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Neuromorphic design of a rhythmic network

Robin Wroblowski, Rodolphe Sepulchre

arXiv:2610.39762v1eess.SYcs.SY

TL;DR

Integrating internally generated rhythms with environmental interaction remains a challenge for oscillator-based CPG design. This paper develops local tuning for event-propagating excitable networks using eDF and ePRC, and reports control of period, phase profiles, and entrainment alongside accommodation of heterogeneity.

  • Problem

    Integrating internally generated rhythms with sensory interaction remains an open design challenge, particularly in asynchronous or event-based control schemes.

    Source: §1 · p. 1

  • Method

    The paper designs CPGs as event-propagating excitable networks, using eDF for nodal delay and ePRC for phase sensitivity to guide local tuning.

    Source: §5 · p. 17

  • Results

    Local control of nodal transmission delays modulates global period and relative phases, while local sensitivity modulation tunes cyclic-network entrainment.

    Source: §5 · p. 17

  • Takeaways & Limitations

    Event-driven excitable architectures accommodate heterogeneity, enabling flexible phase–task maps.

    Source: §5 · p. 17

    The illustrations use periodic sequences in ring networks; general architectures and bursting neurons remain areas for further investigation.

    Source: §5 · p. 17

Abstract

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Central Pattern Generators (CPGs) have long inspired the design of rhythms that combine endogenous generation and exogenous interaction. In neurodynamics and engineering studies, CPGs have been primarily analysed and designed as networks of coupled oscillators. In contrast, the present article regards CPGs as networks that propagate events through excitable rather than oscillatory nodes. The proposed event-based perspective reduces the design of network-level rhythmic properties to the tuning of local nodal input-output characteristics. This reduction enables a distributed and robust design methodology: nodes can be tuned independently, they can be heterogeneous, and the global rhythm results from local event propagation rather than from a specific global tuning of the network parameters. The present article builds on previously introduced concepts that parallel existing characterizations of oscillators: the event describing function (eDF), which characterises the propagation delay of a node, and the event phase response curve (ePRC), which characterises the phase sensitivity of a node. Event-based CPGs made of excitable rather than oscillatory nodes emerge as a principled design framework for robust rhythm generation.

1 Introduction

The introduction reframes CPG design from coupled oscillators to event propagation among excitable nodes, using local tuning to address both rhythmic generation and interaction with the environment.

1 Introduction

  • Coupled limit-cycle oscillators dominate engineering CPG designs, while integrating internally generated rhythms with sensory interaction remains an open challenge, especially in asynchronous or event-based control.Oscillator dynamics and coupling determine synchronization and phase relationships, but movement also depends on sensory feedback, reflexes, body mechanics, and descending commands.Source: §1 · pp. 1–2
  • Event-based CPGs use excitable nodes to represent endogenous rhythm generation and environmental interaction through the same discrete-event framework.The approach treats event-based design as hierarchical, combining rhythm generation and environmental regulation at any scale.Source: §1 · p. 2
  • Unlike oscillators, excitable nodes rest at equilibrium until triggered, so local event processing avoids competing intrinsic frequencies and permits independent tuning despite heterogeneity.This replaces joint balancing of nodal and network parameters with local design rules.Source: §1 · p. 2
  • Directed inhibitory rings can produce sequential rhythms through post-inhibitory rebound while every neuron remains quiescent in isolation.The paper preserves discrete spike-based event propagation rather than representing collective activity with continuous firing-rate dynamics.Source: §1 · p. 2
  • Building on prior work, the article develops eDF to quantify nodal propagation delay and ePRC to quantify phase sensitivity as tools for designing and regulating heterogeneous CPGs.The introduction positions these tools as extensions of oscillator analysis to excitable, non-oscillatory networks.Source: §1 · pp. 2–3

2 Motivating example

The section frames excitable CPGs as event-propagating alternatives to coupled oscillators, with rhythms organized by local node responses. Clock-ring examples motivate robustness to heterogeneity and localized timing control.

2 Motivating example

  • The same Hodgkin–Huxley model can operate as an autonomous oscillator or as an excitable system that rests until an input event triggers an output event.The two operating regimes differ by constant input current, but have distinct input-output properties.Source: §2 · p. 3, Figure 1 · p. 3

2.1 Network design of a clock: sequential ring activation

  • A unidirectional ring propagates events sequentially as each active node triggers its downstream neighbour; inhibitory release can trigger post-inhibitory rebound spikes.The four-node ring is used as a clock-like design example, and its principles extend to rings of any size N.Source: §2.1 · pp. 3–4, Figure 2 · p. 4
  • Excitatory coupling between oscillatory nodes tends toward synchrony, while inhibitory rings can have multiple stable firing patterns sensitive to initial conditions and perturbations.A winner-take-all inhibitory layer combined with a unidirectional excitatory ring can enforce sequence, but oscillatory networks still require precise global tuning.Source: §2.1 · p. 4

