Over-Synchronization in Neural Networks: When Too Much Connectivity Kills Oscillations

February 10, 2025 | Marcel Blattner

Introduction

Neural networks are fascinating systems where the balance between individual neuron activity and collective behavior is crucial. While synchronization is essential for normal brain function, too much synchronization can be problematic. This post explores how excessive connectivity in neural networks can lead to a phenomenon called "oscillation suppression" or "amplitude death," where strong coupling between neurons actually quenches their individual oscillatory behavior.
You can find the code here: https://github.com/marcelbtec/gapjunctionNet

The Model: Izhikevich Neurons on an Erdős–Rényi Network

I studied a network of Izhikevich neurons, which are described by these equations:

$
\frac{dv_i}{dt} = 0.04v_i^2 + 5v_i + 140 - u_i + I_{ext} + I_{coupling,i}
$

$$
\frac{du_i}{dt} = a(bv_i - u_i)
$$

Where:
- $v_i$: membrane potential of neuron $i$
- $u_i$: recovery variable of neuron $i$
- $a, b, c, d$: parameters governing intrinsic dynamics
- $I_{ext}$: external current
- $I_{coupling,i}$: coupling current from other neurons

The neurons are connected in an Erdős–Rényi random graph, where each pair of neurons has a probability $p$ of being connected. The coupling between neurons is diffusive:

$$
I_{coupling,i} = \frac{g}{k_i} \sum_{j=1}^N A_{ij}(v_j - v_i)
$$

Where:
- $g$: coupling strength
- $k_i$: degree of neuron $i$
- $A_{ij}$: adjacency matrix (1 if connected, 0 otherwise)

Key Findings

1. Mean-Field Behavior

When the network is densely connected, I can describe its collective behavior using mean-field variables:

$$
V(t) = \frac{1}{N} \sum_{i=1}^N v_i(t) \quad \text{(Average voltage)}
$$

$$
U(t) = \frac{1}{N} \sum_{i=1}^N u_i(t) \quad \text{(Average recovery variable)}
$$

In the synchronized state, the mean-field equations reduce to:

$$
\frac{dV}{dt} = 0.04V^2 + 5V + 140 - U + I_{ext}
$$

$$
\frac{dU}{dt} = a(bV - U)
$$

2. Stability Analysis

The stability of the synchronized state depends on the coupling strength and network connectivity. Through linear stability analysis, I found that perturbations decay when:

$$
\Lambda_\perp = A_V - g'\lambda_2 < 0
$$

Where:
- $A_V$: voltage stability coefficient
- $g'$: effective coupling strength
- $\lambda_2$: second smallest eigenvalue of the graph Laplacian

3. The Over-Synchronization Threshold

The analysis revealed a critical threshold beyond which increasing connectivity leads to oscillation suppression. This happens because:

1. Strong diffusive coupling forces neurons to follow the average state
2. Individual oscillatory trajectories are suppressed
3. The network settles into a quiescent state

Conclusion

The study provides a theoretical framework for understanding how excessive connectivity can lead to oscillation suppression in neural networks. The findings suggest that biological networks maintain sparse connectivity not just for efficiency, but to preserve their dynamic functionality.

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References

[1] Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Transactions on Neural Networks, 14(6), 1569–1572.

[2] Ermentrout, G. B., & Kopell, N. (1990). Oscillator death in systems of coupled neural oscillators. SIAM Journal on Applied Mathematics, 50(1), 125–146.

[3] Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 143(1–4), 1–20.

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