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.