2.2 Robustness to heterogeneity

  • Above h > 10%, oscillatory networks show pronounced fragility, while excitable networks retain functional rhythms across a much wider heterogeneity range.Success required a sustained sequential pattern across 400 trials at each heterogeneity level.Source: Figure 3 · p. 5, §2.2 · p. 5

2.3 Local modulation

  • A local synaptic change alters only the targeted delay in the excitable ring, but shifts effective delays across all nodes in the oscillatory network.The excitable network's local delay is increased from 20 ms to 40 ms in the modulation experiment.Source: §2.3 · p. 5, Figure 4 · p. 6

2.4 Local analysis

  • Because event propagation reduces global dynamics to isolated local transitions, network behaviour can be analyzed and designed through individual nodes’ input-output responses.The section presents this locality as the basis for a distributed framework to analyze, design, and tune event-driven rhythms.Source: §2.4 · pp. 5–6

3 Endogenous rhythm generation

Endogenous rhythms are generated by inhibitory rebound neurons that propagate events, with each node acting as a tunable delay. The section develops local event-map design and shows how nodal parameters can shape network period, phase profile, and task timing.

3.1 Mechanism of rhythm generation

  • Inhibitory rebound neurons can propagate events with adjustable delays, and the paper focuses on this mechanism for rhythm generation.Release from inhibition can elicit an event; the model also uses IA to control spike-generation latency.Source: §3.1 · p. 6

3.2.1 The event map and event describing function

  • The event map converts input events through synaptic conductance and neuronal membrane potential into output events; the eDF maps input period to steady-state nodal delay.The eDF assumes 1:1 phase locking, with each input event triggering exactly one output event.Source: §3.2.1 · p. 7

3.2.2 The eDF curve

  • The eDF is computed by sweeping input periods and extracting steady-state delay; above Tr, delay reaches ∆∞, while sufficiently fast inputs can break 1:1 entrainment.When Ttot > Tr, the fully reset regime lets the exact steady-state delay be obtained from a single isolated transient response.Source: §3.2.2 · pp. 7–8

3.3 Tuning the event map

  • τsyn provides fine, linear delay tuning, whereas gA provides high-dynamic-range exponential tuning through control of rebound latency.The paper distinguishes external synaptic tuning from internal neuronal tuning.Source: §3.3 · p. 8

3.4.1 Tuning the global period

  • τsyn tunes the network period smoothly and nearly linearly, while gA spans a wider range with exponential scaling; transitions remain phase-locked and stable.These control effects are shown for homogeneous and single-node heterogeneous modulation.Source: §3.4.1 · p. 9

3.4.2 Tuning the phase profile

  • Local parameter changes reshape the phase profile, including by shortening n0 and n2 delays while preserving n1 and n3 delays, without destabilizing event generation.The example changes gA and τsyn to reach the target profile; independently modulated delays also allow duty-cycle reshaping.Source: §3.4.2 · p. 10

3.4.3 Modular design and phase clustering

  • Clustering multiple nodes per task exposes initiation, termination, or sub-phases to local timing control, allowing duty-cycle adjustment and overlapping activations.A two-node cluster can set task onset and offset separately; three nodes can represent heel-strike, mid-stance, and toe-off.Source: §3.4.3 · p. 11

3.5 Robustness and failure modes of the event map

  • The design assumes sparse enough input events (Ttot > Tmin) and relies on inhibitory rebound; robustness depends on locally designing rebound dynamics.The authors propose making nodal delay sensitive to a few currents for tunability and insensitive to most for robustness.Source: §3.5 · p. 11

4 Exogenous rhythm regulation

The event phase response curve (ePRC) measures how periodic external events shift an excitable node’s timing, linking local phase sensitivity to entrainment. Covarying intrinsic and synaptic parameters can tune that sensitivity while preserving nominal timing, and heterogeneous tuning can shape which network phases respond to feedback.

