Bridging Scales: Spectral Theory Reveals How Local Connectivity Rules Sculpt Global Neural Dynamics in Spatially Extended Networks
Authors: Yuhan Huang
Keren Gao
Dongping Yang
Sen Song
Guozhang Chen
Venue: NeurIPS 2025 poster ()
Keywords: Neural Dynamics
Computational Neuroscience
Spectral Theory
Random Matrix
Effective Connectivity
Recurrent Neural Networks
Primary area: neuroscience_and_cognitive_science

Abstract: The brain's diverse spatiotemporal activity patterns are fundamental to cognition and consciousness, yet how these macroscopic dynamics emerge from microscopic neural circuitry remains a critical challenge. We take a step in this direction by developing a spatially extended neural network model integrated with a spectral theory of its connectivity matrix. Our theory quantitatively demonstrates how local structural parameters, such as E/I neuron projection ranges, connection strengths, and density determine distinct features of the eigenvalue spectrum, specifically outlier eigenvalues and a bulk disk. These spectral signatures, in turn, precisely predict the network's emergent global dynamical regime, encompassing asynchronous states, synchronous states, oscillations, localized activity bumps, traveling waves, and chaos. Motivated by observations of shifting cortical dynamics in mice across arousal states, our framework not only provides a possible explanation for repertoire of behaviors but also offers a principled starting point for inferring underlying effective connectivity changes from macroscopic brain activity. By mechanistically linking neural structure to dynamics, this work advances a principled framework for dissecting how large-scale activity patterns—central to cognition and open questions in consciousness research—arise from, and constrain, local circuitry. The implementation code is available at https://github.com/huang-yh20/spatial-linear-project.

Bridging Scales: Spectral Theory Reveals How Local Connectivity Rules Sculpt Global Neural Dynamics in Spatially Extended Networks
Yuhan Huang 1.2
Keren Gao
Dongping Yang" Sen Song*
Guozhang Chen 1+
School of Computer Science, Peking University, China School of Information Science and Technology, University of Tokyo, Japan Zhejiang Lab, China
Department of Biomedical Engincering, Tsinghua University, China guozhang- chen@pku.eeee....mm edu. cn
Abstract
The brain's 's diverse spatiotemporal activity patterns are fundamental cognition and consciousness, yet how these macroscopic dynamics emerge from microscopic neural circuitry remains a critical challenge. We take a step in this direction by developing spatially extended neural network model integrated with spec- tral theory its connectivity matrix. Our theory quantitatively demonstrates how local structural parameters, such E/I neuron projection ranges, connec- tion strengths, and density determine distinct features of the eigenvalue spectrum, specifically outlier eigenvalues and bulk disk. These spectral signatures, in turn, precisely predict the network's 's emergent global dynamical regime, encom- passing asynchronous states, synchronous states, oscillations, localized activity bumps, traveling waves, and chaos. Motivated by observations of shifting cor- tical dynamics in mice across arousal states, framework only provides a possible explanation for repertoire of behaviors but also offers principled starting point for inferring underlying effective connectivity changes from macro- scopic brain activity. By mechanistically linking neural structure to dynamics, this work advances a principled framework for dissecting how large-scale activity patterns central to cognition and open questions in consciousness research- arise from, and constrain, local eircuitry. The implementation code is available com cww.coo///ooo//e www..oanggahooeeo.oeeo y000 spatial linear -rojject,
Introduction
The brain's 's activity is remarkably diverse, forming complex spatiotemporal patterns that vary with an organism's 's cognitive state and level of conseiousness 114 Propagating waves of neural activity, for example, are observed across numerous brain regions and species, from insects to humans 6). Recent high-resolution imaging in mice, as they transition from anesthesia wakefulness, has highlighted this complexity, revealing large-scale cortical waves during anesthesia and a shift towards more localized, intricate spatiotemporal patterns upon awakening (4.. Such observations emphasize that the spatial organization of neural activity is fundamental aspect of brain function 10 Consequently, key challenge to understand how these macroscopic dynamical states, which distinguish brain states, arise from the underlying microscopic neural architecture and effective connectivity rules. Since many critical brain dynamics are inherently spatial, involving coordinated
"Corresponding author: baleaaann..hen@pheee edu.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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activity across neural tissue, models must incorporate spatial extent to capture these phenomena faithfully [TT]
Previous developments in neural field theory have revealed a variety of rich dynamical phenomena in spatially distributed neural networks, such traveling waves and bump-like activity patterns [8] Meanwhile, previous development of dynamical mean--ieel theory (DMFT) [9H25] has uncovered the emergence of chaotic neural activity in large-scale brain networks. However, A comprehensive framework that quantitatively links a broad range of local network structures to the full spectrum of emergent global dynamics in spatially extended systems is still developing. Current theories often explain particular aspects, but a general theory predicting the emergence of, and transitions between, diverse spatiotemporal patterns from fundamental structural properties across wide parameter space remains critical need. For detailed comparison with prior approaches, including neural field theory, numerical simulation and random matrix theory, please refer to Appendix A.
To address this, we present a unifying theoretical framework centered on a spatially extended recurrent neural network model with excitatory (E) and inhibitory (I) populations. The core of our approach is a spectral analysis of the network's s connectivity matrix. Building upon theories of random matrices [26] 30), we provide an analytical formulation for the spectral bulk arising from connection heterogeneity and, critically, demonstrate that the inherent spatial organization of the network itself constitutes a low-rank structure. This allows to characterize a rich set of outlier eigenvalues, reflecting specific spatiotemporal modes determined by the network's 's geometry. We show that this comprehensive "spectral blueprint" encompassing both the derived bulk and the spatially determined outliers quantitatively links key local structural parameters (spatial reach of E/T connections, relative strengths, local density, weight variability) to full repertoire of global dynamical phases. This framework offers predictive power, enabling to anticipate the network's 's spatiotemporal behavior from its effective connectivity structure. These predicted phases include stable asynchronous states, global synchrony, oscillations, localized bumps, traveling waves, and chaotic dynamics. Our work thus provides a principled understanding of how network structure dictates emergent neural activity.
We first define the spatial E/I network (Section 2) then analytically derive its bulk-plus- outlier spectrum (Section 3.2 Appendix A.13). The spectrum predicts six dynamical phases (Section
Road-map
3.3 Appendix A. & 141. The comparison to experimental data can be found in Section 3.5 3.6 Parameters and numerical validation are in Appendix A.5
A.10
2
Spatial Extended Neural Networks
2.1 Model Definition
To explore the link between structure, dynamics, and the eigenvalue spectrum, we constructed a minimal rate-based neural network model with biologically realistic features. The network consists of excitatory and inhibitory neurons, sparse connectivity that decays with distance, and synaptic weights drawn from a Gaussian distribution (Fig. D
We consider a network consisting of Ng excitatory neurons and Nj inhibitory neurons, where NE .N 4 1. Both excitatory neurons and inhibitory neurons evenly distributed in the region [0, 1) x [0, 1). The dynamics of the neurons are described by the following equation:
dh dt - ΣΣ
Jo o0(h)) (t).
Here, h indicates the membrane potential neurons, p(-) is the activation function of neurons, E a,3 {E,1) }} denotes the neuronal populations, 5" represents the external input received by the
population a .nd J represents the synaptic weights between neurons. The external input is modeled as independent white noise with intensity &u 0.1.
The connections between neurons sparse. The probability of connection between neuron located at x; and 33 neuron located at æj is given by.
Lout Na
p- (x: Iz; Vi v3)
g(0; - zÿ; das)g(us Mÿ; yj; das),

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(a)
Anesthetized
Fully awake
-0.00
(c)
Spatially Extended Neural Networks
(d)
Synaptic Weights
(e)
Connection Probability
Neuron Location (Y)
Exc
Connection Probability p3
2nd No
Probability
JoBl
das
9as/ ad
Synaptic Weight
Neuron location (X)
-0.5
Distance
0.0
0.5
Figure I: (a) and (b) Mesoscopic optical imaging of the anesthetized and full awake mouse cortex [4]
Model schematic. (c) Spatially extended neural network with excitatory (E) and inhibitory (I) neurons embedded in 2D space. (d) Synaptic weights follow distance-dependent Gaussian distribution. (e)
Connection probability follow a distance-dependent wrapped Gaussian distribution.
where
(P1-2,+0)
g(r; - daz) =
2 ==
This indicates that the probability of connection between neurons decays according to wrapped Gaussian profile with characteristic decay length of da Here, koout represents the average out- degree, namely the number of neurons to which single neuron projects.
Two neurons are connected with probability of pe ([z; - zz.).. two neurons are indeed connected, the connection weight follows independent Gaussian distribution with mean of Jas/ /ont and
variance / kap. namely, J Lid. N N kas (kag/)
The activation function o6-) can be arbitrary. In our experiment, we chose the activation function (x(x) = tanh(z) for excitatory neurons and (r() 5 tanh(z/5) for inhibitory neurons for the reason that the saturated firing rate of inhibitory neurons substantially higher than that of excitatory m3l However, the choice of activation function has little impact on the conclusions, and we discuss this detail in the Appendix A.6 In the Appendix A.. we also show the results with the activation function as ( 10 Rt (z) and ør (x) = Rx)
2.2 A Repertoire of Emergent Dynamical Phases
Asynchronous State work's synchronization and stability. When inhibition dominates or excitation and inhibition are balanced (Fig. a)),,, the network exhibits low synchronization and small deviations from the steady firing rate. We define this regime the asynchronous state (Fig. c)). In the asynchronous state, neuronal membrane potentials fluctuate around the fixed point (zero) due to external input. Neural activity remains weakly correlated across neurons, and no spatial patterns emerge. This regime aligns with the classical asynchronous state described by (21]
The balance between excitatory and inhibitory interactions shapes the net-
Synchronous State When excitation dominates (Fig. 2(a)), the network can enter highly synchro- nized regime with large deviations from the steady-state firing rate, which we term synchronous state phase. In this phase, the firing rates of all neurons shift away from the fixed point, and strong global synchronization emerges across the network (Fig (c)). This globally synchronized activity resembles pathological brain states such epileptic seizures 32

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Chaos-related
(a)
dr/dn =
dj/ds 1.4
(b)
phase diagram
1.8
1.8
1.1
Oscilation
Wave
Chaos
Async,
10
Async. State
Bump
Asymc,
0.7
1.2
1.7
0.7
0.9
1.1
0.3
(6v//
(c)
Neural Activity
t 0 ms
80 ms
160 ms
x10
Neural Activity
Async. State
Membrans Potential
Exc.
Neuren lozation (Y)
0.0
300
Time(ms)
Nauran location (X)
Sync. State
Oscillation
100
200
300
Bump
0.0
-1.5
100
200
300
12.
Wave
-6
3.0
Chaos
115
300
600
900
Figure 2: Dynamical regimes in spatially extended networks. (a.b) Phase diagrams under different = Jas, see Appendix A.14 (c) Representative
parameters (In simulation we set ğEL = ğII タ1. OEE Appendix 10. Wave-chaos phase boundaries determined neural activity showing temporal evolution (left) and spatial patterns (right) with shared colorbar; enhanced versions in Appendix A. A.12 show the asynchronous state with individual scaling.
σgI

