progress exists, but the coupled question remains open.

Background

Activity-dependent synaptic plasticity - long-term potentiation (LTP) and long-term depression (LTD), the persistent strengthening and weakening of synaptic connections - is widely held to underlie learning, memory, and the development of cortical circuits. A large body of experimental work shows that the direction and magnitude of synaptic change depend on the postsynaptic calcium concentration: near-basal calcium produces no change, moderate elevations induce depression, and larger elevations induce potentiation. Shouval, Bear and Cooper (2002) turned this into an explicit calcium-dependent plasticity (CaDP) rule in which the calcium signal is carried by NMDA receptors (NMDARs), glutamate-gated channels whose calcium permeability is blocked at rest and increases with depolarization, so that calcium enters most effectively when presynaptic glutamate release coincides with postsynaptic depolarization - the biophysical basis of Hebbian association (rules that strengthen synapses when pre- and postsynaptic activity coincide). Pure Hebbian rules of this kind are unstable: potentiation raises excitability, which favors further potentiation, so synaptic strengths run away to saturation. Classical remedies include the Bienenstock-Cooper-Munro (BCM) rule, in which a slowly sliding modification threshold makes further potentiation harder as activity rises, and the experimentally observed phenomenon of synaptic scaling, the multiplicative up- or down-regulation of a neuron's synaptic strengths after chronic decreases or increases in activity; activity-dependent adjustment of the plasticity thresholds themselves is called metaplasticity.

Yeung, Shouval, Blais and Cooper (2004) proposed a unified mechanism for these phenomena in a spiking point-neuron simulation (an integrate-and-fire model, in which the membrane potential sums input currents and emits a stereotyped spike upon crossing threshold) with 100 excitatory and 20 inhibitory synapses. The fast CaDP learning rule updates each synapse's weight according to a U-shaped function Ω([Ca])\Omega([\mathrm{Ca}]) of locally integrated postsynaptic calcium (no change at basal calcium, depression at moderate, potentiation at high levels), and this rule was stabilized by a slow homeostatic regulation of the calcium influx itself, implemented as an activity-dependent regulation of NMDAR permeability driven by the time-integrated membrane voltage. Simulations showed stable, finite synaptic weights with scaling-like fixed points, BCM-like metaplasticity, and synaptic competition: when subsets of inputs carried correlated spike trains or patterned rates, one channel was selectively potentiated while the others were depressed, and when two statistically equivalent correlated channels competed, the winner differed randomly from run to run - spontaneous symmetry breaking (the dynamics itself selects one of two equivalent alternatives). The authors also predicted that transient LTP-inducing stimulation should elevate dendritic calcium, whereas sustained stimulation should bring calcium back toward baseline.

The stabilization mechanism, however, was chosen 'for convenience and clarity': calcium enters only through NMDARs. The authors themselves noted that the experimental evidence for an NMDAR-specific role in stabilization was conflicting - Leslie, Nelson and Turrigiano (2001) had shown that homeostatic scaling is still detectable when NMDARs are pharmacologically blocked - and that calcium can be recruited through numerous cellular pathways acting in parallel, including voltage-gated calcium channels (membrane channels that admit calcium in response to depolarization) and release from intracellular calcium stores (organelles such as the endoplasmic reticulum that sequester calcium and release it back into the cytoplasm on demand). They argued that their results depend only on calcium being the signal for both fast and homeostatic plasticity, and asserted that additional calcium sources 'can be readily incorporated' into the model; this incorporation, however, was never carried out. Which calcium entry pathways actually carry the homeostatic signal in different preparations, and whether the joint emergence of synaptic stability and input selectivity survives a biophysically realistic multi-source instantiation of the framework, is the unresolved question.

Problem Statement

Within the CaDP framework - fast synaptic weight updates driven by a U-shaped function of locally integrated postsynaptic calcium, stabilized by a slow homeostatic regulation of calcium influx - construct and analyze the multi-source instantiation in which postsynaptic calcium is the sum of several parallel entry pathways (NMDA-receptor-mediated influx, voltage-gated calcium channels, and calcium release from or entry into intracellular stores), with the slow homeostatic regulation acting on one or more of these pathways rather than only on NMDAR permeability. The common objective is to establish whether the unified account of synaptic homeostasis and input selectivity survives biologically realistic calcium entry, with three jointly supporting questions: (i) Preservation of stability and selectivity - determine whether bounded synaptic-weight dynamics with scaling-like fixed points and metaplasticity, and the selective potentiation of correlated or patterned input channels with the observed symmetry breaking between equivalent channels, are preserved, lost, or qualitatively changed as the number, kinetics, and relative weights of calcium sources and their regulation sites vary; (ii) Pathway dominance - determine which calcium entry pathway carries the homeostatic error signal under which activity conditions (input-rate regime, correlation structure, chronic inactivity or hyperactivity), and reconcile the result with the experimental record that scaling is NMDAR-independent in some preparations, dependent on NMDAR calcium permeability in others, and sensed by distinct calcium-dependent enzymes for excitatory versus inhibitory synapses; (iii) Model-experiment mapping - derive the predicted dendritic calcium dynamics (transient elevation under potentiation-inducing stimulation, returning toward baseline under sustained stimulation) in the multi-source model and specify how they compare with direct measurements of dendritic calcium during homeostatic plasticity.

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