4.1.1 Definition

  • The ePRC maps a periodic perturbing event sequence to the steady-state timing shift of an excitable node, and its amplitude bounds entrainment under period mismatch.Stable phase-locking offsets satisfy ePRC(δp) = Tp − Tnom; detuning beyond the curve’s peak prevents phase-locking.Source: §4.1 · p. 11, §4.1.1 · p. 12

4.1.2 Contrast to phase response curves analysis

  • Unlike iPRCs and iIRCs, the ePRC maps responses to an independent periodic input sequence into an open-loop input–output timing relation for forced phase-locking.The iPRC concerns pulses to autonomous limit-cycle oscillators, while the iIRC describes continuous-time shifts after an isolated state perturbation.Source: §4.1.2 · p. 12

4.1.3 Excitatory vs. inhibitory ePRC

  • Excitatory perturbations advance timing and inhibitory perturbations delay it in the modeled neuron and synapse, enabling bidirectional tuning of network phase shifts.The authors caution that this pattern is model-dependent; other neuronal models or parameter choices may show regions with the opposite sign.Source: §4.1.3 · pp. 12–13, Figure 11 · p. 13

4.1.4 Phase resetting and entrainment dynamics

  • A period-mismatch shift of the ePRC predicts stable phase-locking offsets, and the excitatory entrainment region matches the curve’s maximum, about 5.5 for gsyn = 0.05.The example has Tp − Tnom = −4 ms and locks at δp ≈ 9; Figure 12 presents the sample raster and entrainment region.Source: §4.1.4 · pp. 13–14, Figure 12 · p. 14

4.2.1 Single-parameter tuning

  • Increasing gA delays, deepens, and widens the ePRC exponentially, whereas increasing τsyn delays the output and shifts the curve’s peak to later offsets.The gA-related peak stays at essentially the same phase offset; τsyn lengthens the fully inhibited interval, when the node is insensitive.Source: §4.2.1 · p. 14, Figure 13 · p. 14

4.2.2 Covaried tuning

  • Covarying increased gA with decreased τsyn deepens phase advance while keeping the ePRC’s right boundary fixed, separating external sensitivity from nominal onset timing.The covariation uses a Hill-type sigmoid for gA and a linear decrease in τsyn.Source: §4.2.2 · p. 15, Figure 14 · p. 15

4.3.1 Homogeneous tuning

  • Uniform covaried tuning lets a previously insensitive network entrain to periodic inputs, while equal sensitivity across nodes yields N phase-shifted entrainment equilibria.Figure 15 contrasts the temporal nodal delays under homogeneous tuning.Source: §4.3.1 · p. 16, Figure 15 · p. 16

4.3.2 Single-node heterogeneous tuning; 4.3.3 Clustered and functional tuning

  • Making only n0 sensitive lets the network lock through a single phase, while clustered tuning can reserve distinct phases or execution windows for sensitive transitions and insensitive durations.Functionally grouped nodes can gate environmental feedback to selected phases without altering network connectivity.Source: §4.3.2 · p. 16, §4.3.3 · p. 17

5 Conclusion

The paper presents a distributed event-based framework for generating rhythms with excitable, non-oscillatory CPG neurons. It uses local transmission delays and sensitivities to tune rhythmic sequences and their interaction with the environment, while identifying the ring-network examples as a scope limitation.

5 Conclusion

  • The framework generates CPG rhythms through event propagation in excitable, non-oscillatory neurons rather than coupling oscillators.The authors aim for modular, robust, and scalable analysis and design of rhythmic sequences.Source: §5 · p. 17
  • The eDF tunes global period and relative phases through local nodal transmission delays, while the ePRC tunes network entrainment through local nodal sensitivity.These address internal rhythm generation and interaction with the external environment, respectively.Source: §5 · p. 17
  • The illustrations cover periodic sequence generation in a ring network; the authors identify bursting neurons and excitation-inhibition topologies as areas for further study.The paper describes its ring-network restriction as illustrative and says the methodology should be investigated in more general architectures.Source: §5 · p. 17

Funding

The research received funding from the European Research Council through the Advanced ERC Grant SpikyControl.

Funding

  • The European Research Council’s Advanced ERC Grant SpikyControl n.101054323 funded the research leading to these results.Source: Funding · p. 17

A.1 Discrete-event node

Each discrete-event node converts incoming event timings into synaptic current, drives an excitable neuron, and detects threshold-crossing output events.

A.1 Discrete-event node

  • The node pipeline converts input events into synaptic current, drives a conductance-based neuron, and emits an event when membrane potential crosses a predefined threshold.The three stages are an event-to-analog synapse, a continuous-time neuron, and an analog-to-event detector.Source: §A.1 · p. 18

A.2 Conductance-based mixed-feedback model

The network simulations use a reduced two-dimensional conductance-based neuron with fast membrane-potential and slower recovery variables. Its current components and gating functions provide separately tunable excitability and rebound behavior.