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Oscillatory Phase Neural oscillations emerge when highly synchronized networks (Fig. Z(a). We term this dynamical regime the oscillatory phase, char- acterized by periodic membrane potential changes and synchronized neural activity (Fig. 2c)). Our identified excitation-inhibition loop mechanism aligns with established experimental and theoretical work. This consistency supports the biological plausibility of our model, as evidenced by prior studies on neural oscillations [[]
coupling strengthens in
Localized Bump Phase Mismatched projection ranges between neuron types generate structured spatial activity patterns. When excitatory neurons project locally while inhibitory neurons extend farther, the system exhibits bump phases and wave phases (Fig. [()).. The bump phase produces localized spatial patterns dependent on activation functions. Neural activity synchronizes within discrete regions, forming stripes (Fig. (c). Rectified linear and threshold power-law functions yield spot-like patterns (see Appendix [A.6), aligning with previous simulation results of spiking neural networks 15 [7.. Similar wavelength- -specccpppciccc patterns occur in juvenile visual cortex (34), with persistent long-range correlations in adulthood [35].
Wave Phase The wave phase features propagating oscillations with location-dependent phase shifts (Fig. (e)) This mirrors biological observations of traveling alpha/gamma waves 6 [5] suggesting our model captures essential mechanisms of spatial-temporal dynamics.
Chaotic Phase High connection sparsity and weight variance drive neural activity into chaotic regime (Fig. bb) and Fig. 9 a)). We term this disordered state the chaos phase, where spatial patterns collapse due to uncorrelated neural firing. Neurons exhibit large-amplitude, weakly correlated fluctuations in the chaos phase (Fig. Z(e)). Membrane potentials vary erratically, matching the second type of asynchronous state described in spiking networks (2I). This aligns with dynamic mean-field -ield theory predictions of chaotic dynamics rate-bat based neural networks (19].
3
The Spectral Blueprint: Decoding Dynamics from Connectivity Structure
3.1 Effective Connectivity and its Eigenvalues
Building on the neuroscience perspective of effective connectivity (EC) model-based measure of directed interactions (36], we propose that the linearized Jacobian matrix -I+ Jo'("") )derivvee from the recurrent neural network's fixed-point dynamics-serves an analogous EC matrix. Here, structural connectivity J is dynamically modulated by nonlinear gains ø' (x"), mirroring how anatom- ical constraints and state-dependent plasticity jointly shape brain network interactions. To unravel how such connectivity shapes collective behavior, we analyze its eigenvalue spectrum, which governs stability and activity patterns. In the following sections, we employ random matrix theory charac- terize this spectrum, revealing universal dynamical regimes emergent from its structure. In the main text, we use the tanh activation function, so the connectivity matrix J can be regarded as the effective interaction matrix. We also present results with alternative activation functions in the appendix A.6 and A.
3.2 Eigenvalue Spectrum of Spatially Extended Networks
The eigenvalue spectrum of spatially distributed neural networks' connectivity matrix J comprises two distinct components: bulk disk region and a set of spectral outliers, which respectively reflect heterogeneous neuron interactions and population-averaee trix decomposition J = J + 6J, where J denotes the expectation matrix and òJ represents zero-mean random fluctuations, we observe that 5J governs the bulk spectrum through its stochastic components while J generates spectral outliers through its low-rank structure. This aligns with the perturbation framework established by (27]. wherein low-rank modifications to random matrices predominantly affect outlier positioning while preserving the original bulk spectral radius (see Appendix 13.2 for details). The dichotomy between these spectral components provides a mathematical characterization of neural network dynamics- the deterministic outliers capture macroscopic interaction features, whereas the bulk spectrum encodes microscopic connection variability.
connectivity patterns, Following the ma-
5

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(a)
|||]||
11.
|
2m
III.
|
2v 2m
Bulk disk
0.3
Bulk disk
Neuron location
mm(A)
0.0
-0.3
0.0
0.4
0.8
Re(A)
Neuron location
(b)
Dominant
Eigenvalues
Eigenvector
Neural Activity
Order Parameters
Real Part
Neural Activity
Async. state
Neuran localion (Y)
x18
0.8
imlA)
-0.8
-4
4
0.9 Mean Act.
Local Sync.
Moran's Index
Osc, Irdex
Re(A)
Neuron (X)
Sync. state
10
Oscillation
2.5
0.0
-2.5
Bump
0.0
-2.5
-3
3
Wave
-6
Chaos
1.5
-1.5
22
0
2
Figure 3: Spectral signatures and spatial patterns of network dynamics. (a) Eigenvalue spectra and corresponding eigenvectors across spatial modes. (b) Example eigenvalue distributions (blue triangles: eigenvalues with the largest real part), leading eigenvectors, neural activity, and order parameters (blue point: data points) for distinct dynamical states. The four order parameters (rightmost column) indicate, from left to right: neural activity fluctuations. local synchrony, spatial patterning, and oscillations. Detailed definitions are given in the Appendix A.4

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3.2.1 Outliers
The matrix J determines the outlier part of the eigenvalue spectrum 27 The expected connectivity between neurons exhibits spatial translational invariance. can decompose the activities neurons into different spatial Fourier modes (See Appendix A.14.).. For each spatial Fourier mode, the effective connectivity matrix is
[| 2 || di 2
M1 2dl 2
J(A) =
DEE exp
DIE exp
(3)
11 de
ØEl exp
911 exp
where K = (2 2mmy): R2: Rvy = 0, #2,
The eigenvalue spectrum of the connectivity matrix reveals distinct outliers generated by spatial Fourier modes different wave vectors k. Each outlier corresponds collective population dynamies mediated by k-specific interaction submatrices. These submatrices encode spatially modulated couplings between excitatory and inhibitory populations, with effective weights determined by three components: (1) baseline inter-population connectivity (Jag between populations and (),, (2)
k-dependent modulation of spatial interaction range exp (1141|2d2.
3.2.2 Bulk Disk
The matrix 8J governs the bulk disk part of the cigenvalue spectrum, of which the radius is determined by the local sparsity of neuronal connections and the variance of the weights. Analytical tools developed in [28 29 characterize the spectral distribution of random matrices with independent, zero-mean, finite-variance entries. Applying these results, we can determine that the eigenvalues are extended within a circle of a specific radius r = / //max (A(M)), where the elements of the matrix M represent the variances of the elements of the connectivity matrix J. The variance of the elements of the connectivity matrix is given by (See Appendix A.13.5 for details),
PaH 2
M E (
0 (Ix, x,1) (1 10-PP- 8 (1===x;1)) -
+ Pe
(|x: 2z1)
Tag houu 0,
kout? 0,
The eigenvalues of this matrix are equivalent to those of the reduced matrix, where the elements rep- resent the heterogeneity of connections between different types of neurons. For the two-dimensional case, this reduced matrix is
NB N
Pap
bout
Map3
+ Gap
kout
4Tdaa
No
Based on the above equations, the radius of the circular part of the eigenvalue spectrum is given by,
MEE + Mn V(MEE M11)² + 4ME
(6)
2
We can observe that the heterogeneous matrix Ma, is composed of two parts: sparsity and variability in synaptic weights. The left half of Equation (Srattion/Srepreeennt the heterogeneity brought about by the sparse connections between neurons, while the right half (qquation[)repressee due the variability in the weights of the neuronal connections. As the local sparsity of neuronal connections and the variance of weights increase, the radius of the bulk disk part of the eigenvalue spectrum also increases.
the heterogeneity
3.3 Linking Spectral Features Dynamical Phases
The eigenvalue spectrum of the effective connectivity matrix Jø'(x") governs network dynamics: the dominant eigenvalue (largest real part) determines stability near fixed point. real part exceeding the critical threshold (Re(Asom > 1) quantifies deviation magnitude, while the dominant eigenvector specifies the spatial activity pattern. We thus classify neural networks into distinct dynamical phases based on their eigenvalue spectrum, summarized in the table I (See Appendix A.1l for details),
7

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Table 1: Dynamical Phases and Spectral Features
Phase
R (dddmm) Condition
Adom Type
Wavenumber (k)
Asynchronous State
R( Adom)
Synchronized State
9 Adom)
2
1
Outliers, Real
k = 0
Oscillatory Phase
R Adom
1
Outliers, Complex
k 0
Bump Phase
R A
1
Outliers, Real
k * 0
Wave Phase
R Adomm)
Outliers, Complex
k o 0
Chaotic Phase
R Adom)
Bulk Disk
3.4
The Role of Key Structural Parameters in Shaping the Spectrum
Balance The magnitude of excitatory and inhibitory interaction affects the magnitude of the real parts of the outlier eigenvalues, which in turn influences the degree of deviation of neural activity from the fixed point and the level of synchronization. As shown in Eq. E] the magnitude of excitatory and inhibitory interactions influences the elements of the effective interaction matrix. As shown Fig. 2a), the more excitatory interaction there is, the larger the real part of the outlier eigenvalues, the greater the degree to which neural activity deviates from the fixed point, and the more synchronized the neural activity becomes; the opposite is also true.
Loop is considered crucial component for the emergence of neural oscillations II 33 Our theory explains this from the perspective of the eigenvalue spectrum. As shown in Fig. a)) the greater the magnitude of the interaction between excitatory and inhibitory neurons, the more likely it is for the outlier eigenvalues to have an imaginary part, causing neural oscillations to emerge(Eq. [ This because 9Iæ and DEL are located on the off-diagonal elements of the effective interaction matrix, and an increase in græ and gEs can lead to the appearance of an imaginary part in the outliers.
DocittationInhibiitee Loop The Excitation-Inhibite
The mismatch of Excitation/Iniibition projection range The mismatch of projection range dap of types of neurons causes the emergence of spatial patterns. As shown in Eq. 3 the elements of the effective matrix ((A) contain decay factors exp (-11k(/d,,,/2)) If all the projection ranges das are the same and let's denote day = d, the eigenvalues would follow the relationship as (k) (0) exp (=] In this case, only the eigenvalues corresponding to the wave vector k = 0 can have the largest real part. However, if the projection range das of types of neurons is mismatched, eigenvalues corresponding to wave vector ls * 0 can have the largest real part.
common way for a spatial pattern to emerge is when the projection range of inhibitory neurons is greater than that of excitatory neurons, which is known as lateral inhibition [35 34]. As shown in the comparison between two figures of Fig. aaa), this mechanism is also applicable in our model. Our theory explains this from the perspective of the eigenvalue spectrum. This is because the elements of
the effective matrix ((h) contain decay factors exp (-11 2 /2) and the elements decay faster with & if da is larger. Therefore, the inhibitory elements of the effective matrix may decay faster than excitatory ones. In some wave vectors k the excitation exceeds the inhibition and the real part of these eigenvalues may exceed I and spatial patterns of neural activity emerge.
Local sparsity Local sparsity, rather than the overall sparsity, plays significant role in chaotic neural activity. In a spatially extended neural network, the number of neurons connected to given neuron certainly sparse compared to the total number of neurons. However, Eq. 5 indicates that what truly determines the dynamics is the ratio of the number of connections a neuron has to the number of neurons within its projection range kaut (TTda NN). namely the "local sparsity" that plays role in the radius of the bulk disk part, which is different from "standard sparsity" kodt /No in the situation without spatial distribution. This suggests that even under globally sparse conditions, the relative density of local connections between neurons in the brain may the reason it can generate synchronized activities such as traveling waves. Besides, the concept of "local sparsity" also demonstrates many phenomena in numerical simulation, including both spike-based and rate- based models. In most numerical simulations that produce neural activity with spatial patterns,
8

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(a)
(c)
Mode
(d)
1.8
XX
x
Wave
Async. State
X
Bump
Mode
1.0
0.7
(b)
More
10
10"
101
10
10"
Variance (%)
10P
Mode
10
10:1
10.2
100
10
102
10a
10
SVD Modes
Figure (a) Phase diagram with SVD analysis on combined time series from marked parameter sets. (b) Projections onto SVD modes for: asynchrony center (blue), wave center (pink), and boundaries (circles/triangles). (c) SVD modes 1,3,4 of combined series. (d) Selected experimental SVD modes. Colored boxes highlight matching phase velocity patterns: green (planar waves), orange and red (clock wise & counter clock wise spirals, respectively).
the connection between neurons is often very dense, sometimes even with full connection locally 8 9 23 while numerical simulations that produce neural activity without spatial patterns are often sparse 337.
3.5
Analyzing Dynamical Transitions
To elucidate phase transitions between distinct dynamical phases, analyze the spatiotemporal organization of emergent activity patterns. We derive phase velocity fields from network activity to capture the local direction and speed of patterns like propagating waves. By applying Singular Value Decomposition (SVD) to these velocity fields, we identify dominant spatiotemporal modes of activity systematically study how these modes reconfigure the system traverses
flow. This allows different dynamical regimes and critical boundaries between them, offering quantitative window into the nature of these transitions.
Our analysis reveals key phenomenon at phase boundaries: "mode mixing", where SVD modes characteristic of both adjacent pure phases significantly contribute. indicating dynamically hybrid states consistent with underlying spectral properties near instability (Fig. 4 ())b)). Crucially, the dominant spatial SVD modes identified in model (e.g., plane waves, spirals) exhibit compelling qualitative similarities to patterns observed in mesoscopic optical imaging of the mouse cortex across different arousal states (Fig. (cd))) [4]. This correspondence suggests our model captures salient principles of spatiotemporal pattern formation and transition relevant to real brain dynamics. See Appendix A.3 for methods and supplementary results.
3.6 The corresponding phase of different degrees of consciousness
Having seen the similarity between patterns in our model and the experiment, we aim to establish relationship between different degrees of consciousness and the corresponding phases in our model. Although the phase might be multivalued function of brain states (multiple phase patterns may coexist in one brain state), within the phase space we have searched and for a limited number of experimental samples, by comparing order parameters (Fig. 5 see Appendix A.9 for details),

