Spike-timing-dependent Plasticity Curve Research Articles

A new pre-conditioned STDP rule and its hardware implementation in neuromorphic crossbar array

This paper proposes a new pre-conditioned spike-timing-dependent plasticity (STDP) learning rule as well as an efficient system-level time-domain circuit modeling and simulation method for the whole crossbar array structure. First, the new pre-conditioned STDP rule is conceived by exercising a slight perturbation to the conventional STDP curve. Software-wise, it can be described in spiking neural network (SNN) codes, while hardware-wise, it is realized by the memristor crossbar array in conjunction with specific spike signals, just as the conventional STDP. The new STDP rule enables the neural network in a neuromorphic chip to achieve better performance both in terms of software and hardware implementation compared with the conventional STDP rule. Second, a time-domain circuit modeling method is proposed for the simulation of the whole crossbar array. Numerical experiments demonstrate that the new pre-conditioned STDP is superior to the conventional STDP both in terms of accuracy and low power consumption. At the software level, a 784 × 100 spiking neural network, trained by the pre-conditioned STDP, for the MNIST handwritten digit recognition achieves an accuracy of 85.90%, which is 3.09% higher than that trained by the conventional STDP. At the hardware level, a downscaled 196 × 10 crossbar array is simulated and trained. The crossbar array trained by the new STDP consumes 21.5% less power than that trained by the conventional STDP. Moreover, it does not compromise the original robustness when subject to non-ideal characteristics such as variation of conductance and resistance of the interconnect in the crossbar array.

Energy- and Area-Efficient CMOS Synapse and Neuron for Spiking Neural Networks With STDP Learning

This paper proposes CMOS synapse and neuron for use in spiking neural networks to perform cognitive functions in a bio-inspired manner. The proposed synapse can trace the eligibility of the timing relationship between pre- and post-synaptic spikes, supporting a bio-plausible local learning rule called the spike timing-dependent plasticity (STDP) in an energy- and area-efficient manner. The proposed neuron can support neural functions such as synaptic current integration, threshold-based firing, neuronal leaking, membrane potential resetting, and adjustable refractory period with improved energy and area efficiency. The STDP curve shape of the synapse and the firing rate of the neuron can be adjusted as desired. Their variability due to process, voltage, and temperature (PVT) variations can also be minimized. The proposed CMOS neuron and synapse circuits were designed in a 28-nm CMOS process. The performance evaluation results indicate that the proposed synapse reduces energy consumption and area by up to 94% and 43% compared to conventional CMOS synapses. They also indicate that the proposed neuron achieves energy and area reductions of 37% and 23%, respectively, compared to conventional CMOS neurons. An associative neural network composed of the proposed neuron and synapse was designed to verify that they together work well for performing a cognitive function of associative learning and inferencing.

Engineering synaptic plasticity through the control of oxygen vacancy concentration for the improvement of learning accuracy in a Ta2O5 memristor

Ta2O5 memristors exhibit bipolar switching properties attributable to the growth and destruction of oxygen vacancy filaments (OVFs). The transmission properties of biological synapse are mimicked in these memristors. The Ta2O5 memristor that contains numerous oxygen vacancies (OVs) is heated under N2 at 10 Torr, and it shows high conductance modulation linearity (CML) because the variation of OVF is governed by the redox reaction. The recognition accuracy of artificial neural networks (ANNs) is affected significantly by the CML of the memristor. Simulation using a convolutional neural network reveals that this Ta2O5 memristor exhibits a high learning accuracy of 93% because of its high CML. Spike-timing-dependent plasticity (STDP) was realized in Ta2O5 memristors. The change rate of synaptic weight variation in the STDP curve, which is also related to the learning accuracy of ANNs, is large in the Ta2O5 memristor heated under N2 at 10 Torr; this confirms that this memristor has a good learning accuracy. Spike rate-dependent plasticity and the transition from short-term plasticity to long-term plasticity are observed in Ta2O5 memristors. Further, they were obtained at a small potentiation spike in a Ta2O5 memristor heated under N2 at 10 Torr because numerous OVs exist in this memristor.

