Hybrid Coupling in Neural Dynamics

• Hybrid coupling models are frameworks that integrate local electrical and non-local chemical synapses with field effects to elucidate complex dynamical regimes in neural networks.
• They employ both diffusive electric coupling and sigmoidal chemical mechanisms to generate phenomena such as coherence, incoherence, and chimera states.
• The models offer actionable insights for neuromodulation strategies by controlling synchronization patterns through parameter tuning and targeted external stimuli.

Hybrid coupling models constitute a foundational framework for understanding complex dynamics in thermosensitive neuronal networks where both local electrical and non-local chemical synapses are present. By integrating multiple coupling modalities with intrinsic and extrinsic electric field effects, these models allow for the systematic exploration of phenomena such as coherence, incoherence, and chimera states in spiking neuron populations. The following sections provide a detailed exposition of the mathematical structure, dynamic regimes, and modulatory effects relevant to these hybrid-coupled networks, as reported in recent analyses of thermosensitive networks under diverse electrical field conditions (Nguessap et al., 18 Sep 2025).

1. Mathematical Formulation of Hybrid-Coupled Thermosensitive Networks

At the single-neuron level, each node in the network is modeled using a thermosensitive FitzHugh-Nagumo (FHN) system extended with an intrinsic electric field dynamic. The governing equations for neuron $i$ are: $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$ where $x_i$ is the membrane potential, $y_i$ the ion current displacement, $E_i$ the intrinsic local field, and $A\cos(\omega t)$ specifies periodic external forcing. The coupling terms are critical:

• Electrical coupling (diffusive, nearest-neighbor):

$J_i = d( x_{i+1} + x_{i-1} - 2x_i )$

• Chemical coupling (non-local, sigmoidal):

$$C_i = \frac{\varepsilon}{2p-2}(x_s - x_i) \left\{ \sum_{j=i-p}^{i+p} \Gamma(x_j) - \sum_{j=i-1}^{i+1} \Gamma(x_j) \right\}$$

where

$$\Gamma(x_j) = \frac{1}{1 + \exp[ -\Lambda (x_j - \theta_s) ]}$$

Parameters $d$ and $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$0 govern the strengths of electrical and chemical coupling, respectively; $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$1 is the cell-size-dependent coefficient for intrinsic field influence; $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$2 models time-dependent external field application.

2. Multichannel Coupling: Electrical and Chemical Synapses

The hybrid network topology results from the simultaneous action of:

• Electrical synapses: Fast, local, symmetric interactions promoting rapid voltage equalization amongst immediate neighbors (diffusive Laplacian term).
• Chemical synapses: Slower, non-local, asymmetric interactions mediated via sigmoidal response windows, supporting propagation and signaling over broader spatial neighborhoods. The choice of $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$3 controls the effective coupling range.

Empirically, the interplay of these synaptic types enables the emergence of both simple (fully coherent or incoherent) and complex (chimera, traveling chimera) states.

3. Effects of Intrinsic and Extrinsic Electric Fields

Intrinsic Electric Field (Cell Size, $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$4)

• With $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$5, the intrinsic field vanishes; the system's dynamics are dominated by synaptic coupling.
• For $$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$6, intrinsic local field feedback can induce single-neuron chaotic behavior and desynchronize the network in the absence of compensating chemical coupling.
• Reintroducing strong chemical coupling ($$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$7) stabilizes the system, recovering coherent or traveling chimera states even in the presence of this endogenous heterogeneity.

External Electric Field ($$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$8)

• Low-frequency forcing ($$\begin{aligned} \frac{dx_i}{dt} &= x_i(1 - \xi) - \frac{1}{3}x_i^3 - y_i + I + A\cos(\omega t) + J_i + C_i \ \frac{dy_i}{dt} &= c \left[x_i + a - b\exp(1/T) y_i\right] + r E_i \ \frac{dE_i}{dt} &= k y_i + E_{\mathrm{ext}} \end{aligned}$$9 small): Induces spatially localized synchronization (network “freezing”) in the subset of neurons exposed directly to the external field, while the remainder sustains asynchronized or chimera-like activity. This spatial heterogeneity, when the stimulus is applied to a subregion, enables precise control of the occurrence and localization of chimera and multichimera states.
• High-frequency forcing: Exhibits negligible effect on the underlying neuronal or network dynamics, with neither local coherence nor inhibition of incoherence emerging.

4. Dynamical Regimes: From Coherence to Chimera States

Hybrid coupling induces a suite of dynamical phenomena:

• Full coherence: Synchronization throughout the network, typically in the strong coupling limit.
• Incoherence: No discernible order; each neuron evolves roughly independently.
• Chimera and traveling chimera: Synchronous (coherent) and asynchronous (incoherent) clusters coexist stably; “traveling” chimeras show mobile incoherence boundaries or phase-lagged wave patterns.
• Transition regimes: Tuning the chemical coupling $x_i$0 or electrical coupling $x_i$1 can switch the network between these states.

Diagnostic measures are used to quantify these regimes:

• Local order parameter $x_i$2 measures local synchrony.
• Strength of incoherence (SI): $x_i$3 typically signals chimera, SI$x_i$4 for full coherence, SI$x_i$5 for full incoherence.
• Discontinuity measure (DM): DM=1 for single chimera boundary, DM$x_i$6 for multichimera states.

5. Functional Implications and Computational Modulation

Chemical synapses are demonstrated to be essential for stabilizing traveling chimeras, especially when intrinsic cell properties destabilize activity. Hybrid coupling acts as a control lever:

• Network-level modulation by adjusting chemical/electrical strengths or the frequency and spatial distribution of external fields.
• Induction or suppression of traveling waves and targeted synchronization or desynchronization (potential application to neuromodulation, e.g., epilepsy suppression).

By balancing cell-intrinsic properties (cell size $x_i$7) and global inputs (external field $x_i$8), chimera-like features can be induced or erased; the region of stimulus application, together with $x_i$9 and $y_i$0, governs spatial characteristics of these states.

6. Applications and Outlook

Hybrid coupling models with explicit inclusion of field effects are primed for:

• Studying neuromodulation strategies via selective spatial and temporal field application.
• Investigating biophysical constraints relevant to pathological rhythms and their suppression.
• Designing bio-inspired computational networks with robust heterogeneity management.
• Building theoretical foundations for more comprehensive models that integrate hybrid synaptic/field coupling with plasticity, adaptation, or metabolic feedback.

The current formulation supports further extensions to high-dimensional networks, heterogeneous cell size distributions, and more realistic synaptic kinetics. It lays the groundwork for simulation-guided intervention strategies in real neural systems and neuromorphic platforms.

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Table: Effects of Hybrid Coupling and Field Parameters on Network States

Parameter Changed	Dynamic Regime Impacted	Characteristic State
Increase $y_i$1	Transition incoherent $y_i$2 traveling chimera/coherence	Traveling or phase-shifted synchronization
Increase $y_i$3	Promotes local coherence; may not suppress chaos from large $y_i$4	Coherence (with small $y_i$5), persists in chaos (large $y_i$6)
Cell size $y_i$7 ($y_i$8)	Drives chaotic single-neuron activity	Incoherence unless $y_i$9 high
Low-frequency $E_i$0	Induces spatially localized freezing	Localized coherence (chimera)
High-frequency $E_i$1	Minimal effect	Native network state retained

This comprehensive mapping of the hybrid coupling dynamics underscores the interplay between network topology, synaptic coupling modality, intrinsic neuronal heterogeneity, and external driving, offering a rigorous platform for the investigation, control, and theoretical understanding of complex collective phenomena in neural ensembles (Nguessap et al., 18 Sep 2025).