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This aligns well with [8), which discovered that stronger E-to-I coupling in a spatially extended network led to propagating waves (corresponding to the wave phase in our model), while weaker one led to bump phase. Pure propagating waves in [8] could not be modulated by external stimuli and had lower decoding accuracy, aligning with the physiological properties of the anesthetized Wave-Bump boundary
Wave
9010)
80
1 16
1.72
1.68
1.64
Loca Sync.
Ouc. Index
0.75
0.90
12.5
0.50
0.85
0.25
Anesthelized weken Fully awake
Figure 5: Blue curves represent order parameters of different degrees of consciousness of the same mouse (1 trial). Red curves represent order parameters of parameter sets in the wave phase (5 trials) with (0Æ=//rl 1.04 and @grz/rrl varying from from the wave phase to wave-bump phase boundary.
found preliminary evidence that from the anesthetized to the fully awake state (increasing degree of consciousness), the corresponding E-to-I coupling strength decreased, moving from the interior of the wave phase towards the wave-bump phase boundary.
state. At the bume- phase boundary, the network entered a critical state where the modulation effect of stimuli was maximized, aligning with the fully awake state.
Conclusions and Discussions
We introduced a spectral theory for spatially extended neural networks, quantitatively linking local connectivity (E/I projection ranges, strengths, and local sparsity) to global dynamics via the eigenvalue spectrum of the connectivity matrix. This spectral blucprint, characterized by outlier modes and bulk disk, accurately predicts rich repertoire of emergent behaviors including asynchronous states, oscillations, bumps, waves, and chaos, providing a mechanistic bridge from structure to dynamics.
Besides, unlike classical neural field models[ 14), approach makes assumption of ho- mogencous connectivity or the continuum limit, enabling the emergence of chaotic dynamics that traditional neural field theory cannot capture. While such chaotic regimes have been extensively characterized non-spatial networks using dynamical mean-field (1)221 their counterparts in spatially structured systems remain largely unexplored. Our RM'T-based analysis bridges this gap, providing a unified and elegant perspective: the outlier cigenvalues correspond to Fourier modes, as in neural field theory, whereas the bulk spectrum reflects DMFT-like statisties. This connection highlights how RMT can serve as powerful theoretical lens for integrating spatial structure and randomness in large-scale neural dynamics.
This framework contextualizes how observed brain dynamics, such as state dependettndn patterns 44. arise. It aligns with neural field theories [4] regarding pattern formation and offers refined insights into chaos generation compared to globally coupled models [9)] highlighting the role of local density. Crucially, posit that model's 's static connectivity represents time-varying effective connectivity in the brain, which is constantly reshaped by neuromodulation, stimuli, and attention [8] (39]. Thus, the identified dynamical phases might offer within a larger phase space that the brain potentially traverses, possibly corresponding to different states of consciousness-aa an idea open experimental testing.
perspective: they could serve candidate states
Key limitations guide future work Extending the theory nonlinear neural/node's dynamics 21 and incorporating synaptic plasticity to model adaptive spectral changes and learning [40 are paramount. Addressing structural complexities beyond isotropic connectivity, modeling the explicit time-variance of effective connectivity, and robustly inferring spectral features from empirical data
41 are also critical. Also, a more direct characterization of chaos needs to done in future works. Furthermore, elucidating the direct computational roles of these spectrally-defined dynamical regimes remains vital pursuit 42 Besides, multiple phase patterns may superimpose or coexist simultancously due nonlinear mode-coupling, which need to be further investigated.
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A Appendixes
A.1 Supplementary discussion
Our calculation of the bulk disk radius builds upon foundational random matrix theory concerning the circular law, applicable even to non-Gaussian weight distributions (26), and methods for structured random networks 33 [9] 28). The emergence of outlier eigenvalues from low-rank perturbations to such random matrices is also well-established [27]. Our novelty lies in systematically deriving both these spectral components for spatially extended E/I networks and explicitly linking them to distinct dynamical phases. While prior work has explored outlier eigenvalues in non-spatial networks, often focusing on single global outliers [43, population modes [44 without considering spatial, or randomly distributed local outliers that can be removed by zero-sum constrain [30 45 and while some studies noted the presence of spatially-organized eigenvalues
secondary observation
without direct calculation or dynamic linkage [46], our framework uniquely connects the full set of ppatially-indexee outlier eigenvalues (for wave vectors) and the bulk disk radius the emergence of diverse spatiotemporal patterns and chaos, respectively.
The explicit calculation of eigenvalues associated with specific wave vectors is crucial for under- standing spatially patterned activity. While some models of cortical networks have implicitly or explicitly involved such wave numbers, they often relied simplifications from neural field theory for analytical tractability [8] 17 [23] or focused on conditions for specific instabilities, such as how E/T imbalance lead to pattern formation at particular wave numbers [15]. Our approach provides a more general matrix-based spectral method that directly ties the parameters of the spatially explicit network (projection ranges, strengths) the entire outlier spectrum without necessarily reducing to continuum field limit, thereby offering a direct bridge from discrete network structure to emergent spatial dynamics like bumps and waves.
A.2 Experimental Data of different degrees of consciousness of mice
The experiment 4 chttps://doi.or//
sittenodo.7774777 used mesoscopic optical imaging of mice expressing genetically encoded voltage indicator in cortical pyramidal neurons, to access spontaneous population voltage activity across both hemispheres of the dorsal cortex. "Anesthetized" refers to that the mice underwent light anesthesia induced by bolus injection of pentobarbiturate. "Post woken" refers to that the mice woke up from anesthesia as indicated by occasional spontaneous coordinated whisker and body movements. "Fully awake" refers to that the mice were well habituated to the imaging conditions and had been free of anesthesia for at least 3 days prior to the imaging session.
A.3 Phase Velocity Field analysis and Wave Pattern Detection
Signal Pre-processing for Phase Extraction: To analyze wave patterns from time-series data recorded at multiple spatial sites (c.g.. from optical imaging or electrophysiological arrays), we first extract the instantaneous phase from each recording site. This process involves two main steps:
The raw signal from each site, denoted as Syaw(t), is first band-pass filtered to isolate activity within frequency range of interest. This step serves to remove high-frqquency noise and focus the analysis on specific neural oscillations (e.g., delta: 0.5-4 Hz; theta: 4-8 Hz. [47]; alpha: 8-13 Hz) or on frequency bands containing significant signal power as determined by power spectral analysis of the recordings. The choice of frequency band is also constrained by the temporal resolution of the recording technique. Let the filtered signal be (1).
Band-pass Filtering:
2.
Phase Extraction via Hilbert Transform: The instantaneous phase ((l) is extracted from the filtered signal x(t) at each site using the Hilbert transform. The analytic signal 2(t) is constructed as,
=tt)= = =(t) + m(((())) = (0))
(7)
where H(x(t)) is the Hilbert transform of xxt), A(t)) is the instantaneous amplitude, and i is the imaginary unit. The instantaneous phase ((t) is then obtained as,
(()) arctan2(H(2(t)), xt)))
(8)
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where arctan2(y, 2) is the two-argument aretangent function that correctly resolves the phase into all four quadrants. The result of this pre-processing a phase time series (t(() for each spatial recording site j.
Optical Flow Method for Phase Velocity Field Estimation: To quantify the propagation of phase patterns across the spatial recording array, we employ an optical flow method, specifically the Horn-Schunck algorithm, adapted for phase data. This method estimates 2D velocity field (U, y, t)), by (z. t)) that describes the motion of surfaces of constant phase.
Let (ar, y. 1) o(ar, y, I) represent the instantaneous phase spatial location (ar, y) and time l. The core assumption of optical flow is brightness constancy, which for phase translates to phase constancy along a trajectory: [[ + vydl, y + vydt, + dt) [2r, y. t). A first-order Taylor expansion yields the optical flow constraint equation:
L + Ty + I 0,
where L. field.
01 Oz Iy
01 and L, By
01
the spatial and temporal partial derivatives of the phase di
To solve for the two unknown velocity components oy) from this single equation, the Horn- Schunck method introduces global smoothness constraint, minimizing energy functional E:
F If [[Tar T,yy 1.)2 a² 2 V1 (()uyl1)) drdy,
(10)
where a² is a regularization parameter that weights the smoothness term. Minimization of this functional leads system of coupled partial differential equations for us and Vy. Discretizing these equations (e.g., using finite differences for derivatives and the Laplacian 2 results in a large system of linear equations:
(LaVe + Ln + 12) V 0, (I20, + I, + I.) a 2² = 0,
(I
(11b)
where us and by now represent values at discrete grid points (I., My). The Laplacian terms V and Viv,, are typically approximated using a five-point stencil, e.g., V2 v) Rs (x(
(Ar/2)*
where u, (2, v) is the average of Ur at the four cardinal neighbors of (z, y), and Ar Ay is grid spacing.
This system is solved iteratively for each time step t:
Ia (Lx Dp
+ 1, + It)
(k+1) (k)
(12a)
12 + 12 + X3
Iy U1
Dz (k) + I, Dyy (k) + +1,)
(k+1) v
0y (k)
(12b)
12 + 13 X2
where (k) denotes the iteration number, and X3 (related to a² and grid spacing Ar. Ay. e.g., X² Rz )z) encapsulates the smoothness constraint. Iterations proceed until convergence or for a fixed number of steps.
Implementation Details:
Partial Derivatives:
Spatial derivatives La, Ly were computed from the phase maps o(x, y, t) at each time using a Sobel filter or a five-point central difference scheme, averaged between two consecutive time frames and t + dt. The temporal derivative I was computed using forward difference between o(x, y, t + dt) and Φ(z, y, t). potentially after local spatial averaging to reduce noise.
Boundary Conditions:
For calculating spatial derivatives near boundaries, appropriate schemes (e.g. Neumann boundary conditions where derivatives are zero) were applied. For the averaging terms Ug Fy in the iterative update, Neumann or zero-padding (Dirichlet-like) boundary conditions were typically used for the velocity components outside the defined spatial grid. The specific choice of derivative computation and boundary conditions was validated on test data to ensure reasonable velocity fields.
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Regularization Parameter at The value of a (or A²) was chosen empirically to balance adherence to the optical flow constraint with the smoothness of the resulting velocity field, often by visual inspection of results on synthetic or sample data.
This procedure yields a phase velocity field (M, y, t), (x, y, f)) for each time point, which can then be further analyzed to characterize wave properties like direction, speed, and coherence.
Supplementary Results of Phase Velocity Field Analysis Our spectral understanding of how network structure dictates distinct dynamical phases can also illuminate the nature of transitions between these phases. To quantitatively characterize these transitions, particularly at phase boundaries and triple points where multiple dynamical tendencies may coexist or compete, we analyze the spatiotemporal structure of phase velocity fields. Analyzing the instantaneous phase Ø(z, y, t) offers several advantages for understanding organized spatiotemporal patterns. The phase captures the relative timing of oscillatory activity across different spatial locations, making it particularly sensitive to propagating waves, synchronized domains, and their complex interactions. By then computing the optical flow of these phase maps, we obtain a velocity field (v, y, t), Byy (y y. t)) that directly quantifies the local direction and speed of these emergent patterns. This provides a rich, time- varying representation of the network's 's collective spatiotemporal organization, which is amenable to techniques like Singular Value Decomposition (SVD) for identifying dominant modes of activity flow.
The methodology for obtaining phase velocity fields from neural activity (either model- gener-eeee or experimental) involves band-pass filtering (e.g., 0-40 Hz, based on signal power spectrum), Hilbert transform extract instantaneous phase o(z, y, t). and an optical flow algorithm to compute the velocity vectors of phase propagation. Full details of this pre-processing and optical flow computation are provided in Appendix [A.3]
To analyze the structure of these time-varying phase velocity fields Vneu (z. y, t), we employ SVD. The velocity field data across all spatial sites (both Up and by components) and time points is arranged into a matrix X, where rows typically represent time and columns represent flattened spatial components, SVD decomposes this matrix as X = UEV*, where columns of U are temporal modes, columns of V are spatial modes, and E contains the singular values indicating the contribution of each mode.
To study phase transitions, we first identify a set of common spatial modes by performing SVD on a combined dataset of phase velocity fields from parameter sets spanning different phases, phase boundaries, and triple points (see markers in Fig. 44). Then, for each individual parameter set (numbered markers at phase boundary in Fig. 4.). its phase velocity field time series is projected onto these common spatial modes. The resulting projection weights quantify how much each common spatial mode contributes to the dynamics of that specific parameter set. By examining the profile of these projection weights (variance explained by each mode) for parameter sets systematically chosen along a path crossing a phase boundary, we can characterize the transition.
Fig. 4b illustrates this for points near the asynchrony- wavee phase boundary. Parameter sets located on a phase boundary exhibit projection profiles that are hybrid, sharing features with the profiles of the pure phases they separate. SVD modes characteristic of both pure asynchrony and pure wave states contribute significantly. At a triple point, the dynamics reflect a richer mixture, with contributions from modes associated with all three converging phases (see Fig(o). This suggests that at these critical junctures in parameter space, the system's is dynamics are not committed to a single attractor but can explore or blend features of multiple underlying dynamical regimes.
This "mode mixing" at boundarics and triple points can be intuitively understood from our spectral theory. Near these critical regions, the eigenvalue spectrum of the connectivity matrix may exhibit near-degeneracies, where multiple eigenvalues (corresponding to different potential dynamical pat- terns, e.g., a k = 0 oscillatory mode and a k A 0 wave mode) have comparable real parts close to the instability threshold. Small amounts of perturbation can then cause the system to fluctuate between, or simultaneously express aspects of, these competing dynamical modes. This is consistent with experimental observations where brain activity can show transient or mixed features, especially
during state transitions 48 49
Encouragingly, the dominant SVD spatial modes extracted from our model's phase velocity fields, such as plane waves or spiral patterns (Fig. 4b), show qualitative similarities to modes extracted
23