SPICE Study of STDP Characteristics in a Drift and Diffusive Memristor-Based Synapse for Neuromorphic Computing

Neuromorphic hardware is a system with massive potential to enable efficient computing by mimicking the human brain. The novel system processes information using neuron spikes (Action Potentials) and the synaptic connections between neurons are trained using biologically plausible methods like spike-timing-dependent plasticity (STDP). Memristor is one of the promising candidates to implement such neuromorphic hardware. Two types of memristors, diffusive and drift, have been proposed to form a synapse showing faithful emulation of STDP, where the diffusion effect is used to trace the spike timing history crucial for STDP and the drift memristor keeps the weight information in a longer time scale. The purpose of this paper is to systematically investigate STDP characteristics in such a synapse with serially connected two memristors using SPICE models. The results show that STDP properties are strongly dependent on device parameters and even the shape of STDP curves is modified. Different shapes of the STDP curve were identified. The results and analysis could support the design of emerging device-based synapses, which can faithfully mimic biological STDP characteristics for future neuromorphic systems.

Photonic implementation of spike timing dependent plasticity with weight-dependent learning window based on VCSOA

Based on the well-known Fabry–Pérot approach, after taking into account the variation of bias current of the vertical-cavity semiconductor optical amplifier (VCSOA) according to the present synapse weight, we implement the optical spike timing dependent plasticity (STDP) with weight-dependent learning window in a VCSOA with double optical spike injections, and numerically investigate the corresponding weight-dependent STDP characteristics. The simulation results show that, the bias current of VCSOA has significant effect on the optical STDP curve. After introducing an adaptive variation of the bias current according to the present synapse weight, the optical weight-dependent STDP based on VCSOA can be realized. Moreover, the weight training based on the optical weight-dependent STDP can be effectively controlled by adjusting some typical external or intrinsic parameters and the excessive adjusting of synaptic weight is avoided, which can be used to balance the stability and competition among synapses and pave a way for the future large-scale energy efficient optical spiking neural networks based on the weight-dependent STDP learning mechanism.

Theoretical investigation of weight-dependent optical spike timing dependent plasticity based on VCSOA

We propose an implement scheme of the weight-dependent optical spike timing dependent plasticity (STDP) based on the vertical-cavity semiconductor optical amplifiers (VCSOAs) subject to double optical spike injections, and numerically investigate the effects of the internal and external parameters of VCSOAs on the optical STDP curve by the well-known Fabry-Pérot approach. Additionally, the optical weight-dependent STDP is realized by introducing the various bias current according to the feedback signal. The simulation results show that, the width and height of the optical STDP curve window can be effectively controlled by adjusting the relevant parameters of VCSOAs. Moreover, the optical weight-dependent STDP is analogous to the biological STDP synapses and can be used to balance the stability and competition among synapses. These results can offer great potential for constructing a large-scale energy efficient optical spiking neural networks (SNNs).

Real-time optical spike-timing dependent plasticity in a single VCSEL with dual-polarized pulsed optical injection

We propose and numerically realize an optical spike-timing dependent plasticity (STDP) scheme by using a single vertical-cavity surface-emitting laser (VCSEL). In the scheme, the VCSEL is subjected to an orthogonally-polarized continuous-wave optical injection (OPCWOI) and dual-polarized pulsed optical injections (DPPOI). Based on the widely used spin-flip model, the response spiking dynamics of VCSEL is numerically studied, and then the optical STDP in a single VCSEL is explored. The roles of bias current, the strength of OPCWOI and DPPOI, and the frequency detuning on the optical STDP curve are numerically analyzed. It is found that, by simultaneously utilizing the response spiking dynamics in two orthogonal polarization modes, an optical STDP could be achieved by using a single VCSEL. Furthermore, the weight update of STDP curve can be calculated in real-time. Additionally, the STDP curves can also be controlled by adjusting some controllable parameters. The real-time optical STDP based on a single VCSEL is numerically realized for the first time, which paves the way towards fully VCSELs-based photonic neuromorphic systems with low power consumption.

A calcium-influx-dependent plasticity model exhibiting multiple STDP curves.

Hebbian plasticity means that if the firing of two neurons is correlated, then their connection is strengthened. Conversely, uncorrelated firing causes a decrease in synaptic strength. Spike-timing-dependent plasticity (STDP) represents one instantiation of Hebbian plasticity. Under STDP, synaptic changes depend on the relative timing of the pre- and post-synaptic firing. By inducing pre- and post-synaptic firing at different relative times the STDP curves of many neurons have been determined, and it has been found that there are different curves for different neuron types or synaptic sites. Biophysically, strengthening (long-term potentiation, LTP) or weakening (long-term depression, LTD) of glutamatergic synapses depends on the post-synaptic influx of calcium (Ca2+): weak influx leads to LTD, while strong, transient influx causes LTP. The voltage-dependent NMDA receptors are the main source of Ca2+ influx, but they will only open if a post-synaptic depolarisation coincides with pre-synaptic neurotransmitter release. Here we present a computational mechanism for Ca2+-dependent plasticity in which the interplay between the pre-synaptic neurotransmitter release and the post-synaptic membrane potential leads to distinct Ca2+ time-courses, which in turn lead to the change in synaptic strength. It is shown that the model complies with classic STDP results, as well as with results obtained with triplets of spikes. Furthermore, the model is capable of displaying different shapes of STDP curves, as observed in different experimental studies.