A.2 Conductance-based mixed-feedback model

  • The reduced two-dimensional node model uses fast membrane potential Vf and a slower recovery variable Vs.Source: §A.2 · p. 18
  • The membrane equation balances intrinsic ionic, synaptic, and applied currents against capacitance-scaled change in fast voltage.The prose identifies Cm as membrane capacitance, Iapp as externally applied current, and Isyn as incoming synaptic currents.Source: §A.2 · p. 18
  • Separating transient sodium current into high-threshold INa and low-threshold rebound INa,Reb allows independent tuning of direct excitability and post-inhibitory rebound.Source: §A.2 · p. 18
  • Gating variables use a saturating rectified linear unit and independent bias and slope parameters, with gates assigned to fast or slow voltage variables.The paper uses SatReLU for linear control boundaries and specifies which gates depend on Vf or Vs.Source: §A.2 · p. 18
  • The baseline sets Cm = 1.0 µF/cm2 and τs = 5.0 ms, while gA is typically 0.0 and exceeds 1.0 when tuning transmission delays.For oscillatory-regime experiments, the paper disables rebound conductance, sets gA = 5.076 for N = 4, and applies Iapp = 4.0 µA/cm2.Source: §A.2 · pp. 18–19
  • The listed gating kinetics include K activation bias 10.0 with slope 0.05 and Na activation bias 10.0 with slope 0.02.The Na inactivation bias is 30.0 with slope 0.05.Source: §A.2 · p. 18

A.3 Hodgkin-Huxley neuron model

The comparison of continuous and event-driven dynamics uses a Hodgkin–Huxley model augmented with an A-type potassium current. Its operating regime is set by tonic current or discrete synaptic triggers.

A.3 Hodgkin-Huxley neuron model

  • The comparison uses a standard Hodgkin–Huxley model augmented with an A-type potassium current.The model is used to compare classical continuous dynamics with event-driven dynamics in Figure 1.Source: §A.3 · p. 19
  • The oscillatory regime uses Iapp = 15.0 µA/cm2, while the excitable regime uses Iapp = 0.0 with discrete synaptic event triggers.The gating variables follow classical α/β exponential kinetics shifted by +65 mV to place the nominal resting potential near 0 mV.Source: §A.3 · p. 19
  • The model specifies maximal conductances GNa = 120.0, GK = 36.0, GA = 10.0, and GL = 0.3 mS/cm2.Reversal potentials are ENa = 115.0, EK = −12.0, and EL = 10.6 mV.Source: §A.3 · p. 19

A.4 Event-based synapse

The event-based synapse uses a logistic-sigmoid drive based on presynaptic membrane potential and a scaled rise-and-decay mechanism. Network interactions are inhibitory by default, with separate excitatory and inhibitory settings for sensory perturbations.

A.4 Event-based synapse

  • The synaptic drive is a logistic sigmoid based on presynaptic membrane potential Vpre.This choice is described as supporting smooth integration while retaining switch-like activation.Source: §A.4 · p. 19
  • The drive activates synaptic gating variable s through a scaled rise-and-decay mechanism.The applied-current description begins in the supplied passage but is incomplete.Source: §A.4 · p. 19
  • Default network coupling is inhibitory, with gsyn = 3.0 mS/cm2 and Erev = −20.0 mV.Sensory perturbations use gsyn = 0.05 mS/cm2 for excitatory inputs and gsyn = 0.15 mS/cm2 for inhibitory inputs.Source: §A.4 · p. 19

A.5 Event detection

The analog-to-event stage registers an output event when the fast membrane potential crosses a detection threshold while rising. The resulting timestamps form the outgoing event sequence for downstream nodes.

A.5 Event detection

  • An output event is registered when fast membrane potential Vf(t) crosses Vdetect with a strictly positive derivative.The detection threshold is 30.0 mV, or 50.0 mV depending on the parameter regime.Source: §A.5 · p. 19
  • The detected timestamps form outgoing event sequence Eout, which propagates to downstream nodes.This is the final stage of the nodal analog-to-event conversion pipeline.Source: §A.5 · p. 19
  • The detection threshold depends on the amplitude of the chosen parameter regime.The passage gives 30.0 mV and 50.0 mV as the threshold values.Source: §A.5 · p. 19

Appendix B Simulation details

The simulations use custom Python scripts and Forward Euler integration, with time-step size tailored to the analysis. Event timings are refined by interpolating threshold crossings between integration steps.

Appendix B Simulation details

  • The eDF and ePRC characterizations use dt = 0.001 or 0.0001 ms to capture sub-millisecond phase shifts.Extended network simulations use dt = 0.01 ms, or up to 0.025 ms where stability permits.Source: Appendix B · p. 19
  • Threshold-crossing times are calculated between integration steps by linear interpolation to eliminate discretization artifacts in event registration.The event threshold is Vdetect.Source: Appendix B · p. 19
  • All neuronal models and network architectures are simulated with custom Python scripts using Forward Euler integration.Simulation routines are accelerated with Just-In-Time compilation using Numba.Source: Appendix B · p. 19
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