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(a)
(b) 1
12
(c) 2.1
Wave
Chaos
12:311
1.3-
Async: State
18:00
2.44
Asyne State X
2 0.7
Sync,
Bump
56
56
Wave
Async. Stade 20
0.8
[dwil)
0.05
de
0.00
0.32
Asynchrony-Oeciletioon boundary
Asynchrony-Wave boundary
Asynchrony- boundary
10"
10a
102
1
10
10
101
101
10
Varlanaa (%)
10
10.9
10
10
10.1
10
10
10
10"
10"0
10
10
10
10
10a
10a
10
10
10
10
10
10
10"
10
10
SVD modes
Synctrony-Oeallstiion boundary
Bump-Waave boundery
Wave-Chaos boundary
10
10
10
10
10
10
101
10
to
10
10
16a
10"
10
10-1
10
10
5
10
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10
10
10.2
10%
10:3
10-1
10
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10?
10
Asynchromy- Sarreeoyyyyyyyyyyyyyyy boundary
Asynchrony Bump boundary
Asymchrony boundary
10
10
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102
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10
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10
10-1
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10a
10
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101
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10
10
Saanncrory-Syncchooyy-oeeee triple point
Asynchrory-Bump-WWwr triple point
Asynchrony-Waav--Chao triple point
Nayal
10
10
102
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101
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10"
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103
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109 10
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1
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10a 101
104
10
10"
phase Figure 6: Analysis of all the phase boundaries and triple points. (a) Ssynchhony-bump-ooe
diagram, and projection curves onto SVD modes of the combined time series of the parameter sets marked by crossings in the phase diagram. Sky blue line refers to asynchrony center, green line refers to oscillation center, dark blue line refers to synchrony center. Circles, triangle and squares refers to the corresponding parameter sets marked by numbers in the phase diagram. The inset shows the projection curves onto the first ten modes. (b) Same as (a), pink line refers to wave center, dark red line refers to bump center. (c) Same as (a), orange line refers to chaos center.
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from mesoscopic optical imaging of mouse cortex 4 across different arousal states (Fig. Ei). This suggests that the underlying principles governing the formation and transition of spatiotemporal patterns in our model may capture salient aspects of real brain dynamics.
A.4 Order Parameters
In order numerically validate the correctness of the prediction of phase. We introduce order parameters CV. (Coefficient of Variation) and Mcan Acti. (Mean Activity) to detect the magnitude of the neural activity fluctuation, introduce Osc. index(Oscillation Index) detect the neural oscillation, introduce Local Sync. (Local Synchronization) and Moran's Index to detect the spatial patterns of neural activity.
Mean Acti. (Mean Activity) This order parameter calculates the magnitude of neurons' firing rate. This order parameter is useful for neural networks with hyperbolic tangent activation functions, given by
Mean Acti = (lo(h,))));
CV. (Coefficient of Variation) This order parameter describes the magnitude of neural activity fluctuation. It calculates the ratio between the standard deviation and mean of the neural firing rate. This order parameter is useful for neural networks with rectified linear, supra-linear and etc. activation functions:
CV. Var(o(h B(o(h))
Ose. Index (Oscillation Index) This parameter describes the magnitude of neural oscillation. Similar to [50], we define this order parameter the fraction of the Fourier spectrum energy of the peak concentrated at oscillation frequency. The Fourier spectrum calculated by averaging the Fourier spectrum of all the excitatory neurons.
Local Syne. (Local Synchronization) This parameter describes the degree of local synchronization of neural activity. It calculates the ratio between the mean of the firing rate and the mean of the absolute value of the firing rate of local neurons. We define local neurons as the neurons within a square with side length of 10 times the inter- spacing. We chose this side length so that it's similar to the detection range local field potential. This order parameter is useful for neural networks with hyperbolic tangent activation functions:
(6 (1)),, Local (b (t))1),
Local Sync.
(1))),) loca!
Moran's Index This parameter detects the existence spatial patterns. It calculates the ratio between the mean of correlation between local neurons and the mean of correlation between all neurons. The definition of "local neurons" " is the same to Local Sync:
Σ NE (o(hF(t) o(hE(t)) Ejens (0(hf(t)) (t))
Moran' Index
Nub. E.VE (d(h)(t) ((h
PH. (Persistent Homology) This parameter is used to characterize the degree of spatial localization 2D scalar field f(x y). The method is to set threshold / and increase from the minimum to the maximum value of [x, y). and then recognize connected components in the sublevel sets X, = {(z, fxlf(x.y) y) I t} at each threshold t. As we sweep through t, new connected components (blobs) are born, and existing ones merge or vanish. Each such event is recorded
birth-death pair (b,, d.), representing the threshold at which a component appears and disappears respectively. The lifetime of blob is defined as d, b, The persistent homology of f(x is defined as the sum of the lifetime of all the blobs. Larger persistent homology means a higher degree of pattern localization. Persistent homology is independent of the size of the field (number of grid points here), and is only dependent on the spatial pattern and the contrast of the pattern.
Before calculating PH., we first smooth the image for both model and experiment using ndim- age. age.generic_fie filter from scipy. The kernel size respectively, proportional to their field size (200 x 200 for model, 20 x 40 for each hemisphere
20 and for model and experiment
25