Spike-shape dependence of the spike-timing dependent synaptic plasticity in ferroelectric-tunnel-junction synapses

Resistive switching (RS) devices have attracted increasing attention for artificial synapse applications in neural networks because of their nonvolatile and analogue resistance changes. Among the neural networks, a spiking neural network (SNN) based on spike-timing-dependent plasticity (STDP) is highly energy efficient. To implement STDP in resistive switching devices, several types of voltage spikes have been proposed to date, but there have been few reports on the relationship between the STDP characteristics and spike types. Here, we report the STDP characteristics implemented in ferroelectric tunnel junctions (FTJs) by several types of spikes. Based on simulated time evolutions of superimposed spikes and taking the nonlinear current-voltage (I-V) characteristics of FTJs into account, we propose equations for simulating the STDP curve parameters of a magnitude of the conductance change (ΔGmax) and a time window (τC) from the spike parameters of a peak amplitude (Vpeak) and time durations (tp and td) for three spike types: triangle-triangle, rectangular-triangle, and rectangular-rectangular. The power consumption experiments of the STDP revealed that the power consumption under the inactive-synapse condition (spike timing |Δt| > τC) was as large as 50–82% of that under the active-synapse condition (|Δt| < τC). This finding indicates that the power consumption under the inactive-synapse condition should be reduced to minimize the total power consumption of an SNN implemented by using FTJs as synapses.

Numerical Implementation of Wavelength-Dependent Photonic Spike Timing Dependent Plasticity Based on VCSOA

We propose to realize photonic spike timing dependent plasticity (STDP) by using a vertical-cavity semiconductor optical amplifier (VCSOA) subject to dual optical pulse injections. The computational model of the photonic STDP is presented for the first time based on the well-known Fabry–Perot approach. Through numerical simulations, the dependences of photonic STDP on the bias current of VCSOA and the input powers are analyzed carefully. Besides, the effect of the initial wavelength detuning on the photonic STDP is also explored. It is found that, the current scheme requires much lower bias current and input power to obtain controllable STDP curve when compared with the previously reported photonic STDP circuits; the initial wavelength detuning is an effectively controllable parameter to realize wavelength-dependent photonic STDP. The computational model of the photonic STDP based on a VCSOA is interesting and valuable for numerically simulating of large-scale photonic spiking neural network, and provides a guideline to design low power consumption photonic neuromorphic systems.

Interplay of multiple pathways and activity-dependent rules in STDP.

Hebbian plasticity describes a basic mechanism for synaptic plasticity whereby synaptic weights evolve depending on the relative timing of paired activity of the pre- and postsynaptic neurons. Spike-timing-dependent plasticity (STDP) constitutes a central experimental and theoretical synaptic Hebbian learning rule. Various mechanisms, mostly calcium-based, account for the induction and maintenance of STDP. Classically STDP is assumed to gradually emerge in a monotonic way as the number of pairings increases. However, non-monotonic STDP accounting for fast associative learning led us to challenge this monotonicity hypothesis and explore how the existence of multiple plasticity pathways affects the dynamical establishment of plasticity. To account for distinct forms of STDP emerging from increasing numbers of pairings and the variety of signaling pathways involved, we developed a general class of simple mathematical models of plasticity based on calcium transients and accommodating various calcium-based plasticity mechanisms. These mechanisms can either compete or cooperate for the establishment of long-term potentiation (LTP) and depression (LTD), that emerge depending on past calcium activity. Our model reproduces accurately the striatal STDP that involves endocannabinoid and NMDAR signaling pathways. Moreover, we predict how stimulus frequency alters plasticity, and how triplet rules are affected by the number of pairings. We further investigate the general model with an arbitrary number of pathways and show that depending on those pathways and their properties, a variety of plasticities may emerge upon variation of the number and/or the frequency of pairings, even when the outcome after large numbers of pairings is identical. These findings, built upon a biologically realistic example and generalized to other applications, argue that in order to fully describe synaptic plasticity it is not sufficient to record STDP curves at fixed pairing numbers and frequencies. In fact, considering the whole spectrum of activity-dependent parameters could have a great impact on the description of plasticity, and a better understanding of the engram.

