Coherent Coupling of Critical Neurons

• The paper demonstrates that renormalized field theory captures fluctuation-induced coherent coupling beyond mean-field approximations.
• It shows that population-level slow fluctuations induce dual timescales in autocorrelation functions of both single neurons and the overall network.
• The analysis links microscopic neuronal heterogeneity to macroscopic, scale-free dynamics, providing quantitative predictions for critical slowing down and memory.

Coherent coupling of critical neurons refers to the emergence of strong, network-mediated correlations and long-lived fluctuations among neurons operating near a critical point in dynamical neural network models. This phenomenon is grounded in the failure of mean-field descriptions to capture fluctuation-driven collective dynamics, instead requiring renormalized theories or population-level field-theoretic approaches to systematically explain how macroscopic modes couple to microscopic heterogeneity. Coherent coupling is central to understanding scale-free activity, dynamic memory, and complex correlations seen in both theoretical models and empirical brain data.

1. Theoretical Foundations: From Mean-Field to Renormalized Field Theory

Traditional mean-field approaches efficiently characterize average network behaviors and phase boundaries but inherently suppress fluctuations—precisely the degrees of freedom that become dominant at or near phase transitions. In neural systems, such fluctuations underlie critical slowing down, multiscale memory, and complex computation.

To address this, renormalized field theory methods are employed. For example, in the context of the Sompolinsky-Crisanti-Sommers (SCS) model for recurrent rate networks,

$$\dot{x}_i + x_i = \sum_{j=1}^N J_{ij} \phi(x_j) + \xi_i,$$

where $J_{ij}$ is a random Gaussian matrix and $\phi$ is a nonlinear activation, the macroscopic activity and its fluctuations are captured by introducing collective auxiliary fields, namely:

• Population activity: $R(t) = \frac{\bar{g}}{N} \sum_{j=1}^N \phi_j(t)$
• Population autocorrelation: $$Q(s, t) = \frac{g^2}{N} \sum_{j=1}^N \phi_j(s) \phi_j(t)$$

Renormalized field theory, via the Martin-Siggia-Rose/Janssen-De Dominicis path integral and a double Legendre transformation, yields a functional of the cumulants (means/covariances) of these fields, generating self-consistency equations for the full, fluctuation-resummed Green's functions.

2. Self-Consistent Green's Functions and Fluctuation Effects

The effective theory is defined via an effective action $\Gamma[\beta_1,\beta_2]$ with self-consistency conditions,

$$\frac{\delta \Gamma}{\delta \beta_1} = j + \beta_1 k = \beta_1 N K,$$

$$\frac{\delta \Gamma}{\delta \beta_2} = \frac{1}{2} k = \frac{1}{2} N K,$$

where $\beta_1$ and $\beta_2$ denote auxiliary field means and covariances, and $K$ encodes quadratic couplings. The solution gives access to:

• Population activity autocorrelation $$\beta_{11}(t,s) = \langle\langle R(t)R(s)\rangle\rangle$$
• Single-unit-induced corrections to both autocorrelation and cross-correlation
• Linear and nonlinear response functions

Crucially, this framework describes how collective network modes—population-level slow fluctuations—couple back to and shape the local statistics of individual neurons, in a way that cannot be factorized as in mean field.

3. Emergence of Coherent Coupling Near Criticality

An essential signature of coherent coupling among critical neurons is the emergence of long-lived, slow-decaying autocorrelations in both population-averaged and single-unit activities as the system approaches the critical point (e.g., $\bar{g} \to 1$ in SCS). Specifically, the renormalized population autocorrelation in the frequency domain,

$$\beta_{11}(\omega) = \frac{1+\omega^2}{(1 - g \phi')^2 + \omega^2} \cdot \frac{g^2}{N} \phi,\phi_*(\omega),$$

develops a slow mode (pole at low frequency) as $g\phi' \to 1$, defining critical slowing down.

This nontrivial, non-self-averaging fluctuation propagates into single-neuron statistics: $$Q^*(s,t) = g^2 \langle \phi^2(s,t) \rangle_* + \frac{1}{2}g^2 \iint \langle \phi^2(s,t), x(u), x(v)\rangle_* \beta_{11}(u,v) + \dots$$ The mean-field term is modulated by a renormalized correction reflecting the collective population fluctuation. As a result, the single-neuron autocorrelation and pairwise cross-correlation functions both develop dual timescales: a fast, private/mean-field decay and a slow, collective/network-induced decay.

Pairwise cross-correlations for single neurons relate directly to population autocorrelation: $$C_{\phi\phi}^x(t-s) = \frac{\beta_{11}(s,t)}{\bar{g}^2 - Q(s,t)/(N g^2)}$$ This demonstrates mathematically that, even in large but finite-size recurrent networks, neurons are coherently coupled via population-level fluctuations and remain correlated over macroscopically long times.

4. Multiscale Dynamics: Temporal Structure and Scaling

A defining feature of coherent coupling is the presence of multiple temporal scales in both collective and local single-neuron observables. The autocorrelation functions exhibit a rapid initial decay (mean-field-like, reflecting private or local noise) followed by an extended tail or plateau (slowly decaying, due to coherent population-level modes). This structure is captured quantitatively only by the renormalized Green’s functions.

As the critical point is approached, the slow timescale (associated with critical slowing down) diverges, and the network becomes increasingly susceptible to small perturbations, supporting memory and integration over extended times.

Finite-size effects are non-negligible even when $N \gg 1$; fluctuation corrections scale as $1/N$, but their integrated effects over time are amplified near criticality due to the divergence of correlation time.

5. Implications: Analytical Bridge, Computation, and Broader Context

The renormalized theory constructed in this framework establishes a direct analytical bridge between macroscopic network-level fluctuations and microscopic single-neuron heterogeneity. It explains several phenomena observed empirically and in simulations that are not captured at the mean-field level:

• Coherent coupling is not self-averaging away, even for large networks, but persists through population modes.
• Population-level critical slowing down drives single-neuron cross-correlation and prolonged response.
• Single-neuron variability (and the pairwise correlation structure) is directly modulated by global, collective fluctuations.

This approach is not limited to the SCS model but generalizes to other recurrent architectures where critical points are approached, especially those involving dynamical phase transitions or rich correlation structures.

6. Mathematical Summary Table

Quantity/Function	Description	Scaling/Critical Point Behavior
$\beta_{11}(t,s)$	Population activity autocorrelation	Slow decay, diverging timescale at criticality
$Q^*(s, t)$	Single-neuron autocorrelation (renormalized)	Dual timescales: fast (local), slow (network)
$C_{\phi\phi}^x(t-s)$	Cross-correlation between two neurons	Proportional to population autocorrelation
$\beta_{11}(\omega)$	Population autocorrelation in frequency domain	Pole near 0 as $g\phi' \to 1$

7. Conclusion and Research Significance

The coherent coupling among critical neurons is a collective network effect derived from fluctuation-induced correlations emerging near phase transitions in recurrent networks. Renormalized field theory, through explicit self-consistency equations for means and large-scale covariance (Green's functions), enables a rigorous mathematical connection between population dynamics and single-neuron heterogeneity. These results imply that at criticality, neuronal dynamics reflect both local and global influences in a non-factorizable manner, supporting phenomena such as long-lived correlations, dynamic memory, and scale-free activity structure. The theory yields concrete, quantitative predictions for autocorrelation and cross-correlation functions at multiple scales, facilitating a unified understanding of criticality and computation in large, yet finite, neural networks (Dick et al., 2023).