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of the cortex). After smoothing, we take the zcsoreed image by subtracting the mean from it and then dividing by the standard deviation. When calculating PH. of the z-scored image, we remove the birth-death pains whose lifetime is less than 2% of the range of the 7-scored image to further remove high-frequency noise. Specifically, for the experimental data, we first take the z-scored image with two hemispheres as a whole, and then calculate PH. separately for each hemisphere and finally take the mean value.
(a)
(b)
Membrand patential
Death
Liretime
Ho
D
-2
Birth
Figure 7: (a) Smoothed network pattern of the center of the wave phase, marked by black star in Fig.
(b) Red dots represent blobs found. (c) Death and birth values of blobs. (d) Lifetime blobs.
A.5 Numerical Simulation
For the theoretical prediction of phase, we predict 61 x 61 gird points for single phase diagram. For numerical simulation of phase prediction, the grid points independently initialize the connectivity matrix and conduct 5 times simulations with total simulation time of 100 times time constant membrane potential т and a step length of 0.01r. We only begin to calculate the order parameters after 25r of simulation in order to avoid the influence of the transient process. Our numerical experiments were conducted on a computing cluster consisting of 16 nodes, each equipped with Intel(R) Xeon(R) CPU E5-2407 0 @ 2.20GHz.
21. For each grid point,
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A.6 Numerical Experiments with Alternative Activation Functions
(a)
Async. State
Osc.
Chaos
-0.3
-0.38
-0.55
(b)
Neural Activity
t= 0 ms
t= 50 ms
t= 100 ms
Firing Rate(Hz)
Async. State
Exc.
Neuron location (Y)
RazalM2)
39
Inh.
32.0
36-
33
31.2
100
200
300
Timel(s)
Oscillation
30
15
0
45
Bump
100
30
50
15
300
Wave
20
Chaos
150
200
100
300
600
900
Figure 8: Dynamical regimes in spatially extended networks with alternative activation functions. (a) The Phase diagram under alternative activation functions. (b) Representative neural activity and spatial patterns of excitatory neurons in different dynamical phases over time.
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A.7
Numerical Results of Order Parameters and Phase Diagrams
Phase diagram
Mean Acti.
Local Sync.
Osc. Index
Moran's Index
(a)
0.7
Async.
0.6
0.40
0.3
Chacs
0
0.08
0.12
(b)
1.1
Chaos
0.05
0.08
0.12
(c)
1.8
Opsilation
1.3
Asymc:
0.0,7
1.7
19w=/0.1
(d)
1.8
Wove
1.4
State
Bump
1
[dee/AA)
(e)
0.2
Wove
6 0.12
Dscilation
Bump
0.04
4,00
Phase diagram
CV. of Firing Rate
Osc. Index
Moran's Index
(f)
0.6
State
0.5
0.4
Bump
-0.8
-0.45
0.0
0.0
0.0
Figure
Phase diagrams and order parameters under different parameters.
A.8 Numerical Results of Order Parameters Calculated Using Membrane Potential
When comparing order parameters the model and the experiment, because the experimental data are the spontaneous population membrane voltage fluctuations of pyramidal neurons, we should correspondingly use membrane potential of excitatory neurons when calculating order parameters of our model. The results about order parameters above are calculated using neuron activity (by applying an activation function the membrane potential), and below in Fig. are results calculated using membrane potential of excitatory neurons.
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Local Sync.
Osc. Index
PH.
(8)
2
64
Myrs Stara
01
12
(b)
1601
Armc: State
0.2
(c)
54
10
46 14
24
10
Figure 10: Order parameters calculated with membrane potential.
A.9 Find the Corresponding Phase of Different Degrees of Consciousness
Firstly, we can determine which phases the experimental states are by comparing order parameters such as local sync. and osc. index of our model and the experiment. As is shown in Fig. [ and Fig. T0 all three states of the experiment have high local sync.; thus we can exclude the asynchrony and chaos phases. By comparing osc. index, we can exclude the bump and synchrony phases. To distinguish between the oscillation and wave phases, we do not use Moran's index as before because the essential difference between unconscious and conscious states reflected in Fig. is the degree of localization of patterns, which cannot be characterized by Moran's index, Instead, we use PH. (see Appendix A.44). which can quantify the degree of pattern localization and distinguish between the oscillation and wave phases at the same time. The oscillation phase has PH. generally lower than all states of experiment, thus can also be excluded. To conclude, the three consciousness states are all in the wave phase.
Next we try to identify their difference of location in the wave phase. In the region of the wave phase we searched the phase diagram varying dı and 0 (Fig (O))). the osc. index too high to be region of interest to the wave phase in the
compatible with the experiment. Therefore we confine phase diagram varying 0r gr| and IøÆz/ğrl (O(b). We cannot find the exact coordinate of a brain state in the phase diagram, because the correspondence between network structural parameters and the order parameters we used is not bijection. Instead, each brain state corresponds to a region in the phase diagram. In addition, the precise region location cannot be determined because grids are not dense enough. Therefore we are only concerned with the changing trend the phase diagram from the anesthetized to the fully awake state.
We can find a trajectory from the interior of the wave phase the wave-bump phase boundary, as is shown in Fig. [5] where the changing trends of local sync., osc. index and PH. are all similar between model and experiment. The corresponding varying structure parameter is |ğræ/ k/ğrl. decreasing from unconscious to conscious state, indicating that higher degree of consciousness is associated with smaller coupling strength from excitatory to inhibitory neurons.
As for the limitation that a precise corresponding region location cannot be determined, future research can utilize more order parameters to narrow down the possible region of brain states on the phase diagram, and use denser grids to determine the exact region location. Additionally, our experimental results are limited one trial of the same mouse (there only one mouse who has data of all three degrees of consciousness in the open source dataset introduced in A.2). Future validation on different datasets with more mice and trials remains be done.
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A.10
Parameters of Phase Diagrams
Parameter
Fig. 9(a)
Fig. 9(b)
Fig. ((c)/(())
Fig. (d))2(())
Fig. 9(e)
Fig. 9(f)
N
40000
40000
40000
40000
10000
40000
N₁
10000
10000
10000
10000
2500
10000
Lout
62.8-439.8
596.90
1169.93
596.90
596.90
321.70
EE
kou
15.7-110.0
149.23
292.48
149.23
149.23
80.42
EE
Lout
62.8-439.8
596.90
1169.93
1169.93
596.90
321.70
Lou
15.7-110.0
149.2-859.5
292.48
292.48
23.9-596.9
29.0-205.9
dEE
0.05
0.05
0.07
0.05
0.10
0.04
dIE
0.05
0.05
0.07
0.05
0.10
0.04
dEl
0.05
0.05
0.07
0.07
0.10
0.04
d1r
0.05-0.12
0.05-0.12
0,07
0.07
0.0-0.2
0.0-0.1
gEe
5.50
5.50
6.3-15.3
6.3-9.9
5.50
0.57
gIn
5
7.2-16.2
9.0-16.2
5
0.12
gEI
-5
-5
-9
-9
-5
-1.90
911
-4.5
-4.25
-9
-9
-1.0-7.00
-0.3 -0.
OEE
0.55
0.3-1.1
0.10
0.10
0.10
σEl
0.55
0.3-1.1
0.10
0.10
0.10
dIE
0.55
0.3-1.1
0.10
0.10
0.10
0.55
0.3-1.1
0.10
0.10
0.10
0
A.11
Relation between Eigenvalues and Dynamics
We want understand the relationship between connectivity structure and dynamics of spatial distributed neural networks. To begin with, we first consider a simple linear neural network, which dynamics follows:
dh; dt
h4 + Jight + E(()-
(13)
Because it's linear dynamical system, the dynamics can be decomposed into different independent modes. Let's assume the connectivity matrix is diagonalizable. J -1 where A is composed of cigenvectors of connectivity matrix, A [p1, V2, .NN, and A [λl Ag. ANv] is composed of eigenvalues of connectivity matrix. The dynamical equation can be rewritten as,
d (A - h (t)).
(14)
Therefore, we can consider the dynamics of each eigenvector component to be independent. The activity neurons superposition of components different directions. Let h(() = Σ, (()). is the magnitude of independent components, which satisfies,
de; dt
(A; 1)c: + {T))(
(15)
For a component with Re(A:) < 1, the magnitude is bounded and fluctuates around 0. For component with Re(A,) 2 the magnitude increases over time as a rate of Re(A)))- Therefore, the component with eigenvalue with the largest real part dominates the dynamics of neural networks.
For non-linear neural network, can also use eigenvalues to understand the relationship between connectivity structure and eigenvalues. We only need to perform linear expansion around the fixed point of neural activity. Let's denote the fixed point of membrane potential as h", and the deviation from the fixed point as 5h. The neural activity follows,
dh, -h; + Jos((h))
(16)
dt
We can perform linear expansion around the fixed point h". The deviation 8h from fixed point follows,
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d 8h; - + (k)o(hh + E((t).
(17)
dt
Therefore, can consider nonlinear neural network equivalent to linear neural network with an effective connectivity matrix J., (hī).
The dynamical regime of neural networks can be characterized through spectral analysis of the effective connectivity matrix Jsy. When all eigenvalues satisfy Re(A,) < 1, neural activity remains bounded near the fixed point, exhibiting small- -mplltude fluctuations around the steady-state firing rate. This regime corresponds to asynchronous irregular activity due to the absence of dominant eigenmodes.
When spectral outliers emerge with Re(A,) 2 1, the corresponding eigenmodes dominate network dynamics. These regimes can be systematically classified [ based on two spectral properties of the dominant eigenvalues: 1) temporal frequency (real vs. complex eigenvalues) and 2) spatial frequency (wave vector k of eigenvectors). Spatial organization in eigenvectors generates distinet spatiotemporal patterns, enabling classification into four phases: synchronized state, oscillatory phase, bump attractor, and traveling wave.
In contrast, when dominant eigenvalues reside within the bulk spectral disk, the system enters a chaotic phase characterized by: (i) spatially unstructured eigenvectors, (ii) large-amplitude fluctuations, and (iii) weak neuronal correlations. Intuitively, the absence of spatial patterning in neural activity stems from the structural homogeneity of dominant spectral bulk eigenmodes. Weak inter-neuronal correlations emerge from high-dimensional superposition of components satisfying Re(A; 2 1). While spectral analysis provides initial insights, analytical determination of phase boundaries requires dynamical mean-field -ield theory, as detailed Appendix A.14
This method is not fully mathematically rigorous. First, it cannot fully deal with the case of multiple fixed points. Second, it cannot fully predict the dynamic behavior when the network activity goes far from a fixed point. Besides, it cannot fully characterize the dynamic behavior where the fixed point is heterogeneous between different neurons.
Therefore, to verify the validity of our theory under nonlinear conditions, we performed many numerical experiments aforementioned to demonstrate that theory is indeed correct under nonlinear conditions and is useful and informative in understanding the relationship between neural networks' dynamics and their connectivity structure.
A.12 Neural Activity of the Asynchronous State
(a)
Neural Activity
(b)
ms
ms
160 ms
Neural Activity
Memorane Potential
Jecuron location (Y)
Exc.
0.03
0.00
-0.03
100
200
300
Time/ms)
Neuron location (X)
Figure 11: Neural activity of asynchronous state. (a) temporal evolution of membrane potentials of excitatory neurons. (b) spatial patterns of excitatory neurons' firing rate.
As illustrated in Fig. neural populations the asynchronous state exhibit weak pairwise cor- relations, resulting in the absence of emergent spatial patterns. Recent theoretical advances by 25), however, demonstrate that low-rank connectivity structures can induce spectral outliers in the long-time window covariance matrix. This finding motivates systematic investigation of spectral properties of covariance matrices in spatially extended neural networks promising direction for future research.
Notably, weak deviations from the fixed point emerge through external input modulation. In our experimental paradigm, network input originates from independent white noise sources with low
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amplitude, resulting in small deviations from the fixed point. The neural activity under other types of external imput needs be further explored.
A.13
Eigenvalues and Eigenvectors of Spatially Distributed Neural Networks
A.13.1 Circular Law
classical result of the random matrix the circular law. If you take an x matrix with independent
and identically distributed (i.i.d.) entries Jıj v.s.d. NOO, =) then n grows large, the eigenvalues of the matrix become uniformly distributed inside the disk with radius a in the complex plane [5T]. This result can be further generalized to more general random matrices where the i.i.d distribution is not Gaussian but has finite variance.
Theorem Let An be the x n random matrix whose entries are ii.d complex random variables with mean and variance The empirical spectral distribution of 1/ // then converges (both in probability and in the almost
sense) to the uniform distribution on the unit disk [26].
This theorem means that the circular law can only be used in ideal Gaussian distribution but also can be used in the case of other distributions like sparse connection and biologically plausible log-normal distribution, etc. However, it still requires the distribution to be identical independent distribution, while there are multiple types of neurons and connection probabilities that decay with the distance between neurons. Therefore, we still can use this theorem in biologically plausible spatially distributed neural networks.
1
Im(A)
-1
-1
0
1
Re(A)
Figure 12:
The eigenspectrum of random matrix with elements from i.i.d Gaussian distribution with unit variance and zero mean.
Thanks to techniques of free probability theory (28) we can deal with the case that the distribution independent, zero-mean but not identical. Besides, using dynamical mean-field theory can also derive similar results [29].
Theorem Let An = (org)
) be an n deterministic matrix, and Xn, (Xi,) ) be an n random matrix with i.d. centered entries of unit variance. Define the rescaled matrix:
1
Y,
/n -An X
where o denotes the Hadamard product. Let μij denote the empirical spectral distribution (ESD) of Yn. For variance profiles o7 = ( n 3) Pn has positive density on the centered disc of radius VV(V,,, where V, 1 and (Va)) is its spectral radius. (28)
Using this theorem, we can deal with biologically plausible neural networks with multiple types of neurons and spatial distribution. Although the connection strength distribution between different types of neurons are different, and the distribution also varies with distance between neurons, we only need to calculate profile matrix og above determine the radius of a cigenspectrum disk.
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A.13.2 Outliers of Eigenspectrum
The theorems mentioned above are all required of zero-mean condition. However, in biologically plausible case, due to Dale's Law, the distribution of connection strength between certain types of neurons cannot have zero-mean. For example, the connection strength between excitatory neurons must have a positive mean.
43 considers a special case of non-zero-mean distributions. In their model, neurons are sparsely connected with a certain probability p for both excitatory and inhibitory neurons. The connection strength are the same among excitatory synapses and inhibitory synapses separately. They showed that the eigenspectrum of the connectivity matrix composed of a bulk disk part and an outlier. This outlier lies in the position of the mean connection strength of all synapses. Namely, the non-zero mean of distribution creates outliers.
The example above is actually a special case of low-rank perturbation on a zero-mean random matrix. We can gain the following intuition. The connectivity matrix can be decomposed into two parts: a determined mean part and zero-mean random matrix part. The zero-mean random matrix part creates bulk disk eigenspectrum. And the determined mean part is actually low-rank matrix because its rank is less than the number of types of neurons. This low-rank perturbation creates outliers in eigenspectrums [27) provided rigorous mathematical theorem on outliers and low-rank perturbation.
Theorem Let Xn be an iid random matrix with finite fourth moment, and for each n, let Cn: be deterministic matrix with rank (1) and operator norm (1). Assume that for large n. Cn has no eigenvalues in {z E
O(1) eigenvalues in {2 C: |2| 2 1 + 3c}. Then, almost surely, for large n, vB Xn + Cn has exactly eigenvalues in [2 C:
+ < |z| < + +3} and has
C: : |2| 2 1 + 2z}. and these eigenvalues satisfy X
1 Xn + Cx)) = A,(CC)) + o(1) as vn
n + oo for each S S j 27
With this theorem, can finally characterize the eigenspectrum of biologically plausible neural networks. The sparsity and variance of connection strength both contributed to the zero-mean random part, which creates bulk disk part of eigenspectrum. And the Dale's law force the connectivity matrix have a determined part. This part is often low-rank, thus creating outliers of the eigensepctrum.
A.13.3 Eigenspectrum of connectivity matrices of spatially extended neural networks
As mentioned above, the eigenvalues of the connectivity matrix J consist of a circular bulk part and set of outliers. (27) noted that for random matrix subjected low-rank perturbation, the eigenvalues of the new matrix largely remain within the original circle, with any outliers located where the eigenvalues of the perturbation matrix lie. We can conceptualize the connectivity matrix J as comprising the expected values of each element, J, and the deviations from this expectation, 8J. The matrix JJ corresponds to the aforementioned random matrix, while corresponds to low-rank perturbation matrix. Therefore, áJ dictates the bulk part of the eigenvalue spectrum, while determines the outlier part.
A.13.4 Outliers
Eigenvectors and the Spatial Translation Invariance of Neural Networks The matrix deter- mines the outlier part of the eigenvalue spectrum (27). In order calculate the eigenvalues of the matrix J, we need to utilize its property of spatial translation invariance
Let's start with relatively simple mie-emmnnioon case.
The expected component of the connectivity matrix, J, can be expressed as a block matrix:
EE
El
J
E
7ll
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where J represents the expected connectivity from neurons of type neurons type a, forming a matrix of size No x Ng. It satisfies the following relation,
Jog
Jas kout aB
Paid (1x2- x.y).
The element Jij depends only the distance between neurons and j and their respective types. In our model, the proportion of excitatory to inhibitory neurons satisfies NE N, = 4 1. We consider four excitatory neurons and inhibitory neuron as forming small unit, and the neural network consists of repeated instances of this unit. When the neural network is collectively translated by several unit distances in physical space, the matrix J remains unchanged, indicating its translational invariance.
We define the following block matrix P, which satisfies:
pE
0 pl
P
where pa is a matrix of size No x No:
P 6
P.J = di.-11
The matrix P represents the translation of the neural network by one small unit and has the property:
PJp-1= = J.
This implies that the matrices P and J share common eigenvectors.
The eigenvalues P satisfy A, = e , where 2nn, n = 0, #l.... =| N1 Each eigenvalue
2
An is fivefold degenerate, with the eigenspace spanned by the vectors {य) satisfy the following properties.
((())) = 0, .,. 4]. which
(k) (k) (T)T, where ug is vector of length Ne and unt
(k)
(k)
These vectors be expressed as ui [ "1 a vector of length Ni.
For / = 0, ,.....3:
17 (k)
8ijmodd
Ny
LtJ
(k)
O.
VNi
For 1 4:
F1 15 O, [u/
(k))
(K)
e ST.
NNi
Thus, in the basis: {1) (1t² = 0, ..... 4; h: 2mn, written as block diagonal matrix, where each block corresponds to an effective connectivity matrix for specific Fourier mode. The effective connectivity matrix for Fourier mode with wave vector is given by:
,..., N2 1), the matrix J can be
[J0m]
(K)) Ju = =4
ΣΣ
Joa
uv
Σ exp( (xp(-iikas))
Jae hour
hou
exp
(_ (x] x,)²
x exp(ikor)
N₁
N
22rda
2d12 a/9
N1 [ dx N1 ( dx' N₁ Dad
1
1
(x; - x22)
+ ik((:; - Iz))
No
x
No
exp
2d
(18)
Ni 11
Jap exp
x2
No J
dx
dag
202
+
ikr
.N.
x Paß exp
12
No daß
2
34