Symmetric spike timing-dependent plasticity at CA3-CA3 synapses optimizes storage and recall in autoassociative networks.

CA3–CA3 recurrent excitatory synapses are thought to play a key role in memory storage and pattern completion. Whether the plasticity properties of these synapses are consistent with their proposed network functions remains unclear. Here, we examine the properties of spike timing-dependent plasticity (STDP) at CA3–CA3 synapses. Low-frequency pairing of excitatory postsynaptic potentials (EPSPs) and action potentials (APs) induces long-term potentiation (LTP), independent of temporal order. The STDP curve is symmetric and broad (half-width ∼150 ms). Consistent with these STDP induction properties, AP–EPSP sequences lead to supralinear summation of spine [Ca2+] transients. Furthermore, afterdepolarizations (ADPs) following APs efficiently propagate into dendrites of CA3 pyramidal neurons, and EPSPs summate with dendritic ADPs. In autoassociative network models, storage and recall are more robust with symmetric than with asymmetric STDP rules. Thus, a specialized STDP induction rule allows reliable storage and recall of information in the hippocampal CA3 network.

Local activity in dendrites controls STDP by altering NMDA receptor kinetics

Spike timing-dependent plasticity (STDP) is a temporally asymmetric form of Hebbian plasticity in which the sign and magnitude of synaptic strength is determined by the precise timing between pre- and postsynaptic spikes. Postsynaptic NMDA receptors (NMDARs) play an important role in STDP due to their characteristic magnesium ion (Mg2+) gating process [1]. The Mg2+ gate is opened by near coincident presynaptic glutamate release and postsynaptic spiking. Contrary to previous belief, this gating process is not instantaneous but has distinct fast and slow time constants, which might influence the time window over which spike-EPSP coincidence occurs [2,3]. The aim of this study is to investigate how local activity in dendrites alters NMDAR kinetics and in turn controls the time window over which STDP occurs. Using a detailed biophysical model of an NR2A subunit, we find the NMDAR current to be temporally asymmetric with a faster block than unblock process. This is due to the voltage-dependent fast and slow time constants of Mg2+ gating. These time constants were on the order of 0.8 ms and 5 ms respectively and changed nonlinearly with membrane potential. By varying both amplitude and duration of a model action potential (AP), we find that dendritic Ca2+ spikes cause Mg2+ unblock to be dominated by its fast time constant. Conversely, somatic Na+ spikes elicit a slower unblocking. Since backpropagating APs (BAPs) attenuate along the dendritic arbor, our results suggest that NMDAR kinetics vary depending on synapse location. Next, we incorporated this biophysical model into a compartmentalized model of a neocortical pyramidal cell in order to investigate the effect that dendritic inhibition and synapse location has on STDP. NMDAR kinetics is significantly different in the basal and apical compartments, which gives rise to a heterogeneous set of STDP curves along the length of the dendrite. Distal synapses had a broader potentiation window than apical synapses, suggesting that distal branches are less selective to correlated inputs. This synapse-location dependent STDP learning rule agrees with previous findings [4]. Dendritic inhibition plays an important role in shaping the computations performed by pyramidal neurons in the neocortex. However, the role that inhibition plays in controlling synaptic plasticity is less clear. By incorporating inhibitory synapses in our model, we find that varying the timing of inhibition relative to the EPSP shifted the peak of the STDP curve, resulting in a reversal of the temporal order of depression and potentiation. Inhibition altered NMDAR kinetics by increasing its blocking rate. As a result, changing the balance between excitation and inhibition in the dendritic branch results in a sliding threshold between LTP and LTD. In summary, we show that NMDAR kinetics can be altered in an activity-dependent manner resulting in different STDP learning rules. When taken in context of the BCM theory, our finding implies that dendritic inhibition can dynamically alter the threshold between LTP and LTD by altering the time constants of the NMDAR. Our model provides a novel framework for understanding how STDP in different dendritic compartments selects inputs alters information processing in individual neurons.

