New Tools Predict How Neurons Sync in Networks

A new framework from researchers at KU Leuven and the University of Cambridge adapts classical engineering tools to better understand and predict rhythmic behaviors in networks of excitable neurons, which are fundamental to biological systems like the brain and have applications in robotics and neuromorphic engineering. Traditionally, s like the describing function (DF) and phase response curve (PRC) have been used to analyze oscillations in systems with smooth, harmonic dynamics, but they fall short when applied to excitable neurons that generate discrete spikes in response to stimuli rather than continuous oscillations. This limitation has hindered progress in designing reliable rhythmic circuits, such as central pattern generators for locomotion, where precise timing is crucial. The new approach shifts focus from harmonic oscillations to event-based interactions, providing a more accurate model for neuronal networks.

The key finding is that by treating excitable systems as discrete-event models, researchers can define an event describing function (eDF) and an event phase response curve (ePRC) that capture the timing relationships between input and output spikes. The eDF measures the relative phase shift between periodic input events and the resulting output events, expressed as ϕ(T) = Δ(T)/T, where Δ(T) is the event delay and T is the input period. For example, in simulations with Hodgkin-Huxley neurons, the eDF shows that inhibitory nodes have a longer event onset compared to excitatory nodes, with specific periods like T = 25 for inhibitory and T = 15 for excitatory cases, as illustrated in Figure 2. This allows predictions about network oscillations, such as in ring networks where the sum of relative phases must equal one for a rhythm to exist.

Ology involves modeling excitable neurons as input-output systems where presynaptic voltage spikes (input events) map to postsynaptic spikes (output events) through a synapse-neuron node, as depicted in Figure 1b. The synapse acts as a discrete-to-analog converter, turning events into currents, while the neuron and a threshold function act as an analog-to-discrete converter, producing output events. This framework assumes 1:1 phase-locking, where each input triggers exactly one output, and uses numerical integration to compute eDF and ePRC from simulations. For instance, the eDF is derived by simulating across a range of input periods, and the ePRC is calculated by applying periodic synaptic perturbations to a nominal event oscillation and measuring the induced phase shifts.

From the paper demonstrate the predictive power of these tools. Figure 3 shows eDF curves for inhibitory and excitatory nodes, revealing that inhibitory nodes have a resting period Tr above which the neuron fully recovers, while excitatory nodes may exhibit other phase-locked modes or phase slips. Table I compares predicted and simulated network periods for various configurations, such as an I-I ring (inhibitory-inhibitory) with a predicted period of 25.31 ms versus a simulated 25.43 ms, showing close alignment. The ePRC analysis in Figure 5b illustrates how excitatory and inhibitory perturbations affect phase shifts, with zero crossings indicating potential phase-locking equilibria, and simulations in Figure 6 confirm entrainment behaviors in ring networks under different forcing frequencies.

Of this work are significant for designing and controlling rhythmic networks in practical applications like neuromorphic engineering and robotics. By providing simple graphical s, such as using eDF curves to predict oscillation periods in ring networks, engineers can more easily design central pattern generators for tasks like locomotion without relying on complex harmonic assumptions. The framework also offers insights into how sensory inputs can fine-tune timing in biological systems, as seen in the ePRC analysis where perturbations adjust spike timing based on phase. This could lead to improved robotic controllers that mimic natural rhythmic behaviors, leveraging the event-based nature of excitable systems for more efficient and adaptive performance.

Limitations of the approach include its focus on 1:1 phase-locking modes, which may not capture all behaviors in excitable networks, such as higher-order locking or sustained inhibition in inhibitory nodes at high input frequencies. The paper notes that predictions can become inaccurate in the nonlinear regime of small periods due to modeling mismatches, like differences in spike width between excitation and rebound spikes, as mentioned in Section IV-B. Additionally, the monotonicity of eDF curves, which aids stability predictions, may not generalize to all excitable models, and future research is needed to extend the framework to burst-excitable systems and more complex architectures. Despite these constraints, the tools offer a foundational step toward better understanding and engineering of rhythmic neuronal networks.