chapter23.md

Entrainment by excitatory input pulses

Overview

Periodic excitatory pulses can make a neuron's spikes adopt the drive's timing. The examples compare LIF and WB neurons, display one-to-one and n-to-one responses, and use phase return maps to distinguish locking from irregular responses.

Core ideas

Entrainment is phase locking to an external period $T$. In one-to-one locking, one spike is associated with each input cycle; n-to-one locking has n input cycles per neuronal cycle or, depending on the event convention, a repeating multi-cycle relationship. A return map takes the phase after one pulse to the phase after the next; a stable fixed point predicts a repeatable phase, while a nonconvergent orbit predicts irregular timing.

Essential model

If $\alpha_k$ is the phase of the kth relevant spike relative to the pulse train, the pulse response gives a map

$$ \alpha_{k+1}=F(\alpha_k)\pmod 1. $$

The detailed WB simulations implement the pulse through a synaptic gate, while the compact map scripts plot $F$ directly.

Code examples

All seven examples now live in one notebook, chapter23.ipynb: simulate_lif_entrainment shows periodic pulses entraining an LIF neuron. plot_f_entrainment plots a phase return map and its intersections with the identity line. simulate_f_entrainment_2/plot_f_entrainment_2 iterate a second return-map construction from a chosen initial phase. simulate_wb_entrainment_intervals measures WB spike intervals under periodic excitation, sweeping the synaptic strength; its inner loop is JIT-compiled with numba. simulate_wb_neuron_entrained gives WB trajectories with a stable locked response and phases relative to the pulse period. simulate_wb_neuron_irregular demonstrates a WB response whose pulse-relative timing does not settle into the same pattern. simulate_wb_neuron_n_to_one illustrates an n-to-one WB entrainment pattern.

What to look for

On a return map, a fixed point lies where $F(\alpha)=\alpha$; repeated iterations should approach it only when it is stable. In the WB plots, compare the sequence of pulse-relative spike phases: a locked response repeats, whereas the irregular trace drifts. For n-to-one locking, count pulse intervals rather than assuming every pulse evokes a spike.

Suggested order

1. Run simulate_lif_entrainment, plot_f_entrainment, and plot_f_entrainment_2.
2. Run simulate_wb_neuron_entrained and simulate_wb_entrainment_intervals.
3. Contrast simulate_wb_neuron_irregular with simulate_wb_neuron_n_to_one.

Prerequisites and related chapters

Chapter 20 supplies the synaptic-gate mechanism and Chapter 7 supplies the LIF reset model. Chapters 25--27 make the same phase-map viewpoint explicit for PRCs, coupled oscillators, and delays.

Running the examples

Open chapter23.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The simulate_wb_entrainment_intervals cell is JIT-compiled with numba, so after the first (one-time compile) call it takes well under a minute instead of the uncompiled sweep's roughly an hour.