Statistical Theory of Long-Term Synaptic Plasticity
(A) Schematic of a synapse with presynaptic (, release probability; blue) and postsynaptic (, quantal amplitude; red) components, both subjected to change via long-term plasticity induction. A common induction protocol of long-term potentiation (LTP) consists of high-frequency stimulation (tetanus protocol; inset bottom right).
(B) A tetanus protocol in hippocampal CA1 excitatory synapses can yield pre- (left panels) or postsynaptic (right panels) modifications ().
(C) In our theoretical framework, the postsynaptic response statistics (black) are optimized to meet a minimum-variance bound (green, here at 1 mV for illustration, see main text for how we interpret and estimate the bound). During long-term synaptic plasticity, the synapse minimizes the difference between the current distribution and its bound (i.e., the Kullback-Leibler divergence, see ) by changing both the release probability (blue) and the quantal amplitude (red).
(D) The theory predicts an optimal direction of change toward a bound (green cross) that depends on the initial and (cf. and ).

Statistical Long-Term Synaptic Plasticity, StatLTSP, Predicts Expression Loci of Synaptic Potentiation in Hippocampus
(A) Long-term potentiation (LTP) experiments in hippocampus using a tetanus protocol (bound estimated with this dataset).
(B) Short-lasting potentiation (SLP) experiments in hippocampus using a tetanus protocol (bound estimated in A).
(C) Short-lasting potentiation (SLP) experiments in hippocampus using a long-step-current protocol (bound estimated in A). (i) Model predictions and observed changes in and parameters (black and purple, respectively). Green cross represents the estimated bound, which is outside the plotted range of . (ii) Predicted and observed changes in both (blue) and (red). There is no significant difference between predicted and observed changes for both (hipp. LTP: p = 0.8; tetanus-SLP: p = 0.54; current-SLP: p = 0.62) and (hipp. LTP: p = 0.96; tetanus-SLP: p = 0.67; current-SLP: p = 0.9). (iii) Distribution of angles (in degrees) between observed and predicted changes for statLTSP (black solid line), a random (orange solid line) and a shortest path model (dark orange dashed line; see ). Predictions for LTP by shortest path model are not different from the predictions by the random path model (p = 0.83). LTP and SLP experiments were reanalyzed from and , respectively.

StatLTSP Predicts Expression Loci of Long-Term Potentiation in Visual Cortex
(A) LTP experiments in visual cortex using spike-timing-dependent plasticity (STDP) protocols ( represents the delay between pre- and postsynaptic spikes; ISI is the inter-spike interval).
(B) LTP experiments in visual cortex using a long-depolarizing step protocol. (i) Model predictions and observed changes in and parameters (black and purple, respectively). (ii) Predicted and observed changes in both (blue) and (red). There is no significant difference between predicted and observed changes for both (STDP-LTP: p = 0.83; dep-LTP: p = 0.6) and (STDP-LTP: p = 0.88; dep-LTP: p = 0.96). (iii) Distribution of angles (in degrees) between observed and predicted changes for statLTSP (black solid line), a random (orange solid line) and a shortest path model (dark orange dashed line; see ). STDP and depolarization-LTP data reanalyzed from and , respectively.

StatLTSP Predicts Expression Loci of Long-Term Depression
(A) Flow field when setting a lower bound mV. Setting either or makes the postsynaptic response equal to zero (the lower bound is represented by the solid green line; see ).
(B) Decreasing toward a lower bound (green), which controls synaptic transmission variance, is statistically more efficient (blue) than decreasing (red).
(C) Change in divergence when changing or alone for the lower bound mV.
(D) Schematic representation of a synapse with an STDP protocol that yields LTD ( represents the delay between pre- and postsynaptic spikes; ISI is the inter-spike interval).
(E) Model predictions and observed changes in and parameters (black and purple, respectively).
(F) Predicted and observed changes in both (blue) and (red). There is no significant difference between predicted and observed changes for both (p = 0.63) and (p = 0.63).
(G) Distribution of angles (in degrees) between observed and predicted changes for statLTSP (black solid line), a random (orange solid line) and a shortest path model (dark orange dashed line; see ). STDP LTD data reanalyzed from . Error bars represent mean ± SEM.

