Stochastic synaptic dynamics under learning

TL;DR

The paper develops a stochastic framework for STDP-driven synaptic dynamics, deriving a Langevin-type drift-diffusion description that incorporates spike-time cross-correlations via neuron response functions. It then maps single-synapse dynamics to a mean-field network to quantify hetero-associative memory storage, forgetting, and memory capacity under sparse coding, highlighting that cross-correlations are crucial for accurate capacity predictions. A key finding is the tradeoff between early accuracy, memory lifetime, and learning speed, with slow learning and homeostatic scaling enabling greater capacity but causing gradual forgetting. The approach yields quantitative predictions for memory capacity, recall performance, and optimal training durations, and points to experimental tests and extensions to recurrent plasticity and intrinsic synaptic noise.

Abstract

Learning is based on synaptic plasticity, which affects and is driven by neural activity. Because pre- and postsynaptic spiking activity is shaped by randomness, the synaptic weights follow a stochastic process, requiring a probabilistic framework to capture the noisy synaptic dynamics. We consider a paradigmatic supervised learning example: a presynaptic neural population impinging in a sequence of episodes on a recurrent network of integrate-and-fire neurons through synapses undergoing spike-timing-dependent plasticity (STDP) with additive potentiation and multiplicative depression. We first analytically compute the drift- and diffusion coefficients for a single synapse within a single episode (microscopic dynamics), mapping the true jump process to a Langevin and the associated Fokker-Planck equations. Leveraging new analytical tools, we include spike-time--resolving cross-correlations between pre- and postsynaptic spikes, which corrects substantial deviations seen in standard theories purely based on firing rates. We then apply this microdynamical description to the network setup in which hetero-associations are trained over one-shot episodes into a feed-forward matrix of STDP synapses connecting to neurons of the recurrent network (macroscopic dynamics). By mapping statistically distinct synaptic populations to instances of the single-synapse process above, we self-consistently determine the joint neural and synaptic dynamics and, ultimately, the time course of memory degradation and the memory capacity. We demonstrate that specifically in the relevant case of sparse coding, our theory can quantitatively capture memory capacities which are strongly overestimated if spike-time--resolving cross-correlations are ignored. [...]

Figures (8)

• Figure 1: We investigate memory properties by studying a single synapse dynamics that effectively captures synaptic population dynamics. (a) A Poisson process $\eta$ drives the neuron $x$ through the synapse $w$. The weight $w$ is plastic, i.e., it dynamically depends on $x$ and $\eta$. We indicated the tracers $A_{\text{pre}}$ and $A_{\text{post}}$ used in the embedding eq:model_exp_local. (b) STDP window eq:plasticity_kernel for $w=0.01$ (light gray) through $w=0.2$ (black). The setup (a,b) is studied in sec:Stochastic-dynamics-of (c) A cue representation $\boldsymbol{q}$ in a layer of Poisson processes (gray) is activated and drives a recurrent neural network through a matrix of plastic synapses. Among the excitatory neurons (red) in the recurrent network, a target representation $\boldsymbol{p}$ receives an additional stimulus. The cue and target subsets are marked by the green arrows and circles. (d) Training procedure to store hetero-associations between cues and targets. The setup (c,d) is studied in sec:Dynamics-of-learning. Parameters $\Delta_{c}=2\times10^{-3}$, $r_{ac}=8\times10^{-3}$, $\tau_{c}=16.8/20$, and $\tau_{ac}=33.7/20$ are matched to Bi1998_10472.
• Figure 2: Stochastic dynamics of single synapses. (a1) Sample trajectories of eq:model_general and (a2) sample trajectories of the corresponding Langevin equation eq:langevin for $\mu=0.6$, $D=0.2$, $\nu=0.1$, $m_{0}=0.1$, $\sqrt{V_{0}}=10^{-3}$ and STDP parameters as in fig:joint_model(b). (b1--b4) Drift coefficient $D^{(1)}$ from simulations (orange circles) and theory [eq:D1 (black line) and single contributions: firing-rate (gray), mean-response (green), and noise-intensity--response (blue)] for different $\mu$ and $D$. The instantaneous firing rate $r$ and the coefficient of variation $C_{v}$ at $w=0.1$ are indicated in the upper center of the four panels. (c1--c4) Finite-time diffusion coefficient $D^{(2)}$ for $\Delta t=10$. Simulation results (orange circles). Theory (black lines) $[V(\Delta t)+(m(\Delta t)-w)^{2}]/(2\Delta t)$, with $m$ from eq:solution_mean_drift and $V$ from eq:solution_diffusion.
• Figure 3: Acceleration-to-deceleration transition. (a) Drift coefficient, theory (red line) and simulation (orange circles), for $\mu=0.75$, $D=0.028$, $r\approx0.16$, $C_{v}\approx0.58$, corresponding to the red cross in (c). (b) Drift coefficient, theory (green line) and simulation (light-green circles), for $\mu=0.71$, $D=0.057$, $r\approx0.23$, $C_{v}\approx0.59$, corresponding to the green cross in (c). (c) Slope of $D^{(1)}(w=0.1)$ obtained from eq:D1 plotted against the rate and $C_{v}$ of the postsynaptic neuron assuming $w=0.1$ fixed. Other parameters as in fig:single_synapse_full.
• Figure 4: Population dynamics during a training session of length $T=50$. Mean $m_{ab}$ and standard deviation $\sqrt{V_{ab}}$ of the four populations from eq:mVdynamics_4pops (black line and gray shading) and from simulations (red errorbars). The separate rightmost errorbars schematically illustrate the effect of homeostasis eq:homeostasis_theory (we do not consider an actual time for homeostasis). Parameters are neuron:$\mu_{E}=0$, $D_{E}=0.1$, $\mu_{I}=0.5$, $D_{I}=0.05$network: $N_{E}=4000$, $N_{I}=1000$, $C_{E}=200$, $C_{I}=50$, $(J,g,h)=(0.01,5,2)$input: $(f_{c},f_{s})=(0.05,0.1)$, $M=200$, $m_{0}=0.05$, $\nu_{\text{hi}}=1$, $\nu_{\text{lo}}=0.1$, $\nu_{s}=64$, $J_{s}=1/80$, STDP: as in fig:joint_model.
• Figure 5: Dynamics of memory traces and memory capacity. (a) Mean synaptic strength from cue to (non-)target neurons (with respect to association 0) after $k$ subsequently stored associations. Simulation results denoted by circles, green for cue-to-target and red for cue-to-non-target, dark colors are averages over $100$ realizations, light colors are single realizations. Theory eq:trace_degrad_sol (black lines) and $m_{0}$ (gray line). (b) Variance of synaptic strength from cue to (non-)-target. Colors as in (a), theory (black) from eq:var_degrad_iter and stationary variance (gray) from eq:stationary_variance. (c) Fraction of correctly activated neurons on average over 100 realizations (dark purple circles), for a single realization (light circles) and theory eq:frac_correct_hits based on the full drift (black solid line) and neglecting cross-correlations (dotted line). (d,e) Memory capacity eq:capacity as a function of input sparseness $f_{c}$ (d) and the input width $M$ (e) with (solid) and without (dotted) cross-correlations. Parameters as in fig:Four-populations-training-session, but $\nu_{\text{lo}}=0$.