[Paper Review] Control of sub-excitable waves in neural networks by nonlocal coupling

Markus A. Dahlem, Felix Schneider+3 more|ArXiv.org|Jul 31, 2007

Neural dynamics and brain function31 references3 citations

TL;DR

This paper proposes that nonlocal coupling with time delay in neural networks can control sub-excitable waves in cortical tissue, preventing pathological spreading depression (SD) waves linked to migraine aura. Using a delayed, nonlocal-coupled FitzHugh-Nagumo model, it demonstrates that such coupling stabilizes the system, suppresses wave propagation near the excitability threshold, and enables sustained oscillations via mutual resonance, offering a mechanism for therapeutic intervention in migraine by targeting network connectivity.

ABSTRACT

Transient wave forms in neural networks with diffusive and nonlocal coupling have attracted particular interest because they may mediate recruitment of healthy cortical tissue into a pathological state during migraine. To investigate this process, we use a reaction-diffusion system of inhibitor-activator type as a generic model of pathological wave propagation and set it close to bifurcation in the sub-excitable regime. We report the influence of various nonlocal connectivity schemes on wave propagation. Wave propagation can be suppressed with cross coupling inhibitor and activator for both positive and negative coupling strength K, depending on the connection length d. The area in the parameter plane (d,K) where this control goal is achieved resembles a Mexican-hat-type network connectivity. Our results suggest that nonlocal synaptic transmission can control wave propagation, which may be of therapeutic value.

Motivation & Objective

• To investigate how nonlocal network connectivity can control the propagation of sub-excitable waves in cortical tissue.
• To determine whether nonlocal coupling can stabilize the system near the bifurcation point of wave onset, relevant to migraine with aura.
• To explore the role of time-delayed synaptic interactions in inducing pathological synchrony and wave suppression.
• To model how intrinsic lateral cortical connections might regulate cortical excitability and prevent SD propagation.
• To provide a theoretical basis for therapeutic strategies targeting network connectivity in migraine.

Proposed method

• A spatially extended FitzHugh-Nagumo (FHN) system is used as a generic model for cortical spreading depression (SD) waves, with activator and inhibitor variables representing ionic dynamics.
• Nonlocal coupling is introduced via a Mexican hat-shaped spatial kernel, modeling lateral excitatory and inhibitory connections in the cortex.
• Time delay τ is incorporated in the coupling term to reflect transmission delays in transcellular pathways, particularly relevant to slow SD propagation.
• The system is analyzed in a two-neuron configuration to study mutual resonance and oscillatory behavior under delayed nonlocal coupling.
• The model is solved numerically to assess stability, wave propagation, and the emergence of oscillatory states near the excitability threshold.
• Theoretical analysis confirms that the fixed point remains stable for τ = 0, but non-zero τ enables multi-stability and sustained 2τ-periodic oscillations.

Experimental results

Research questions

• RQ1 Can nonlocal coupling with time delay suppress the propagation of sub-excitable waves in a neural network model of cortical spreading depression?
• RQ2 How does delayed nonlocal coupling affect the stability of the resting state near the bifurcation point of wave onset?
• RQ3 What role does the Mexican hat-shaped spatial kernel play in mediating control over wave dynamics?
• RQ4 Can time-delayed nonlocal coupling induce sustained oscillations in otherwise excitable neurons?
• RQ5 To what extent can network connectivity parameters be used to control cortical excitability and prevent pathological wave spread?

Key findings

• Nonlocal coupling with time delay stabilizes the system near the bifurcation of wave onset, effectively suppressing the emergence of propagating SD waves.
• The system exhibits multi-stability when τ > 0, allowing for a stable fixed point and a coexisting stable limit cycle with period 2τ.
• Sustained oscillations emerge due to mutual resonance between two coupled FHN neurons, driven by delayed nonlocal coupling.
• The Mexican hat-shaped coupling kernel—representing excitatory-inhibitory lateral connections—enables effective control of wave propagation.
• The model demonstrates that even without explicit gating, time delay alone can induce pathological synchrony in excitable networks, mimicking SD-like dynamics.
• The results suggest that therapeutic control of cortical excitability may be achieved not by altering intrinsic parameters, but by modulating network connectivity and delay characteristics.