chapter24.md

Synchronization by fast recurrent excitation

Overview

Fast recurrent excitation can bring initially dispersed RTM neurons into a common firing rhythm. The examples progress from two-cell and recurrent-network cases to synchronous, splay-state, and heterogeneous networks.

Core ideas

An excitatory pulse advances cells that are susceptible to it, so repeated interactions can contract phase differences. Synchrony means the cells fire together; a splay state distributes their phases around the cycle. Heterogeneous drive or intrinsic properties work against a common rhythm, testing how robust the coupling mechanism is.

Essential model

Each RTM cell receives summed recurrent excitation through synaptic gates:

$$ I_{{\rm syn},i}=g_{\rm syn}\sum_{j\ne i}s_j(v_{\rm syn}-v_i). $$

The scripts build phase-spread initial conditions from the single-cell limit cycle and then integrate the full conductance-based network.

Code examples

All six examples now live in one notebook, chapter24.ipynb: simulate_rtm_e_to_e_network simulates a baseline recurrent E-to-E RTM network with release and synaptic-gate variables; simulate_rtm_e_to_e_network_2 calls the same function for a much longer run initialized with phases spread across the cycle, watching the splay state contract toward synchrony -- its inner loop is JIT-compiled with numba. simulate_rtm_e_to_e_heterogeneous tests synchrony when the network is not identical (random per-neuron drive and per-pair coupling). simulate_rtm_two_cell_network makes the phase interaction visible in a two-cell reciprocally excitatory network. simulate_rtm_sync shows a network started from identical states, giving the synchronous reference raster. simulate_rtm_splay initializes evenly spaced phases to display the asynchronous splay configuration. All the network examples share one rtm_init helper that finds the single-cell RTM limit cycle and interpolates phase-spread initial conditions from it.

What to look for

Compare the aligned raster of simulate_rtm_sync with the evenly staggered spikes of simulate_rtm_splay. In the recurrent network, see whether initially different phases move together over successive cycles. Then inspect the heterogeneous case for phase dispersion or a shifted collective timing rather than assuming perfect coincidence is required.

Suggested order

1. Run simulate_rtm_sync and simulate_rtm_splay as reference initial conditions.
2. Run simulate_rtm_two_cell_network, then simulate_rtm_e_to_e_network and simulate_rtm_e_to_e_network_2.
3. Finish with simulate_rtm_e_to_e_heterogeneous.

Prerequisites and related chapters

Chapter 20 develops the fast chemical synapse used here. Chapter 23 shows how external pulses entrain one neuron; Chapters 25--29 reduce related network questions to PRCs, phase maps, weak coupling, and stability.

Running the examples

Open chapter24.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The simulate_rtm_e_to_e_network_2 cell is JIT-compiled with numba, so after the first (one-time compile) call the long recurrent run takes seconds instead of minutes.