Self-tuning spike-timing dependent plasticity curves to simplify models and improve learning

Plasticity for feed-forward excitation ought to optimally assign credit to which synapses cause a postsynaptic cell to spike. It is common to use a double-exponential fit of the LTP and LTD curves [1]; however, exponential curves are not always optimal and are prone to some pathologies. For example, if there are repeated patterns in the input spikes, learning will degenerate to only the selection of the earliest spikes [2]. Often the parameters for STDP are hand tuned for particular problems and networks. Measuring of the cross-correlogram offline can provide useful insight as to what the optimal STDP curve should be. We propose an adaptive STDP curve that is derived online from the cross-correlogram, and will discuss its

Bursts shape the NMDA-R mediated spike timing dependent plasticity curve: role of burst interspike interval and GABAergic inhibition.

Spike timing dependent plasticity (STDP) is a synaptic learning rule where the relative timing between the presynaptic and postsynaptic action potentials determines the sign and strength of synaptic plasticity. In its basic form STDP has an asymmetric form which incorporates both persistent increases and persistent decreases in synaptic strength. The basic form of STDP, however, is not a fixed property and depends on the dendritic location. An asymmetric curve is observed in the distal dendrites, whereas a symmetrical one is observed in the proximal ones. A recent computational study has shown that the transition from the asymmetry to symmetry is due to inhibition under certain conditions. Synapses have also been observed to be unreliable at generating plasticity when excitatory postsynaptic potentials and single spikes are paired at low frequencies. Bursts of spikes, however, are reliably signaled because transmitter release is facilitated. This article presents a two-compartment model of the CA1 pyramidal cell. The model is neurophysiologically plausible with its dynamics resulting from the interplay of many ionic and synaptic currents. Plasticity is measured by a deterministic Ca(2+) dynamics model which measures the instantaneous calcium level and its time course in the dendrite and change the strength of the synapse accordingly. The model is validated to match the asymmetrical form of STDP from the pairing of a presynaptic (dendritic) and postsynaptic (somatic) spikes as observed experimentally. With the parameter set unchanged the model investigates how pairing of bursts with single spikes and bursts in the presence or absence of inhibition shapes the STDP curve. The model predicts that inhibition strength and frequency are not the only factors of the asymmetry-to-symmetry switch of the STDP curve. Burst interspike interval is another factor. This study is an important first step towards understanding how STDP is affected under natural firing patterns in vivo.

The Effects of NMDA Subunit Composition on Calcium Influx and Spike Timing-Dependent Plasticity in Striatal Medium Spiny Neurons

Calcium through NMDA receptors (NMDARs) is necessary for the long-term potentiation (LTP) of synaptic strength; however, NMDARs differ in several properties that can influence the amount of calcium influx into the spine. These properties, such as sensitivity to magnesium block and conductance decay kinetics, change the receptor's response to spike timing dependent plasticity (STDP) protocols, and thereby shape synaptic integration and information processing. This study investigates the role of GluN2 subunit differences on spine calcium concentration during several STDP protocols in a model of a striatal medium spiny projection neuron (MSPN). The multi-compartment, multi-channel model exhibits firing frequency, spike width, and latency to first spike similar to current clamp data from mouse dorsal striatum MSPN. We find that NMDAR-mediated calcium is dependent on GluN2 subunit type, action potential timing, duration of somatic depolarization, and number of action potentials. Furthermore, the model demonstrates that in MSPNs, GluN2A and GluN2B control which STDP intervals allow for substantial calcium elevation in spines. The model predicts that blocking GluN2B subunits would modulate the range of intervals that cause long term potentiation. We confirmed this prediction experimentally, demonstrating that blocking GluN2B in the striatum, narrows the range of STDP intervals that cause long term potentiation. This ability of the GluN2 subunit to modulate the shape of the STDP curve could underlie the role that GluN2 subunits play in learning and development.

Pyramidal Neuron Conductance State Gates Spike-Timing-Dependent Plasticity

Neocortical neurons in vivo process each of their individual inputs in the context of ongoing synaptic background activity, produced by the thousands of presynaptic partners a typical neuron has. Previous work has shown that background activity affects multiple aspects of neuronal and network function. However, its effect on the induction of spike-timing dependent plasticity (STDP) is not clear. Here we report that injections of simulated background conductances (produced by a dynamic-clamp system) into pyramidal cells in rat brain slices selectively reduced the magnitude of timing-dependent synaptic potentiation while leaving the magnitude of timing-dependent synaptic depression unchanged. The conductance-dependent suppression also sharpened the STDP curve, with reliable synaptic potentiation induced only when EPSPs and action potentials (APs) were paired within 8 ms of each other. Dual somatic and dendritic patch recordings suggested that the deficit in synaptic potentiation arose from shunting of dendritic EPSPs and APs. Using a biophysically detailed computational model, we were not only able to replicate the conductance-dependent shunting of dendritic potentials, but show that synaptic background can truncate calcium dynamics within dendritic spines in a way that affects potentiation more strongly than depression. This conductance-dependent regulation of synaptic plasticity may constitute a novel homeostatic mechanism that can prevent the runaway synaptic potentiation to which Hebbian networks are vulnerable.

