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Gap junctions

Overview

Gap junctions are electrical synapses: each cell receives a current determined by the voltage difference from its neighbours. The examples contrast the resulting synchronization of spiking cells with subthreshold coupling. They also contrast LIF event handling with a continuous WB voltage trace.

Core ideas

For a pair, a gap-junction current is proportional to $v_j-v_i$, so it reduces voltage differences rather than imposing a fixed chemical reversal potential. This promotes synchrony when cells spike, but subthreshold behavior can still be distinctive. In LIF models, a spike reset makes the threshold and reset separation part of the coupling mechanism.

Essential model

With electrical conductance $g_{\rm gap}$, cell $i$ receives

$$ I_{{\rm gap},i}=g_{\rm gap}\sum_{j\ne i}(v_j-v_i). $$

The continuous current is supplemented by the model-specific event rule: an LIF voltage is reset after crossing threshold, whereas the WB network evolves its conductance-based voltage continuously.

Code examples

All four examples now live in one notebook, chapter21.ipynb: simulate_lif_network_with_gj compares two LIF voltage traces under diffusive gap-junction coupling, with and without an additional spike-triggered voltage kick to the other cell (epsilon); vertical marks identify spikes. simulate_reset_threshold integrates a single WB conductance-based voltage trace and marks two reference voltage levels. simulate_wb_network_with_gj integrates two WB neurons with a gap junction to display their voltage alignment. simulate_wb_network_with_gj_subthreshold focuses on voltage-difference coupling before the WB cells spike, taking a gap_gate(v1) callable so the same stepper can run the always-on and subthreshold-only cases.

What to look for

Inspect whether an initial voltage difference shrinks in the subthreshold WB trace and whether spike times become aligned in the network plots. In the LIF comparison, separate diffusive electrical equalization from the additional spike-triggered kick. Relate the discontinuous reset to the voltage gap immediately after a threshold crossing in simulate_lif_network_with_gj; contrast this with the smooth WB trajectories, including simulate_reset_threshold.

Suggested order

  1. Run simulate_lif_network_with_gj, then simulate_reset_threshold as a continuous WB voltage reference.
  2. Run simulate_wb_network_with_gj_subthreshold to isolate electrical equalization.
  3. Run simulate_wb_network_with_gj and compare the spiking case.

Prerequisites and related chapters

Chapter 7 defines LIF threshold-and-reset dynamics, and the WB examples use the conductance-based interneuron model from earlier chapters. Chapter 20 covers chemical synapses; Chapters 24 and 29 return to synchrony and its stability.

Running the examples

Open chapter21.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.