---

[[(]] takes the form:
This expression accurate for Long-wavelength modes. the matrix
(K)
(k)
(k)
EEB
9EE
DE1
J(k)
9
EE
FEE
SET
9IE
91E
911
5x5
where (k) N1 No
das 2- 2
Jag
Fas exp
For the = Fourier mode, the outliers are precisely the two eigenvalues given by T((k) each of which is non-degenerate. For k * 0 modes, since the eigenvalues corresponding to ik coincide, they exhibit twofold degeneracy.
For the two dimensions case, the mathematical derivation is almost the same. In the case of two dimensions, the spatial translation can be done in both directions. Therefore, the eigenvectors are plane waves.
Effective Connectivity Matrix and Eigenvalues For each spatial Fourier mode, we can further simplify the effective connectivity matrix follows:
[| deg 2 d de 2
11 2
J(A)
J
JEE exp
DIE exp
(19)
ell
df
DEI exp
D11 exp
where K- = (2Tn. 2nny). nz: ny = 0, +1, +2,
As takes on different values, the effective connectivity matrices for different spatial Fourier modes yield different eigenvalues. These eigenvalues constitute the outliers in the eigenvalue spectrum of the connectivity matrix. In the case without spatial distribution, the number of outliers corresponds to the number of neuron types; in the presence of spatial distribution, as the spatial scale of the network increases, the number of outliers also increases and is ordered according to their corresponding wave vectors When the spatial scale approaches infinity, these outliers will be arranged along continuous curve (here, we reference the figure from spatial effect).
The two-dimensional F = (2mng, 2TTy), Thy Fly 0, #1, = The eigenvalues the two-dimensional case are also given by the effective connectivity matrix eigenvalues for different & values. The eigenvectors corresponding to the outlier eigenvalues are provided by the spatial Fourier modes indicated by k.
is similar, except that the wave vector l is vector that takes values
A.13.5 Bulk disk part
The matrix 5J governs the circular part of the eigenvalue spectrum. 28 and 29 provide formulations for the eigenvalue distribution of random matrices with independent entries, mean zero, and finite variance. The eigenvalues are distributed within a circle of a specific radius /max (A(M)), where the elements of the matrix M represent the variances the elements of the connectivity matrix J. The variance of the elements of the connectivity matrix J is given by:
M ij
= E. JJ = aa (Ix, - 2z5)))) -p
=po
p.. (lx; x2l)) x
Da."
+ p (1x; - (0:1-1,))
Go.3
kout
hout
ax3
(20)
Similar to the mean part, the matrix M is also spatial translation invariant matrix, because both the variance and connection probability of the synapses between two neurons are only related to the relative distance between these two neurons. Therefore, the variance matrix M can also be diagonalized into series of small block matrices. Let's start with the one dimension case,
35

---

()
ag
0
0
M((km2))
2m)
ag
U-
M
(kmms) ag
0
Because the radius of the bulk disk of an eigenspectrum only depends on the largest eigenvalue of M and all the elements of M are positive, only need consider the zero wave vector submatrix M (k=0)
Zero wave vectors spatially uniform, thus the elements of submatrix M()=0) are the average of original elements of a For simplicity, we denote the matrix M Ma M
(k=0)
For the one-dimensional case, this reduced matrix Mas is:
Ma0 Σ (ongg)
dr N&E (oTge)
(21)
Ng
Pag
hout
+ 0ag
N
kont
2/TduB Na
ag
For the two-dimensional case, the reduced matrix Mag is:
Σ (
Mag
+00
[ (5JJ00)
dx 2nx NoF
(22)
Ng
Fas
hout
+ FaB²
No
Lout
4Tdag"
Na
A.14 Phase Boundary of Chaos Phase
As mentioned above, the theoretical prediction based on linearization and eigenspectrum cannot fully characterize the dynamical behavior far away from a fixed point. The chaos phase one of these cases. The bulk disk of an eigenspectrum corresponds to neural activity without spatial patterns, while outliers are related to spatially ordered neural activity. If both the radius of the bulk disk and the real part of outliers are larger than 1, they will compete against each other and we cannot directly determine whether the neural activity spatially ordered. However, with the tool of dynamical mean-field theory (DMFT) [19, we can mathematically rigorously derive the condition for spatial patterns the chaos phase.
For neural network with the radius of a bulk disk part greater than 1, if the deviation of the mean of the neurons' membrane potential gradually amplifies after a perturbation, eventually the neural activities of different neurons will become synchronized, and the neural network will no longer be in the chaos phase. [52] calculated how the mean and variance of the neuronal membrane potential evolve ve after being subjected to perturbation,
(I- --D+00x = E(gx + 16(()), +
(23)
- I-Ag Ag 0x) = B + Ao(t).
(24)
36

---

where,
Au omlll) + (1
(25)
BRI = 2 20 + (110))),
(26)
DAI = o (of) (
(27)
Eg1 20 (010)).
(28)
Here, taking the average (-) refers to averaging over the distribution of the neuronal membrane potentials. The distribution of the neuronal membrane potentials follows Gaussian distribution, the
mean and variance of which can be theoretically calculated based on dynamic mean-fied field theory [52]
Let's denote the response matrix follows,
Ag Eg
-B I-D
R
(29)
For neural network with the radius of bulk disk part larger than if all the eigenvalues of a response matrix are less than O, the chaotic neural activity is stable and the neural activity is within chaos phase.
We further extended this result to spatially extended neural networks. We can regard a neural network with spatial distribution be composed of lots of populations with AS.. Therefore, neural network with spatial distribution is equivalent to a neural network with infinite populations.
In the model presented in the main text, the activation function for neurons is the hyperbolic tangent function, tanh. The distribution of membrane potential is a Gaussian distribution with zero mean. Therefore, the matrix B and E are matrix. The response matrix R. is block diagonal matrix. The submatrix Ag in the upper left corner represents the mean of membrane potential response to external inputs, and the submatrix I D in the lower right corner represents the variance of membrane potential response to external inputs.
The stability of the mean of membrane potential determines the phase boundary between the chaos phase and other phases with locally synchronized neural activity. Therefore, the largest real part of cigenvalues of the matrix Ag determines the phase boundary of the chaos phase.
Similar to the calculation above, the matrix Ag spatially translation invariant. Therefore, we can diagonalize this matrix into series of submatrices corresponding to different wave vectors k. The elements of submatrices are as follows,
(KE((h))) exp (1 exp
(8,(K)))TEE exp
[MMI 2
(A) = (Ag) Agg(()
(30)
|| 2
|| d2,
(8, exp
Averaging is performed over the distribution of neuronal membrane potentials. Using the tool of dynamical mean-field theory [52], the distribution of neuronal membrane potentials follows a Gaussian distribution. Let's denote the expectation of membrane potential u., the expectation of firing rate ma the variance of membrane potential as Ag, and the autocorrelation of firing rate as Ca. They satisfy,
where π- = (2 2 , n2. my = 0,
Ua Σ Jasms + hg.
(31)
ma (balV Aaz + uk)));
(32)
Aa
N3
Jaß
kout
+ oag
Os+ E
(33)
N
kout
4Tda3² No
Co (Ø²() ( + +
(34)
37

---

where he is the direct current (DC) input to the 3 type of neurons. follows a standard Gaussian distribution with unit variance.
the eigenvalues of all the submatrices are less than 1, the chaotic neural activity stable and the neural network is in the chaos phase. Otherwise, the chaotic neural activity is unstable. We can assign its phase based on the wave vector and whether the eigenvalues of instability are complex to determine which phase it belongs to.
38