StatLTSP Explains Synaptic State Dependence of Expression Loci
Framework predicts a specific post/pre expression of synaptic weight changes for a given combination of baseline and , consistent with experimental findings (see main text; cf. shortest path model in ). Bottom: post/pre ratio predicted by statLTSP for different baseline values of . Postsynaptically silent synapses are represented by minimal baseline . Left: post/pre ratio predicted by statLTSP for different baseline values of . Presynaptically silent synapses are represented by minimal baseline . Green cross represents the bound estimated from hippocampal data (cf.A).

Feedback Control of Expression Loci Requires Endocannabinoid and Nitric Oxide Signaling
Left: schematic of pre- and postsynapse with LTP protocol and pharmacological intervention used (data from ). Middle: scatterplot of observed changes in (i) and (ii) over the predicted divergence in . Right: scatterplot of observed changes in (i) and (ii) over the predicted divergence in .
(A) Nitric oxide (NO) blockade data.
(B) Endocannabinoid (eCB) blockade data.
(C) Summary of blockade experiments. Control LTP (dark red and blue lines) was obtained using the same protocol, but without drug wash-in (see ). Alternative models did not provide a parsimonious explanation for the role of eCB and NO (cf.).

Inhibitory Plasticity Specifically Aims at the Mean Excitatory Input
(A) Statistics of both excitatory (green) and inhibitory (purple) currents were recorded before and after long-term plasticity induction using an STDP protocol. The statistics of inhibitory input (purple Gaussian) can be modified through pre- and postsynaptic long-term plasticity (top; represents the delay between pre- and postsynaptic spikes) to balance out specific statistics of the excitatory input. Such a statistical EI balance can be achieved by inhibition matching the mean (light green; i.e., a reliable bound as in statLTSP) or mean and variance of excitatory responses (dark green Gaussian).
(B) Model predictions and observed changes in and parameters (black and purple, respectively). Solid arrows represent the mean and light areas the standard error of the mean (see for individual data points and bounds). Green cross represents the bound that we consider at inhibitory synapses (i.e., the experimentally observed mean excitatory current across all experiments studied here).
(C) Predicted and observed changes in (blue) and (red). There is no significant difference between predicted and observed changes for both (p = 0.29) and (p = 0.97).
(D) Distribution of angles (in degrees) between observed and predicted changes for statLTSP (black, solid line), a shortest (dark orange, dashed line) and a random path model (orange, solid line). The shortest model also performs worse when analyzing changes in and as in (C) (see main text).
(E–G) StatLTSP with estimated bounds for individual experiments. (E) Correlation between estimated and observed bounds (see main text for details). (F) Predicted and observed changes in (blue) and (red) (similar to C). There is no significant difference between predicted and observed changes for both  (p = 0.36) and (p = 0.71). (G) Distribution of angles (in degrees) between observed and predicted changes for statLTSP (black, solid line), a shortest (dark orange, dashed line) and a random path model (orange, solid line), similar to (D). Data reanalyzed from . Error bars represent mean ± SEM.

Inhibition Aiming for Mean Excitation Yields a Better Statistical Excitation-Inhibition Balance
(A and B) Changes in mean (A) and variance (B) of inhibitory currents for statLTSP (top) and the EI ratio (bottom).
(C) Model selection criteria for a model in which inhibition aims for the excitatory mean current (light green), a model in which inhibition aims for both mean and variance of excitatory responses (dark green), and a model in which inhibition aims for the mean excitation, but its variance is fixed (i.e., does not change; white) (cf.).
(D) A given sample from inhibitory and excitatory postsynaptic responses generates an EI ratio, which we use to estimate the distribution of EI balance. Comparison of distributions of EI balance for two possible views: inhibition response statistics matches both excitatory mean and variance (dark green) or only excitatory mean (light green).
(E) Change in E/I distributions as the variance of excitatory increases for both cases. Dotted lines in (D) represent the mean of the distributions and dotted lines in (E) represent the variance of the excitatory responses used in (D). Error bars represent mean ± SEM.