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Spike Timing-Dependent Plasticity (STDP)

Overview

STDP changes synaptic weights according to the relative timing of pre- and postsynaptic spikes. The examples in chapter40.ipynb begin with the Abbott--Song timing rule and an adapting RTM voltage trace, then progress through a three-cell PING network (first with fixed E-to-E coupling, then with that coupling evolving under STDP), and finish with a full PING network in which every recurrent E-to-E synapse evolves under STDP.

Core ideas

Weight-update sign and size depend on spike order and time difference: a presynaptic spike shortly before a postsynaptic one potentiates the synapse, and the reverse order depresses it. An adaptation-like variable tracks how recently each cell spiked and gates these updates. In three-cell PING, E-to-E coupling changes the E-assembly lag and frequency; once STDP is active, that timing and the weights coevolve.

Essential model

For $\Delta t=t_{\rm post}-t_{\rm pre}$, a representative pairwise rule is

$$ \Delta w = \begin{cases} A_+,e^{-\Delta t/\tau_+} & \Delta t>0\ \text{(post after pre)}\[2pt] -A_-,e^{\Delta t/\tau_-} & \Delta t<0\ \text{(pre after post)} \end{cases} $$

The network implementation approximates this with a smooth, voltage-triggered version: each E-cell carries an adaptation-like trace, and every E-to-E weight is nudged up or down whenever its pre- or postsynaptic cell crosses spike threshold, softly clamped between 0 and a fixed upper bound B.

Code examples

All eight examples live in one notebook, chapter40.ipynb.

simulate_abbott_song/plot_abbott_song cover ABBOTT_SONG: the piecewise-exponential timing rule $F_0$ and a windowed version $F$ that tapers to zero as $|z|\to0$, with tau_plus and K_plus exposed through sliders.

simulate_rtm_voltage_trace_with_a/plot_rtm_voltage_trace_with_a cover RTM_VOLTAGE_TRACE_WITH_A: an RTM voltage trace together with an adaptation-like variable $a$ that relaxes toward 0 while the cell is depolarized, with i_ext and C exposed through sliders.

A shared block of RTM/WB gating functions, tau_peak_function/ tau_d_q_function (synapse rise-time solver), derivative_three_cell_ping, spike_detection, and simulate_three_cell_ping/reciprocal_g_ee covers the fixed-coupling two-E/one-I network reused by four examples:

  • THREE_CELL_PING_1 (baseline, no E-to-E coupling) and THREE_CELL_PING_2 (weak vs. strong reciprocal coupling) are plotted with plot_three_cell_raster, and both have interactive sliders over the coupling strength.
  • sweep_three_cell_ping_ee/plot_ee_sweep cover THREE_CELL_PING_3 (reciprocal coupling sweep) and THREE_CELL_PING_4 (one-way coupling sweep), each reporting the E1-E2 lag and E2 frequency as a function of coupling strength.

simulate_three_cell_ping_5/plot_three_cell_ping_5 cover THREE_CELL_PING_5: the same two-E/one-I network, but with the reciprocal E-to-E weights evolving under STDP, tracked via g_12, g_21 and the shrinking E1-E2 lags. g_ee0 is exposed through a slider. Its 50000-step explicit-Heun loop (_three_cell_ping5_loop) is @njit-compiled.

simulate_ping_with_stdp/plot_ping_with_stdp_raster/ plot_ping_with_stdp_density cover PING_WITH_STDP: 200 RTM E-cells and 50 WB I-cells with every E-to-E synapse plastic under STDP, reporting the raster and the final kernel-density estimate of E-to-E synaptic strengths (vec_g). Its per-timestep update (_stdp_loop, shared with THREE_CELL_PING_5's STDP increment kernel _g_ee_derivative_s) is @njit-compiled; a full run takes a few minutes even compiled, so no interactive slider is provided for it.

What to look for

Read the Abbott--Song curve with signed pre/post timing before interpreting weight traces. In the fixed-coupling three-cell cases, compare E-cell lag and frequency as coupling changes. In the STDP cases, ask whether weight evolution changes the spike order that produced it: in THREE_CELL_PING_5, the faster-driven E-cell reliably leads, so its outgoing synapse potentiates toward its bound while the reverse synapse decays, and the lag itself shrinks as that asymmetry grows. In PING_WITH_STDP, look for a widened, non-degenerate spread in the final E-to-E weight distribution rather than every synapse sitting at its initial value.

Suggested order

  1. Run ABBOTT_SONG and RTM_VOLTAGE_TRACE_WITH_A.
  2. Run THREE_CELL_PING_1 through THREE_CELL_PING_4.
  3. Finish with THREE_CELL_PING_5 and PING_WITH_STDP.

Prerequisites and related chapters

Chapter 9 provides adaptation context, Chapter 20 chemical synapses, Chapter 30 PING, and Chapter 39 short-term plasticity.

Running the examples

Open chapter40.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. NumPy, SciPy, Matplotlib, ipywidgets, and numba are required. The THREE_CELL_PING_3/_4 coupling sweeps and the PING_WITH_STDP example are the slowest cells and can take several minutes; that is expected.