Reviews:
[{"review_id": "5k1q6Xtze9", "reviewer": "Reviewer_yuaB", "summary": "The paper studies a spatially extended neural network model with the goal of linking neural structure to dynamics. It applies spectral theory of random matrices to the analysis of its connectivity matrix. Specifically outlier eigenvalues can be interpreted in the current case. As in earlier work, asynchronous states, synchronous states, oscillations, localized activity bumps, traveling waves are identified, but although the comparison to cortical dynamics is attempted, qualitative comparisons remain difficult, while qualitative similarities are observable in some of the theoretically predicted cases as discussed earlier by (the cited paper by) Ermentrout.", "questions": "Considering that consciousness was used as a motivation in the abstract and introduction and a relevant paper is cited in the appendix, but is is not further discussed in the main paper, can any conclusions regarding consciousness be drawn from the results shown here?\n\nLikewise potential biomarkers for neurological disorders are mentioned. Can you explain how the biomarkers could possibly be related to your results?\n\nCan you comment on the similarity of the obtained patterns and the measurements in neuroscience? It seems there are some similarities which, however, leave other interpretations open. It will be necessary to avoid overstating the explanatory power of the current level of modeling.\n\nCan you add results on the mode coupling that are necessary to understand the formation of the obsevable patterns? There are many examples in the literature related to physics systems but also in the context of neural systems that you can follow. In prinicple, it should be possible to obtain the phase diagrams theoretically, although there are some hysteretic effects that are not easy to obtain, but which have also here not been studied numerically.", "limitations": "Limitations are largely will described, but some points are raised in the questions above.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 1, "strengths_and_weaknesses": "Much is known about the spectral theory of related systems, see the classical work of Swift and Hohenberg or (in the context of neural systems) of Amari in the 1970s, or Brunel and others in later years, which implies that the study of spectra of spatially extended systems based on superposition of Fourier modes is meanwhile a standard procedure. In order to understand relations between stripe-like and hexagonal patters, a second-order stability analysis is needed which is barely started in eq. 18 in the appendix, but it is not developed prodcutively, so that the main results of the paper are numerical although in a good correspondence with the RMT results. It is still a pity that the 20th century standard is not reached in this otherwise well-research study. \n\nA main criticism is the interpretation of results in terms of biology and neuropsychology, see the questions below.\n\nPhase diagrams like in Fig 2a and Fig. 4A have been shown frequently, so that it is not clear what exactly is the new contribution in the present case.", "was_revised": true, "final_justification": "In order for the paper to be accepted, the speculations on consciousness, biomarkers for neurological disorder, or understanding brain functions need to be more reasonably formulated. Although the rebuttal discussed certain limitations in these respects, this needs to be conveyed to the reader of the paper as well in unambiguous statements. In addition, the \"unique sales point\" over papers by Shunichi Amari, Haim Sompolinski, Gregor Shcoener and others needs to be made more clear. The phenomena described here are well known in the literature, but here the argument does indeed \"stops short of drawing a solid conclusion\" regarding the significance of these phenomena for brain function although the reader would expect this from the abstract.", "rebuttal": "Dear Reviewer,\n\nWe sincerely appreciate the time and care you have devoted to evaluating our manuscript. Your incisive questions and constructive suggestions have helped us recognize where the presentation can be sharpened and where additional analysis will strengthen the narrative. Below, we address each point in detail.\n\n**W1:Phase diagrams like in Fig 2a and Fig. 4A have been shown frequently, so that it is not clear what exactly is the new contribution in the present case.**\n\nWe thank the reviewer for pointing out that the role of the phase diagrams (Fig. 2a, 4a) was not sufficiently clear. These panels illustrate how connectivity features (e.g., E/I balance, E-I loop, projection range and so on) determines dynamics and are the direct visual counterparts of Section 3.4. Specifically:\n\n• Fig. 2a: x-axis = excitation–inhibition balance; y-axis = excitation–inhibition loop; comparision between two panels: mismatch of Excitation/Inhibition projection range.\n\n• Fig. 2b: shows how connection-weight variance drives chaos.\n\nWe will add explicit captions and text links to make these relationships transparent in the revised manuscript.\n\n**Q1: \"Considering that consciousness was used as a motivation in the abstract and introduction and a relevant paper is cited in the appendix, but is is not further discussed in the main paper, can any conclusions regarding consciousness be drawn from the results shown here?\"**\n\nThank you for this important question. We hope our framework can draw conclusions regarding consciousness as follows:\n\n1. We hope to learn which dynamical phase corresponds to conscious versus unconscious states.  As sketched in Figure 3(b), each phase is distinguished by a unique set of order parameters; by estimating these parameters from empirical recordings, we hope to label the brain’s state as “conscious” or “unconscious” in a phase-specific manner.\n2. With the random-matrix tools introduced, we can already predict how the spectrum of the effective connectivity matrix J selects a given phase. Besides, we discuss how specific connectivity characteristics select phases in section 3.4. We therefore hope to invert this mapping: once we know which phase is linked to consciousness, we hope to identify the characteristics of the connection structure (such as EI balance, EI loop, projection range) that place the system in that phase.\n3. Ultimately, we hope to isolate the component of effective connections whose preservation keeps the brain in the “conscious” phase and whose disruption pushes it into an unconscious one.\n\nWe tried to draw the above conclusions but they were not shown in the paper because we lacked appropriate methods to compare simulation and experiment. But now we have found a way to establish a preliminary relationship between consciousness degree and the corresponding phase in our model, which is explained below and will be added to in the revised manuscript after section 3.5.\n\n1. By utilizing the order parameters of mean activity, local synchronization, oscillation index, as is used in the paper, we can exclude the possibility of asynchrony, synchrony, chaos, bump phase, and tell that the experimental states of different degree of consciousness can only be in either the oscillation phase or the wave phase. \n   \n2. When distinguishing between oscillation and wave, we didn't use Moran's index as in the paper because we found that the essential difference between unconscious and conscious states is the degree of localization of patterns, which can't be characterized by Moran's index. Instead, we used persistent homology (see *PS* for detailed definition). It can quantify the degree of pattern localization and distinguish between the oscillation and wave phase at the same time.\n\n3. By comparing the persistent homology of the experimental activity and the simulated activity, we can know the corresponding phase of different degrees of consciousness.\n\nHowever, the current work stops short of drawing a solid conclusion for the reason of absence of solid evidence for a Unique-Phase brain: We currently lack solid evidence that, at any given moment, the brain resides in a single, well-defined phase. Instead, multiple phase patterns may superimpose or coexist simultaneously. Numerically, we have yet to devise suitable order parameters to capture such coexistence. Theoretically, by neglecting the mode-coupling effects you rightly highlighted (See Q4), we were unable to predict scenarios in which two phases coexist or superimpose. Consequently, although consciousness served as our motivation, we have not been able to discuss it in the depth it deserves. We hope to fill this gap in our future works.\n\n*PS*: Definition of persistent homology. Persistent homology is used to characterize the degree of spatial localization in a 2D scalar field $f(x, y)$. The method is to set a threshold $t$ and increase it from the minimum to the maximum value of $f(x, y)$, and then recognize connected components in the sublevel sets $X_t=${$(x,y)|f(x,y) \\le t$} at each threshold value. As we sweep through values of $t$, new connected components (blobs) are born, and existing ones merge or vanish. Each such event is recorded as a birth-death pair $(b_i, d_i)$, representing the threshold at which a component appears and disappears respectively. The lifetime of a blob is defined as $d_i-b_i$. The persistent homology of $f(x, y)$ is defined as the sum of the lifetime of all the blobs. Larger persistent homology means a higher degree of pattern localization. The advantage of persistent homology is that it is parameter-free and scale-invariant.\n\n**Q2: \"Likewise potential biomarkers for neurological disorders are mentioned. Can you explain how the biomarkers could possibly be related to your results?\"**\n\nDifferent phases in our model have different corresponding effective connectivity (3.1 & 3.4 in paper).\nWe have found a preliminary relationship between degrees of consciousness and the phases (see in answer to Q1), from which we can establish a connection between effective connectivity and conscious states. This may have the potential to be used as a biomarker for neurological disorders. \n\n**Q3: \"Can you comment on the similarity of the obtained patterns and the measurements in neuroscience? It seems there are some similarities which, however, leave other interpretations open. It will be necessary to avoid overstating the explanatory power of the current level of modeling.\"**\n\nAs is discussed in the answer to Q1, we can use persistent homology to quantitatively characterize the similarity of patterns in our simulation and in experimental data of different degrees of consciousness. However, we indeed recognize that these similarities do not provide a definitive explanation and that other interpretations are possible. We will take care to avoid overstating the explanatory power of our current level of modeling.\n\n**Q4: \"Can you add results on the mode coupling that are necessary to understand the formation of the obsevable patterns?\"**\n\nWe thank the reviewer for the perceptive comments. The suggestion to employ mode-coupling analysis indeed offers a principled route to (i) identify sub-phases within one phase, (ii) predict the phase boundary when several eigenvalue outliers cross the instability threshold simultaneously. Below we outline how we will incorporate these ideas in the revised manuscript and in forthcoming work.\n\n1. Degenerate-mode coupling and sub-phase selection\n   \n    Numerical integration shows that the “bump phase” may adopt either stripe or hexagonal spatial profiles. At present we label all stripe/hexagon states as “bump”. A weakly nonlinear expansion around the homogeneous fixed point will allow us to derive coupled Landau equations for the slowly varying amplitudes $A_k$ of the marginally stable modes k with $|k| = k_c$.\n\n    For the Swift–Hohenberg class, the quadratic coefficient $\\alpha_2$ in the Taylor expansion of the activation function determines the pattern selection: $\\alpha_2 = 0$ selects stripes, whereas $\\alpha_2 \\neq 0$ selects hexagons via a resonant triad $k_1 + k_2 + k_3 = 0$. In our main text the tanh non-linearity has $\\alpha_2 = 0$ and stripe-like neural activity patterns, while the quadratic non-linearity used in Appendix A.5 has $\\alpha_2 \\neq 0$ and has hexagonal neural activity patterns, consistent with this principle. We hope to perform more detailed calculation in future works.\n\n    How to carry out an analogous derivation within the RMT framework remains an open question. Due to a full derivation is beyond the scope of the present revision, we hope to address it in future work.\n\n2. Coupling between outliers of distinct phases\n\n   When multiple eigenvalue outliers satisfy $Re (\\lambda_i) > 1$, our current criterion—assigning the phase associated with the largest $Re(\\lambda_i)$—works empirically but is not derived from first principles. By expanding the tanh activation function up to third order and tracking the time evolution of the amplitudes of two modes, we obtain the following equations,\n\n    $$\\begin{cases}\n    \\partial_t A_1 = \\mu_1 A_1 - g_{1111} |A_1|^2 A_1 - g_{1112} |A_1|^2 A_2 - g_{1122} |A_2|^2 A_1 - g_{1222} |A_2|^2 A_2, \\newline\n    \\partial_t A_2 = \\mu_2 A_2 - g_{2222} |A_2|^2 A_2 - g_{2221} |A_2|^2 A_1  - g_{2112} |A_1|^2 A_2 - g_{2111} |A_1|^2 A_1.\n    \\end{cases}$$\n\n    with $\\mu_i = \\lambda_i − 1$, predicts either (i) one-mode-dominant, (ii) mixed patterns, or (iii) winner-take-all dynamics, depending on the signs and magnitudes of the cross-coupling coefficients $g_{ijkl}$. These coefficients can be computed explicitly from connection matrix parameters. Thus, this approach allows us to obtain the phase boundary when several eigenvalue outliers simultaneously cross the instability threshold.\n    \n    We appreciate the reviewer’s guidance. As a full derivation would require substantial space, we defer it to a future study, where we hope to explore it in greater depth."}, {"review_id": "bhVO7jD96C", "reviewer": "Reviewer_wB1x", "summary": "This paper introduces a theoretical framework linking the microscopic architecture of spatially extended neural networks to their emergent macroscopic dynamics. Using a rate-based model of excitatory and inhibitory neurons on a 2D grid with distance-dependent connectivity, the authors develop a spectral theory, grounded in random matrix theory, for the network’s effective connectivity. They show that local parameters shape the eigenvalue spectrum, comprising a bulk disk and distinct outliers tied to spatial Fourier modes. This spectrum explains a diverse range of dynamics, such as synchrony, oscillations, bumps, waves, and chaos, within a unified model, and aligns qualitatively with cortical dynamics observed in mice.", "questions": "- The paper introduces the key concept of local sparsity, which plays a central role in determining the bulk radius and chaos onset. Could you clarify how this quantitatively refines the standard notion of sparsity in non-spatial random networks? \n\n- The SVD analysis of the phase velocity field (Fig. 4, A.2) offers a strong link to experimental data. Could a more direct quantitative connection be made between the dominant SVD modes and the leading eigenvectors of the theoretical effective connectivity matrix?\n\n-The model assumes isotropic connectivity. How might the spectral theory extend to anisotropic connectivity?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThe theoretical framework is built upon solid foundations in random matrix theory and is meticulously applied to spatially extended networks. The claims are well-supported by a combination of analytical derivations and extensive numerical simulations. The appendix provides a wealth of supporting material that confirms the paper's quality.\n\nWeaknesses: \n\nThe linearization around a fixed point is a necessary simplification for the initial theory. While Appendix A.8 acknowledges this limitation and Appendix A.11 employs DMFT to address it near the chaos boundary, a full theoretical account of dynamics far from the fixed point, such as large-amplitude limit cycles, remains an open challenge, as expected for such complex systems. Some sections of the paper are difficult to follow due to limited context and abrupt transitions. Providing clearer background and improving the narrative flow would significantly enhance readability.", "was_revised": true, "final_justification": "I appreciate the authors’ initial rebuttal and clarifications. As noted in my original review, the paper presents a rigorous theoretical framework grounded in random matrix theory, effectively applied to spatially extended networks, and supported by robust analytical and numerical evidence, with extensive supplementary material reinforcing its quality. My major concerns regarding anisotropic extensions and the influence of outliers were properly addressed.