GABA inhibition modulates NMDA-R mediated spike timing dependent plasticity (STDP) in a biophysical model

Spike timing dependent plasticity (STDP) has been demonstrated in various neural systems of many animals. It has been shown that STDP depends on the target and the location of the synapse and is dynamically regulated by the activity of adjacent synapses, the presence of postsynaptic calcium, presynaptic GABA inhibition or the action of neuromodulators. Recent experimental evidence has reported that the profile of STDP in the CA1 pyramidal neuron can be classified into two types depending on its dendritic location: (1) A symmetric STDP profile in the proximal to the soma dendrites, and (2) an asymmetric one in the distal dendrites. Bicuculline application revealed that GABA(A) is responsible for the symmetry of the STDP curve. We investigate via computer simulations how GABA(A) shapes the STDP profile in the CA1 pyramidal neuron dendrites when it is driven by excitatory spike pairs (doublets). The model constructed uses calcium as the postsynaptic signaling agent for STDP and is shown to be consistent with classical long-term potentiation (LTP) and long-term depression (LTD) induced by several doublet stimulation paradigms in the absence of inhibition. Overall, simulation results provide computational evidence for the first time that the switch between the symmetrical and the asymmetrical STDP operational modes is indeed due to GABA inhibition. Furthermore, gamma frequency inhibition and not theta one is responsible for the transition from asymmetry-to-symmetry. The resulted symmetrical STDP profile is centered at +10 ms with two distinct LTD tails at -10 and +40 ms. Finally, the asymmetry-to-symmetry transition is strongly dependent on the strength (conductance) of inhibition and its relative onset with respect to pre- and postsynaptic spike stimulation.

Closed-Form Treatment of the Interactions between Neuronal Activity and Timing-Dependent Plasticity in Networks of Linear Neurons

Network activity and network connectivity mutually influence each other. Especially for fast processes, like spike-timing-dependent plasticity (STDP), which depends on the interaction of few (two) signals, the question arises how these interactions are continuously altering the behavior and structure of the network. To address this question a time-continuous treatment of plasticity is required. However, this is - even in simple recurrent network structures - currently not possible. Thus, here we develop for a linear differential Hebbian learning system a method by which we can analytically investigate the dynamics and stability of the connections in recurrent networks. We use noisy periodic external input signals, which through the recurrent connections lead to complex actual ongoing inputs and observe that large stable ranges emerge in these networks without boundaries or weight-normalization. Somewhat counter-intuitively, we find that about 40% of these cases are obtained with a long-term potentiation-dominated STDP curve. Noise can reduce stability in some cases, but generally this does not occur. Instead stable domains are often enlarged. This study is a first step toward a better understanding of the ongoing interactions between activity and plasticity in recurrent networks using STDP. The results suggest that stability of (sub-)networks should generically be present also in larger structures.

Discrete States of Synaptic Strength in a Stochastic Model of Spike-Timing-Dependent Plasticity

A stochastic model of spike-timing-dependent plasticity (STDP) postulates that single synapses presented with a single spike pair exhibit all-or-none quantal jumps in synaptic strength. The amplitudes of the jumps are independent of spiking timing, but their probabilities do depend on spiking timing. By making the amplitudes of both upward and downward transitions equal, synapses then occupy only a discrete set of states of synaptic strength. We explore the impact of a finite, discrete set of strength states on our model, finding three principal results. First, a finite set of strength states limits the capacity of a single synapse to express the standard, exponential STDP curve. We derive the expression for the expected change in synaptic strength in response to a standard, experimental spike pair protocol, finding a deviation from exponential behavior. We fit our prediction to recent data from single dendritic spine heads, finding results that are somewhat better than exponential fits. Second, we show that the fixed-point dynamics of our model regulate the upward and downward transition probabilities so that these are on average equal, leading to a uniform distribution of synaptic strength states. However, third, under long-term potentiation (LTP) and long-term depression (LTD) protocols, these probabilities are unequal, skewing the distribution away from uniformity. If the number of states of strength is at least of order 10, then we find that three effective states of synaptic strength appear, consistent with some experimental data on ternary-strength synapses. On this view, LTP and LTD protocols may therefore be saturating protocols.