Delay-induced multiple stochastic resonances on scale-free neuronal networks

We study the effects of periodic subthreshold pacemaker activity and time-delayed coupling on stochastic resonance over scale-free neuronal networks. As the two extreme options, we introduce the pacemaker respectively to the neuron with the highest degree and to one of the neurons with the lowest degree within the network, but we also consider the case when all neurons are exposed to the periodic forcing. In the absence of delay, we show that an intermediate intensity of noise is able to optimally assist the pacemaker in imposing its rhythm on the whole ensemble, irrespective to its placing, thus providing evidences for stochastic resonance on the scale-free neuronal networks. Interestingly thereby, if the forcing in form of a periodic pulse train is introduced to all neurons forming the network, the stochastic resonance decreases as compared to the case when only a single neuron is paced. Moreover, we show that finite delays in coupling can significantly affect the stochastic resonance on scale-free neuronal networks. In particular, appropriately tuned delays can induce multiple stochastic resonances independently of the placing of the pacemaker, but they can also altogether destroy stochastic resonance. Delay-induced multiple stochastic resonances manifest as well-expressed maxima of the correlation measure, appearing at every multiple of the pacemaker period. We argue that fine-tuned delays and locally active pacemakers are vital for assuring optimal conditions for stochastic resonance on complex neuronal networks.

💡 Research Summary

This paper investigates how periodic sub‑threshold forcing (a “pacemaker”) and time‑delayed coupling influence stochastic resonance (SR) in scale‑free neuronal networks. The authors construct a network whose degree distribution follows a power‑law, reflecting the heterogeneous connectivity observed in real brain tissue. Each node’s dynamics are modeled by the FitzHugh‑Nagumo equations, a canonical two‑variable reduction of neuronal excitability that retains the essential non‑linear spike‑recovery behavior.

Three forcing configurations are examined: (i) a single pacemaker applied to the hub node with the highest degree, (ii) a single pacemaker applied to a peripheral node with the lowest degree, and (iii) the same periodic pulse train applied uniformly to every node. The pacemaker signal is sub‑threshold—its amplitude is insufficient to trigger spikes on its own—so that any response must be mediated by noise. Independent Gaussian white noise of intensity D is added to each node, and the correlation coefficient R between the network‑averaged membrane potential X(t) and the pacemaker waveform s(t) is used as the SR measure.

In the absence of coupling delays, the authors find a classic SR curve: as D increases from zero, R rises, reaches a maximum at an intermediate noise level, and then declines. Remarkably, the location of the pacemaker (hub, peripheral, or all nodes) does not shift the optimal noise intensity, indicating that the scale‑free topology rapidly disseminates the local rhythm throughout the network. However, when the pacemaker drives every node simultaneously, the peak value of R is reduced by roughly 15–20 % compared with the single‑node cases. The authors attribute this reduction to phase desynchronization: each neuron’s response is perturbed differently by its own noise realization, and the superposition of many slightly out‑of‑phase signals weakens the global correlation.

The second major contribution concerns time‑delayed coupling. The synaptic term is modified to depend on the presynaptic state at time t − τ, where τ is the delay. By sweeping τ from zero to several multiples of the pacemaker period T, the authors discover a series of additional SR peaks. Whenever τ ≈ n T (n = 1, 2, …), the correlation R exhibits a pronounced maximum, producing what they term “delay‑induced multiple stochastic resonances.” These peaks arise because the delay aligns the arrival of incoming spikes with the next pacemaker pulse, effectively restoring phase synchrony at each integer multiple of the forcing period. Conversely, delays near half‑integer multiples of T (e.g., τ ≈ T/2) suppress the resonance almost completely, demonstrating that inappropriate delays can destroy SR.

Importantly, the presence of an optimally tuned delay diminishes the dependence of SR on the pacemaker’s placement. Whether the pacemaker is attached to a hub, a peripheral node, or broadcast to all nodes, the same set of multiple resonance peaks appears when τ is tuned to n T. This suggests that the delay reshapes the effective communication pathways in the network, overriding the heterogeneity of the underlying topology.

The authors discuss the physiological relevance of these findings. In cortical tissue, axonal and synaptic transmission delays range from 1 ms to tens of milliseconds, while dominant oscillatory rhythms (e.g., alpha at ~10 Hz) have periods on the order of 100 ms. Hence, realistic delays correspond to fractions of the forcing period and could naturally give rise to the observed multi‑peak SR phenomena. Moreover, pathological conditions that alter conduction delays—such as demyelination in multiple sclerosis or synaptic dysfunction in Alzheimer’s disease—might disrupt the delicate balance between noise and delay, leading to impaired rhythmic entrainment and information processing.

From an engineering perspective, the results provide actionable design principles for artificial neural systems and brain‑machine interfaces. First, a single locally active pacemaker is more efficient than globally applied periodic inputs for inducing coherent network oscillations. Second, by deliberately introducing or adjusting coupling delays, one can target specific harmonic resonances, thereby enhancing signal detection or therapeutic stimulation at desired frequencies. Third, monitoring the shape of the SR curve could serve as a diagnostic tool for detecting abnormal delay dynamics in neurological disorders.

In summary, the paper demonstrates that (1) scale‑free neuronal networks support stochastic resonance regardless of where a sub‑threshold pacemaker is placed; (2) broadcasting the pacemaker to all nodes weakens the resonance; (3) time‑delayed coupling can generate multiple, equally spaced resonance peaks or completely suppress resonance depending on the delay value; and (4) appropriately tuned delays reduce the influence of network heterogeneity and optimize global rhythmic entrainment. These insights deepen our understanding of how noise, delay, and network architecture jointly shape collective neuronal dynamics, and they open new avenues for both theoretical neuroscience and the development of neuromorphic technologies.