\n\nAccordingly, I maintain my original rating: 5 – Accept.", "rebuttal": "Dear Reviewer,\n\nThank you for your careful reading and constructive feedback. Below we address each of your specific comments in detail.\n\n**W1: Some sections of the paper are difficult to follow due to limited context and abrupt transitions. Providing clearer background and improving the narrative flow would significantly enhance readability.**\n\nThank you for highlighting the issue of narrative flow. We agree that the original presentation placed the final theoretical statements in the main text while relegating all detailed derivations to the appendices without adequate sign-posting, making the exposition difficult to follow.\n\nTo remedy this, we will implement the following changes:\n\n1. Section 1 (Introduction) will now include a concise “Road-map” paragraph as follows,\n   \n\"We first define the spatial E/I network (Sec 2), then analytically derive its bulk-plus-outlier spectrum (Sec 3.2, App A.10). The spectrum predicts six dynamical phases (Sec 3.3, App A.8 & A.11). Phase transitions are studied via SVD of phase-velocity fields (Sec 3.5, App A.2). Parameters and numerical validation are in App A.4–A.7.\"\n\n2. Provide more specific references to the appendix. For example, in Section 3.2.2 “Bulk Disk,” we will add explicit references to the specific theorems and equations in A.10 so that the derivation is clearer.\n\nWe believe these two revisions will improve readability and coherence.\n\n**Q1: The paper introduces the key concept of local sparsity, which plays a central role in determining the bulk radius and chaos onset. Could you clarify how this quantitatively refines the standard notion of sparsity in non-spatial random networks?**\n\nFor clarity, the key differences are summarized below.\n\n| Aspect                   | Standard sparsity                                                                                                         | Local sparsity                                                                                                            |\n| ------------------------ | ------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------- |\n| Expression               | $\\frac{k^{out}_{\\alpha\\beta}}{N_{\\alpha}}$                                                                  | $\\frac{k^{out}_{\\alpha\\beta}}{\\pi d_{\\alpha\\beta}^{2}\\,N_{\\alpha}}$                                         |\n| Interpretation           | Fraction of connections a neuron makes relative to the **total** number of neurons                                        | Fraction of connections relative to the expected number of neurons **within its axonal spread** $\\pi d_{\\alpha\\beta}^{2}$ |\n| Scaling with system size | Decreases as $1/N_{\\alpha}$ when the tissue area grows while $k^{out}_{\\alpha\\beta}$ is kept fixed | Remains constant, because number of neurons within axonal spread $\\pi d_{\\alpha\\beta}^{2}$ remains constant.                          |\n\n\nTherefore, unlike standard sparsity, local sparsity remains invariant across spatially extended neural networks of different sizes that share identical structural properties, providing a better characterization of network sparsity and prediction of its dynamical behavior.\n\nIn the revised manuscript, we will explicitly present the expression for standard sparsity and clearly delineate its differences from local sparsity to enhance clarity.\n\n**Q2: Could a more direct quantitative connection be made between the dominant SVD modes and the leading eigenvectors of the theoretical effective connectivity matrix?**\n\nThank you for kindly suggesting that we could find a connection between the dominant SVD modes and the leading eigenvectors of the connectivity matrix. If I didn't misunderstand you, the SVD modes are vector fields while the leading eigenvectors are scalar fields, which are hard to compare. Instead, we have established a directly quantitative connection between scalar fields of neural activity of model and experiment, which will be added to the revised manuscript after section 3.5.\n\nTo be specific, we have used order parameters to quantitatively characterize the similarity of simulation and experimental neural activity, finding a preliminary corresponding relationship between different degrees of consciousness and phases in our model. \n\nFirstly, by utilizing the order parameters of mean activity, local synchronization, oscillation index, as is used in the paper, we can exclude the possibility of asynchrony, synchrony, chaos, bump phase, and tell that the experimental states of different degree of consciousness can only be in either the oscillation phase or the wave phase. When distinguishing between these two phases, we didn't use Moran's index as in the paper because we found that the essential difference between unconscious and conscious states is the degree of localization of patterns, which can't be characterized by Moran's index. Instead, we used persistent homology, which is used to characterize the degree of spatial localization in a 2D scalar field $f(x, y)$. Larger persistent homology means a higher degree of pattern localization. \n\nBy comparing the persistent homology of the experimental activity and the simulated activity, we can quantify their degree of similarity and find the corresponding phase of different degrees of consciousness.\n\nIn addition, we also tried two other methods to find a quantitative connection between theory and experiment, but they all have limitations explained below.\n\n1. We tried to compare the SVD modes of simulation and experiment data (Fig.4 (c)(d)). However, we found it tricky because our model does not have a typical scale like the brain. If we want to compare them, we have to consider scaling of the vector fields, but we don't know how much to scale. For example, a sink-pattern vector field will look like plane wave at some distance away from the singularity if it is enlarged to a certain extent, which will lead to fallacies when being compared to the experimental SVD modes.\n\n2. We tried to compare the leading eigenvectors of the connectivity matrix and the leading spatial mode of dynamical mode decomposition (DMD) of the experimental data. DMD can decompose the activity $f(x,y,t)$ into the following linear combination form:\n    $$f(x,y,t) = \\sum_{i=1}^{n} b_i \\phi_i(x,y) \\exp(\\omega_i t),$$\n    where $b_i$ is the amplitude of DMD mode $i$, and $\\phi_i(x,y)$ is the spatial part of the DMD mode.\n\n    We used persistent homology to compare the DMD spatial modes and the leading eigenvectors. However, we found that the leading DMD spatial mode can not well reflect the pattern localization properties of the experimental activity, sometimes matching the property while sometimes not, depending on the specific method of DMD.\n\n**Q3: The model assumes isotropic connectivity. How might the spectral theory extend to anisotropic connectivity?**\n\nWe now provide three possible generalizations of anisotropic connectivity. All derivations assume the connection probability (Eq. 2) is modified while the spatial translation invariance of the mean connectivity is retained.\n\n1. Shift of the projection range center\n    \n   we can introduce anisotropy by shifting of the projection range center, which means that,\n\n   $$ p_c^{\\alpha\\beta}(x_i - x_j; y_i - y_j) = \\frac{k^{out}_{\\alpha\\beta}}{N_{\\alpha}} g(x_i - x_j + \\Delta x; d_{\\alpha\\beta}) g(y_i - y_j + \\Delta y; d_{\\alpha\\beta})$$\n\n    In this case, the connection probability of one neuron projecting to another located at an offset $(\\Delta x, \\Delta y)$ is maxima.\n\n    **a. Bulk disk part**\n    \n    The bulk disk part would remain unchanged under this kind of anisotropy. \n\n    According to Equation 4. and Appendix A.10, the radius of bulk disk part only depends on the variance matrix M. The shift of the projection range center only results cyclic row permutation of variance matrix M, which leaves all eigenvalues of matrix M—and hence the bulk radius—unchanged.\n\n    **b. Outliers**\n\n    Currently, the outliers no longer lie along curves or lines, but instead spread over a certain region of the complex plane.\n\n    According to Equation 3 and Appendix A.10.4, each outlier has a specific wave-vector $\\vec{k} = \\left( 2\\pi n_x, 2\\pi n_y \\right), \\quad n_x, n_y = 0, \\pm 1, \\pm 2, \\cdots$. In the isotropic case, outliers only denpend on the norm of wave-vector $\\Vert k \\Vert$, which is \"one-dimentional\", therefore outliers lie along curves or lines. However, in the anisotropic case, outliers depend on $(k_x, k_y)$, and spread over a certain region of the complex plane.\n\n    The calculation of outliers in the anisotropic case is similar to Appendix A.10.4. Because the expectation part of connection matrix is still spatially-translation invariant, we can still obatain its eigenvectors and eigenvalues following the same procedure of Appendix A.10.4.\n\n2. projection-range anisotropy\n   \n    Another way to introduce anisotropy is to have different projection range at different axis.\n\n    $$p_c^{\\alpha\\beta}(x_i - x_j; y_i - y_j) = \\frac{k^{out}_{\\alpha\\beta}}{N_{\\alpha}} g(x_i - x_j; d^x_{\\alpha\\beta}) g(y_i - y_j; d^y_{\\alpha\\beta})$$\n\n    **a. Bulk disk part**\n\n    The radius of bulk disk part only depends on the variance matrix $M^{\\alpha\\beta}_{ij}$. Therefore, we can follow the same procedure of Appendix A.5.10 to calcualte it.\n\n    **b. Outliers**\n\n    As in case 1, the outliers fill a two-dimensional region labelled by $(k_x​,k_y​)$; their exact positions follow from the same Fourier analysis detailed in Appendix A.10.4. \n  \n\nWe appreciate your patience and helpful suggestions."}, {"review_id": "547FlYaWrr", "reviewer": "Reviewer_eB7W", "summary": "This paper investigates the underlying mechanisms through which macroscopic dynamics arise from microscopic neural circuitry. By integrating a spatially extended neural network with spectral theory, the authors propose a novel framework, which is substantiated by numerical experiments.", "questions": "- What does $J$ represent in Equation (1)? Does it refer to the elements of the connection matrix?  \n- Are there any other related works in this research area? If so, how does this paper differ from or improve upon those existing studies?", "limitations": "See Weaknesses and Questions.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**  \n- The research problem addressed in the paper is both important and novel.  \n- The numerical simulation results appear to be reasonable and support the proposed approach.\n\n**Weaknesses**  \n- Some notations are not clearly defined. For instance, in line 77, it is unclear what the symbol ‘[.]’ denotes—does it represent a rounding function?  \n- While the use of the activation function $\\tanh(x)$ is standard, the appearance of the expression $5\\tanh(x/5)$ is unusual and lacks sufficient explanation or justification.", "was_revised": true, "final_justification": "This paper tackles a research problem that is both significant and novel. The presented numerical simulations are sound and effectively validate the proposed method. All of my previous concerns have been satisfactorily addressed, and I have no further comments.", "rebuttal": "Dear Reviewer,\n\nThank you very much for your careful reading and for the helpful feedback.  We appreciate the positive assessment of the novelty, significance, and technical soundness of our work, as well as the specific suggestions for improving clarity.  Below we address every point that you raised.\n\n**W1: Some notations are not clearly defined. For instance, in line 77, it is unclear what the symbol ‘[.]’ denotes—does it represent a rounding function?” (Line 77)**\n\nThank you for point out that the symbol here lack of clarity. The notation $\n[\\cdot]_+$ means,\n\n$$\n[\\cdot]_+ = \\begin{cases}\n0 & \\text{if } x \\leq 0, \\newline\nx & \\text{if } x > 0.\n\\end{cases}\n$$\n\nThis symbol is common in the computational-neuroscience literature(for example, Kadmon & Sompolinsky, 2015), yet it still requires further clarification. In the final version, we will revise this notation to $ReLU(x)$\n\n**W2: While the use of the activation function $\\tanh(x)$ is standard, the appearance of the expression $5\\tanh(x/5)$ is unusual and lacks sufficient explanation or justification.**\n\nWe use the activaton function of inhibitory neurons $5\\tanh(x/5)$ for its biological plausibility. The saturated firing rate of inhibitory neurons is substantially higher than that of excitatory neurons (Wang B et al., 2016; Zhong P et al., 2011; Sanzeni A et al., 2020). The activation function of inhibitory neurons is $5\\tanh(x/5)$ while that of excitatory neurons is $\\tanh(x)$, so that the saturated firing rate of inhibitory neurons is higher than that of excitatory neurons, which is consistent with experimental evidence. In the final version, we will add the explanation of activation functions.\n\n**Q1: What does J represent in Equation (1)? Does it refer to the elements of the connection matrix?**\n\nYes, it refer to the elements of the connection matrix. To be specific, $J^{\\alpha\\beta}_{ij}$ weight from neuron j to neuron i (positive for excitatory, negative for inhibitory), and neuron i belongs to type $\\alpha$ and neuron j belongs to type $\\beta$. We will further clarify the meaning of $J$ in the final version.\n\n**Q2: Are there any other related works in this research area? If so, how does this paper differ from or improve upon those existing studies?**\n\nThank you for raising the question about related work.  We apologize for the oversight: a full comparison with existing literature is already provided in Appendix A.1, but we failed to direct the reader to it from the main text.\n\nIn the revised manuscript we will:\n\nAdd a forward citation in Section 1 (Introduction) that reads:\n“For a detailed comparison with prior approaches, including neural field theory, numerical simulation and random matrix theory, please refer to Appendix A.1.\"\n\nWe appreciate your patience and the helpful pointer.\n\n\n### reference ###\nKadmon J, Sompolinsky H. Transition to chaos in random neuronal networks[J]. Physical Review X, 2015, 5(4): 041030.\n\nWang B, Ke W, Guang J, et al. Firing frequency maxima of fast-spiking neurons in human, monkey, and mouse neocortex[J]. Frontiers in cellular neuroscience, 2016, 10: 239.\n\nZhong P, Yan Z. Differential regulation of the excitability of prefrontal cortical fast-spiking interneurons and pyramidal neurons by serotonin and fluoxetine[J]. PloS one, 2011, 6(2): e16970.\n\nSanzeni A, Akitake B, Goldbach H C, et al. Inhibition stabilization is a widespread property of cortical networks[J]. elife, 2020, 9: e54875."}]

Author rebuttal:

Meta-review: This work attempts to characterize collective behavior in a set of spatially extended neural network models using Random Matrix Theory (RMT). The virtue of this approach is that it provides a novel approach that does not rely on homogeneous connectivity or a continuum limit. Phase diagrams are derived by considering the outliers of eigenvalue distributions of the connectivity matrices of these networks, with potential application to patterned brain states as observed in, e.g., wide field calcium imaging.

Reviewers appreciated the paper's novel theoretical contribution, which offers a complementary approach to other methods in the field and reproduces their results in overlapping cases. Weaknesses in the work were mostly presentational, with some need for contextualizing ambitious claims about consciousness and differentiating this work relative to prior art in the field. 

During discussion, authors provided additional clarification of novelty and committed to revisions that should enhance the readability of the paper.
Decision: Accept (poster)