Voltage and Spike Timing Interact in STDP - A Unified Model

Abstract

A phenomenological model of synaptic plasticity is able to account for a large body of experimental data on spike-timing-dependent plasticity (STDP). The basic ingredient of the model is the correlation of presynaptic spike arrival with postsynaptic voltage. The local membrane voltage is used twice: a first term accounts for the instantaneous voltage and the second one for a low-pass filtered voltage trace. Spike-timing effects emerge as a special case. We hypothesize that the voltage dependence can explain differential effects of STDP in dendrites, since the amplitude and time course of backpropagating action potentials or dendritic spikes influences the plasticity results in the model. The dendritic effects are simulated by variable choices of voltage time course at the site of the synapse, i.e., without an explicit model of the spatial structure of the neuron.

Keywords: LTD; LTP; STDP; computational neuroscience; frequency; model; synaptic plasticity; voltage.

Figure 1

Schematics of the model. (A) LTD occurs at the time of a presynaptic spike (green) if the low-pass filtered voltage trace ${\overline{u}}_{-}$ (magenta) is above θ₋ (dashed line). The amount of LTD is proportional to the size of the yellow box. If the timing difference between post- and presynaptic spikes is too big, no LTD is induced (bottom). (B) LTP requires three factors: a momentary voltage u (black) above θ₊ (dashed line), the trace $\overline{x}$ (red) left from a previous presynaptic spike above 0, and the trace ${\overline{u}}_{+}$ (blue) of the low-pass filtered voltage above θ₋ (dashed line). The three conditions are met at the moment of the second postsynaptic spike in a post-pre-post triplet (top panel), but not after a single pre-post pair (bottom). The amount of plasticity is proportional to the multiplication of the yellow boxes. (C) Presynaptic stimulation under voltage clamp conditions shows the relevance of the threshold θ₋ for onset of LTD and θ₊ for the onset of the LTP contribution. LTP becomes dominant if the voltage is 10 mV or more above θ₊.

Figure 2

Model depends on spike frequency. (A). Schematics of STDP experiment. Injection of a current pulse in the presynaptic neuron at t = 5 ms leads to an EPSP which is followed t = 15 ms by an action potential triggered by a current pulse into the postsynaptic neuron. (B) If pre-post pairings at lags of 2 ms are repeated, the total amount of weight change (vertical) depends on the repetition frequency (horizontal axis). Model in blue, data redrawn from Markram et al. (1997) in green. A standard pair-based STDP model cannot account for the frequency, whatever the summation scheme (black dashed line: all pairs contribute; red dashed line: only pairs between the nearest spikes contribute to plasticity). (C) The frequency dependence is different, if both pre- and postsynaptic spikes are generated by a Poisson process (box: zoom). (D) The total amount of plasticity depends on the number of pre-post pairings at 2 ms lag. At least two pairs at 20 Hz are necessary. With our set of parameters, saturation at the maximal weight occurs for around seven pairings (model is blue, data redrawn from Senn et al. (2001) in green; in experiments, saturation is maybe already reached after two pairings, dashed green line).

Figure 3

Plasticity results depend on voltage trajectory. (A) Eight 50 Hz pre-post pairings are induced by injection of somatic current pulses (center, schematic). We model the voltage time course at synapses located on the soma (dashed) and basal (solid) dendrite by the sequence of action potentials, shown on the left. The voltage time course at synapses located distally on apical dendrites is modeled as subthreshold response (right, solid line). (B) 50 Hz pre-post pairing leads to LTP (left column) when postsynaptic response consists of spikes as in the basal dendrite (or if the presence of backpropagating action potentials) and to LTD when the postsynaptic response stays subthreshold (right column). Green, data redrawn from Sjöström and Häusser (2006); blue, simulations.

Figure 4

Burst-timing-dependent learning window. A postsynaptic burst of three spikes is paired with a presynaptic spike. (A) Assumed voltage waveform at the basal dendrite. (B) The total weight change plotted as a function of the time between the presynaptic spike and the start of the postsynaptic burst varies. Data redrawn from Kampa et al. (2006) in green, simulations in blue.

Figure 5

Results with extracellular stimulation. (A) Extracellular stimulation of presynaptic fibers followed 10 ms later by postsynaptic stimulation is described by a compound EPSP of 2.5 mV and the upswing of an action potential (schematic). (B) 60 repetitions of pre-post pairs lead to LTP despite a repetition frequency of only 0.2 Hz. The amount of LTP in our model (blue line) is smaller than in the corresponding experiment (green). The reverse firing leads to LTD. (C) Triplets consisting of two post- and one presynaptic spikes in various configurations (see drawing) are repeated at low frequencies. Presynaptic stimulation is extracellular. Line and bars: simulations, green: data redrawn from Froemke and Dan (2002). (D) Same, but triplets consisting of two pre- and one postsynaptic spike. The big error bars in (C,D) indicate that data are very noisy and thus it is only relevant whether triplets induce LTP or LTD in each configurations.

Figure 6

Pairs, triplets, and quadruplets of spikes in cultured hippocampal neurons. (A) Intracellular presynaptic stimulation results in a model EPSP of 7.5 mV (schematic). (B) STDP function in the pair-experiment. (C) Triplet experiments. Δt_1,2 in the post-pre-post configuration is the time between the single pre- and the two postsynaptic spikes (blue histogram bars and schematics) or, in the pre-post-pre experiment the time between the single post- and the two presynaptic spikes (red histogram bars and schematics). The four different post-pre-post triplets are: (a) post-5ms-pre-5ms-post, (b) post-10ms-pre-10ms-post, (c) post-5ms-pre-15ms-post, and (d) post-15ms-pre-5ms-post. The four different pre-post-pre triplets are: (a) pre-5ms-post-5ms-pre, (b) pre-10ms-post-10ms-pre, (c) pre-15ms-post-5ms-pre, and (d) pre-5ms-post-15ms-pre. Lines and bars: simulations. Green circles: data redrawn from Wang et al. (2005). (D) Quadruplet experiment (see main text).