chapter20.md

Chemical synapses

Overview

Chemical synapses turn presynaptic activity into a conductance that drives the postsynaptic voltage toward a reversal potential. These examples use RTM neurons to make the synaptic gate visible, then use that gate for a self-synapse and for the temporal accumulation produced by repeated input.

Core ideas

The synaptic current is conductance based: $I_{\rm syn}=g_{\rm syn}s(v_{\rm syn}-v)$. A release variable can rise rapidly after a spike and feed the gate $s$, which decays more slowly. The rise and decay constants therefore shape both the peak time and the duration of the conductance. The NMDA factor also depends on postsynaptic voltage because magnesium block is relieved by depolarization.

Essential model

For the two-stage synapse used here, release $q$ and activation $s$ evolve as

$$ \dot q=R(v)(1-q)-q/\tau_{d,q},\qquad \dot s=q(1-s)/\tau_r-s/\tau_d. $$

tau_d_q_function numerically chooses $\tau_{d,q}$ to obtain a requested activation peak time. The resulting $s$ multiplies the synaptic conductance in the RTM voltage equation.

Code examples

All eight examples now live in one notebook, chapter20.ipynb: simulate_b_jahr_stevens plots the voltage-dependent NMDA magnesium-block factor from Jahr and Stevens; simulate_rtm_plot_s compares a fast and a slower synaptic-gate rise alongside the RTM voltage trace; simulate_rtm_plot_q separates transmitter-release $q$ from the gate $s$, showing why a two-variable synapse has a delayed profile; simulate_two_stage_synapse (shared by three examples below) uses the explicit release-and-gate system: called directly with fixed time constants for two timing choices, or with tau_d_q_function-solved time constants to hit a prescribed peak time; simulate_rtm_with_autapse_f_i_curve follows forward and backward RTM frequency-current sweeps with an excitatory autapse; simulate_s_buildup shows how closely spaced presynaptic events build up a synaptic gate, called once with a fast decay and once with a slower one (buildup and slow-buildup).

What to look for

In the gate plots, locate the delay between a voltage spike and the maximum of $s$. Increasing the rise or release time moves that maximum and broadens the conductance. Compare the two buildup calls to see that a slow decay retains activation between events. In the autapse sweep, compare the two directions to identify any history dependence of the firing state.

Suggested order

1. Run simulate_rtm_plot_s, simulate_rtm_plot_q, and simulate_two_stage_synapse with the two fixed-timing calls.
2. Run the prescribed-peak-time simulate_two_stage_synapse calls, then simulate_s_buildup with the fast and slow decay parameters.
3. Examine simulate_b_jahr_stevens and simulate_rtm_with_autapse_f_i_curve.

Prerequisites and related chapters

The RTM conductance model is introduced earlier in the single-neuron chapters. Chapter 21 replaces chemical conductance with electrical coupling, while Chapters 23--29 use pulses and phase descriptions to study network timing.

Running the examples

Open chapter20.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The simulate_rtm_with_autapse_f_i_curve cell integrates a forward+backward sweep over 31 values of $I$; its inner loop is JIT-compiled with numba, so after the first (one-time compile) call it takes well under a second instead of several minutes per direction.